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First-order separation of transfinite regular languages

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Rémi Morvan at Laboratoire Bordelais de Recherche en Informatique
Rémi Morvan
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Abstract

We prove that the first-order separability problem for transfinite regular languages—which asks, given two languages of words indexed by countable ordinals and defined by monadic second-order formulæ, if there exists a first-order formula satisfied by every word of one of the two languages and by none of the other—is decidable. To do this, we first give an elementary proof of Henckell’s theorem on pointlike sets, that we obtained by refining Place & Zeitoun’s proof for finite words. We then generalise this result to transfinite languages: we show that pointlike sets for the first-order logic over transfinite words corresponds to the (ordinal) group saturation.
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   
 
     
 
 
   
   
  &  
  &  
   

 
       
    
    
 
    
       
    
       
     
      
      
      
        
    
  
      
       
      
      
      
      
    
    
    
    
    
 
       knowledge     
       
  
            
         
           
          
             
            
          
            
       
        
     
  
   
    
   
     
  
   
  
   
   
  []  
   
  
     
      
  
     
   
    
     
     
   
    
     
    
    
   
   
     
         
       
         
 
  
            
         
          
          
 
          
            
           
                 
             
            
       §      
  §          
          
   
              
 @ens-paris-saclay.frremi.morvan
   
           
           
         
        
            
             
            
      
  
   ℕℕ>0  ℤ         
            
                 
   ∼ ∼        
              
              
  𝜋      
       𝒫        
  𝒫     
 
    𝒫          
    𝒫       
             𝒫
  ⋃𝑥 = 𝑥  
𝑥 ∈ 𝑆  ⋃∶𝒫(𝑆) → 𝑆  
  − ∶ 𝑆 → 𝒫(𝑆)
  𝒫          
        𝒫
       + ∗    
              
         
               
              
      
             
 1 2        
       1     2   
          
          
         
        + 
     
  
     
§   
   
 §  
  
 
     +         
          
     +          −1  
          
              
            
           
   
  § 
§
      
          
            
          
             
     
             
           
            
        §    
             
          §    
  §         
     
    
   
   
     
     
    
     
     
        
    
   
     
    
     
     
    
   
   

   
     
             
 𝑘         
   
             
            
    
               
        ℕ     
𝑘
      𝑘          

   ′ +     ′ 𝑘 
 𝑘′   ′      𝑘    
  𝑤 𝑤′ 
    
  𝑤   𝑤′ 
  𝑘0∈ℕ  
 𝑘 ≥ 𝑘0𝑤≢𝑘𝑤′

         
       ′′+  𝑘′ 𝑘′ 𝑘
′′            
              
           
     +  𝑘   𝑝𝑘𝑞
      +𝑘 𝑘  
     𝑘        
              
 
   
   
     
𝐴+   𝜑<𝑥   
  𝜑  
2< 𝑥     
  
     ℕ>0    +    𝑘  
  𝑘+1       𝑘 
       <𝑥 ≥𝑥   
    
    + 𝑘 +ℕ>0  
             
 +      ℕ>0      
     
       ℕ   𝑘  
+𝑘2𝑛−1𝑝 
     𝑛≡𝑘𝑝     
 ℕ        𝑟   
𝑟 𝑚   𝑝       𝑘 
 𝑚𝑚+1          𝑚 
𝑚≡𝑘𝑚
∗

∗
𝑎,
𝑎𝑎∗


ℤ/2ℤ
     
  𝑆𝐿 𝐿 =
𝑏+(𝑎𝑎)∗
     
   
           
     +∗        
 𝐿            
  1+∗2 + 3 ∗   
       1    2
    3         
        
 ℤℤ   ℤℤ      2 3
   1 2        
  
          2 3  
            3 
  2 +2 3   
       ⇒
 ⇒ ⇒ ⇒  
 12+     
 1 2 
        1 2
 1≡𝑘2   ℕ
 1≡𝑘2≡𝑘  ℕ
 1 2 𝑘  1≡𝑘2≡𝑘  
  11 221𝑘2
      
𝜑 ∶ 𝐴+→ 𝑈    
    
    
    
   𝑘 ∈ ℕ
   
 +        
         
       ℕ  +𝑘   
      
  
𝑈𝑅𝑘(𝜑) 𝐴+/≡𝑘
 (𝑢, 𝐿) ∈ 𝑅𝑘(𝜑)  𝑢 ∈ 𝜑[𝐿]
    𝜑
  
    
(𝑅𝑘(𝜑))𝑘∈ℕ 
    
   𝜑  
   𝑈 

            
            
             
             
        
    1 2    
             1
2
      1 2     
  
      +𝑘   
    ℕ   1 1 2 2    1𝑘2 
12       1 2  1
 2 𝑘
  1 2   𝑙     ℕ
            1
 2     𝑖𝑖       
 1 2       1𝑙2  11
 22  𝑖𝑖 
  
               
           
            
              
              

      
 §   
     
     
    
   
    
   
   𝒜 𝒫    +
grp𝒜 
  𝒮 𝒫 𝒜     
 𝒢𝒮     𝒢𝒮     𝒮  
𝜋𝜋+1 2𝜋−1 𝒮    ∗
grp𝒜   
  +
grp𝒜      
        +
grp     
               +
grp
  +
grp
        +
grp𝐿  
   𝐿    
              
+
grp𝐿             
            
            +
grp𝐿
   
        
∗
{𝒃𝒂,
𝒃𝒂𝒂}∗
 {𝒂,
𝒂𝒂}∗
∗
{𝑎},
{𝑎𝑎}∗
 

     
  +
grp(𝑆𝐿) 
  𝑆𝐿 𝐿 =
𝑏+(𝑎𝑎)∗
  
            
     𝒢𝒮   𝒢𝒮    
              
          
    𝒮 𝒫     
          
               
      𝒮     
 𝒢    𝒮     𝒢𝒮  1𝑛 
  𝒢  𝒢𝑖 𝜋
𝑖𝜋+1
𝑖2𝜋−1
𝑖  𝑖⋅   
 𝑖  𝒢𝑖𝒮      𝒢1𝒢𝑛  𝒮 
 𝑖𝒢𝒢𝑖𝒢𝒢𝒢1𝒢𝑛
  𝑖  𝒢        
  𝒢 𝒮
          
       
          𝐿
      𝐿     
  §
       
 𝐴+= 𝐴 ⋅ 𝐴∗= 𝐴∗⋅ 𝐴
  𝒜𝒫  
+
grp𝒜𝒜∗
grp𝒜∗
grp𝒜𝒜
   𝒜∗
grp𝒜 +
grp𝒜     
           
 +
grp𝒜  𝒜𝒜 𝒜∗
grp𝒜   ′  𝒜∗
grp𝒜
  ′′′ ′ ′′𝒜∗
grp𝒜 
  𝒢       𝒜∗
grp𝒜    𝒢
 𝒢𝒢 𝒢𝒜∗
grp𝒜
     
    
 𝑎 ∈ 𝒜 
𝑎⋅(⋃+
grp(𝒜)) ⊊ ⋃+
grp(𝒜)
   𝑎 ∈ 𝒜 
(⋃+
grp(𝒜)) ⋅ 𝑎 ⊊ ⋃+
grp(𝒜)
 +
grp(𝒜)   
   
    
     
    
    
  
      
       
     
   
               
     §         
          §   
  
     𝒜𝒫  
   𝒜  +
grp𝒜+
grp𝒜 
   𝒜  +
grp𝒜 +
grp𝒜 
 +
grp𝒜  
               
+
grp𝒜   𝒜          
    𝒜⋅  𝒜    𝒜⋅    
 +
grp𝒜           
+
grp𝒜
  
            
            
  
   +          
   +
grp          +
grp 
+
grp+
      +   
       +
grp
          
           
            
  2 3       
            
+
grp𝐿           1
2           
   +
grp𝐿   
          
 
        +
grp 
      ↓+
grp(𝜑)
    
      
      
     
   
   
   
    

  
          
   
     +
grp
          
            
      +          

 +      ≡𝑘    ℕ 
     ′     ℕ 𝑘
 ′′
𝑘  ′𝑘′
𝑘   𝑘   
𝑘′
𝑘    𝑘    ′ 
            
        ℕ     +𝑘    
   ℕ>0    𝑝𝜋   𝑘     
 +𝑝𝑘𝑝+𝑞  ℕ    𝑝𝑝+1 2𝑝−1   
 𝑘  ′   𝜋2𝜋−1 ′  𝜋2𝜋−1
  
     𝒫+𝒫     𝒫  
       +       
              
          
       
  
  
 
       
      +    

++
grp  
  +
        +
𝐿             
        
            + 
            
              2 +
3 ∗      +
grp𝐿    
   
++
grp𝐿 
+𝜑
𝐿
𝜇
+
grp𝐿
            
          
      
      
        +    
         §    
           
        𝒜+𝒜⋅ 𝒜 𝒫  
 𝒜 +
grp𝒜    𝒜       
  +𝑘     𝒜     
   +
grp𝒜       
 +
grp𝒜           
                 
      
    
     
     
    
     
  
     
    
 
             
               
              
  
          + ∗
      
++
grp𝐿  
 
     
       
  §
        
     +
grp
       +  
   
 ++
grp    ℕ  𝑘  𝑈 
     

    
           
          +𝑘 
      ≡𝑘    ′   𝑘′  ̂
𝜑(𝑤) = ̂
𝜑(𝑤′)
    ̂
𝜑−1[̂
𝜑(𝑤)] 
𝑘
 

′ 

   


+
grp
    +
grp          

              
         +  
        −1    
  +               +
grp 
            

  
           
         
         §
   §     
         
         
           
          
   
     
    
  
   
   
    
  
𝜔   
        
         
             
            
            
          
          
            
       §   
       §      
            
   §          
§   §          
    
  
     
§  §
      
     
   
   𝜔1  
    
    
     
     𝜔1 
     
 
           
            𝜄𝜄<𝜅   
                
     𝜄<𝜅𝜄        
         1
            
          
           
            
              
     
  
           
            
 
             
           
    𝗈𝗋𝖽  𝗈𝗋𝖽+     
  ∗ +          
           
                
 
           
         𝗈𝗋𝖽𝗈𝗋𝖽  
      𝖿𝗅𝖺𝗍  𝗈𝗋𝖽       
 𝗈𝗋𝖽𝗈𝗋𝖽         
   𝖿𝗅𝖺𝗍     
          𝗈𝗋𝖽   
         
    
     
      
𝜋 ∶ 𝑀∗→ 𝑀
1 
2𝜄𝖿𝗅𝖺𝗍
𝜄<𝜅𝜄𝜄<𝜅   𝜄𝜄<𝜅 𝗈𝗋𝖽𝗈𝗋𝖽
              
(𝑀𝗈𝗋𝖽)𝗈𝗋𝖽 𝑀𝗈𝗋𝖽
𝑀𝗈𝗋𝖽 𝑀
−𝖿𝗅𝖺𝗍
𝜋𝗈𝗋𝖽 𝜋
𝜋
   

  2         
             
            
            
          
              
          𝗈𝗋𝖽+ 
  1        2𝜄𝖿 𝗅𝖺𝗍
𝜄<𝜅  𝜄𝜄<𝜅
  𝜄𝜄<𝜅 𝗈𝗋𝖽+𝗈𝗋𝖽+     −𝗈𝗋𝖽 
−𝗈𝗋𝖽+     
    −∗ −+ 
   
     
 
    −𝗈𝗋𝖽
 −𝗈𝗋𝖽+    
  
   
    
  §
     
    𝜔
   
𝜔  
§   
    
   
    
 
            1
   
   𝗈𝗋𝖽  𝗈𝗋𝖽+         
           
   1     
     
        ∗     
                
          
            
             
              
   𝜔           𝜔 𝑖<𝜔 
     𝜔     
            
               
𝜔𝜔
     
    

     𝜔
𝜔      


1 
2𝑛𝜔𝜔
3𝜔𝜔
4
5𝜔
        𝜔  
1 2𝑛𝜔𝜔  3𝜔 𝜔
  
      
    
    
      
     ℕ      max ℕ
ℕℕ    ℕℕ     ℕ>0
          𝜔     
      𝜔 
 ℕ     ℕ   𝑛  
             𝑥 ∼𝑛𝑦 𝑥′∼𝑛𝑦′
 max(𝑥, 𝑥′) ∼𝑛max(𝑦, 𝑦′)
 𝑥 ∼𝑛𝑦 𝑆(𝑥) ∼𝑛𝑆(𝑦)
 
𝜔    𝜔𝑛        𝑛
           

1  𝜔

 𝑘−1 𝑘 ℕ>0
 
     
    
     
 
        
𝜔   𝜔     
         2 
             
   𝜔 𝜔        
3     2
        1 1
           
             
        
    
    
  
 
    
ℝ>0       
        𝜄<𝜅 𝜄    
  𝑖∈𝐼 𝑖         

ℝ>0          
              
     
ℝ>0     𝑖<𝜔 
    
    
𝜄<𝜅𝜄 
 𝜄<𝜅 𝜄          
           
  
   
  (2−𝑖)𝑖<𝜔
ℝ>0
ℝ>0
   
     
         
         
           
      

    
     
   
   
   
    
 
         
               
         𝜔
      𝗈𝗋𝖽+      𝜔 
            
     
 §
             
   
           
           
          
       
     
§       
  
𝜔   
    
      
   
   
   
     
   
     
   
 
              
          
        𝜋 ⋅,𝜔  
         
         𝜋⋅,𝜔  
            
   𝜔        
      max 
ℕ>0
           
           
         
          
    
    
    
    
𝜔𝜔     
  
       
              
            
      
          
   
        
         
  𝝎
          𝗈𝗋𝖽   
   [0,𝜔𝜔[
      12   1 2 
  1 2        𝜔
    §
  𝜔       §   𝜔 
 𝜔1           
         1 
       𝜔 1  


                
   
     
  
    
   
   
   
    
  𝜔 
   
   
𝜔  
    
   
     
  

         
     𝗈𝗋𝖽+         𝐿
     ∗    𝐿   
𝗈𝗋𝖽+  𝜔 𝜔       
    + ≥𝜔      
    2    
            
     <𝜔𝑛 𝑛+1   ℕ
           
      
        
          
 
𝜔∗𝜔
∗
𝑎,
𝑎𝑎∗


ℤ/2ℤ


     
    𝐿 =
(𝑎𝑎)𝗈𝗋𝖽+     
 𝑎𝜔   §
 𝗈𝗋𝖽+        
      

  
  
 
     ℕ2𝑝 𝐿2𝑞2𝑝+1 𝐿2𝑞+1  2𝑝 𝐿2𝑞+1 
   ℕ2𝑝 𝐿𝜔 2𝑝  𝜔 𝜔 𝜔  
 2𝑝+1 𝐿𝜔  𝜔     𝜔𝐿𝜔   
        10   𝛼𝐿𝜔 0  
𝛼𝐿𝜔        
       
   
∗
∗
∗
    
    
(𝑎𝑏)𝗈𝗋𝖽+
            𝗈𝗋𝖽+
     
            
 𝗈𝗋𝖽+ 𝗈𝗋𝖽𝗈𝗋𝖽 𝗈𝗋𝖽 𝗈𝗋𝖽    
  
      §
           
    §       
      
         𝒫𝗈𝗋𝖽+ 𝒫
  𝜄𝜄<𝜅𝜄𝜄<𝜅 𝜄𝜄 
      𝒫    

     𝒫     
        𝒫     
           𝒫
       
    
    
    
 
         
          
 
𝜔𝜔 ⋅
             

     
             ℒ
ℛ𝒥 ℋ      §    
          
             
  ℒ𝒥    ℒ     ℛ  
  ℋ   
                
 𝒟   
    𝒟=𝒥
     
  
  
             
    𝒥     𝒥      
               
      ℋ      
             
             
          
           
        ℒ ℛ 
 
        
 𝒥          
   
    𝜔𝜋𝜔      
           
   ℛ   𝜔𝜔
  ℛ      𝜔 𝜔𝜔
𝜔𝜔 3
            
 𝜔 𝜔  ℛ      
   𝜔
      𝒥𝒥𝒥  𝜔ℒ𝜔

    𝜋   𝜋    𝒥   
ℒ  ℒ𝑛    ℕ>0     𝒥 
𝒥  𝒥         
     𝜔 𝜔 𝜔 𝜔
        𝗈𝗋𝖽+     
       𝒥  𝜔 ℒ𝜔
     ⇒ ⇒
      ⇒
     
    
  
    𝒥        
  
 01   𝑖𝑖<𝜔 𝜔
 𝜔 
 𝜔     
              𝐺 
  2 𝜔 𝜋𝜔 𝜔
𝐺        
              
   
         𝒥  ℋ
     𝒥     ℋ   2𝒥2
𝒥𝒥𝒥    𝜔ℒ𝜔  ℋ𝒥𝜔𝒥𝜔
   𝜔ℛ  𝜔𝜔   𝜔ℛ 
     𝜔   𝜔 𝜔  𝑒≤ℋ𝑓 
 𝑒𝑓 = 𝑒 = 𝑓𝑒  
ℋ    


       
       𝒥       
ℒ            
   ℒ𝜔
           
  𝜔     ℒ𝜔  𝜔𝒥
   𝜔  𝜔   𝜔ℛ  𝜔𝒥   
      𝜔𝒥     
    𝜔𝒥 𝜔ℛ   
    ℒ𝜔 𝜔ℒ𝜔   ℋ 
      
            
    𝒥       ℒ ℒ𝜔
        𝗈𝗋𝖽+     
  𝒥          
𝜔𝒥 𝜔𝒥     𝜔 𝜔     
         𝒥   
              ℛ 
      ℒ    𝜔 𝜔

  
          𝜋 
   𝜋 𝜋𝜋       
   𝜋𝜔𝜋        
            
          
  §         
    𝛼   
          
  𝜔
            
  𝜔    𝜔 𝜔  
 𝛼𝜋   𝜋
⋅,𝜔   
  𝜔
       𝑛    
     
             
   𝛼    
            
     𝜌     
 𝛼𝜌 |𝑆| 𝜔
          
     𝗈𝗋𝖽+          
         |𝑆|  
       |𝑆|     §
             
 𝛼 𝛽         
          
𝜔  1+𝜔  
𝛼𝛼𝛾𝛼𝛾 𝛾𝛽𝛾 𝛽𝛾 𝛽
      𝛼𝛼<𝜔1      ℛ
       𝛼𝛼<𝜔1    
              𝛽
            𝛼𝜋
 𝛼𝜔𝜋         ℋ ℒ𝜔
     𝛼𝜋     
 
𝑠𝜔𝑖⋅𝑛
∗
𝑠𝜔𝑖⋅2
𝑠𝜔𝑖
𝑠𝜔𝑖−1
∗
𝑠𝜔
∗
𝑠∗


 𝑠𝜔𝑖
     
    
      
             ℕ
 𝜔𝑛      𝜔𝑛⋅    
  §
     

     
            
            
      𝗈𝗋𝖽+    
           𝑘
     𝑘 𝑘    ℕ 
′ 𝑘′   ′      
 𝑘         𝑘
         

        
  
 𝗈𝗋𝖽+  ℕ    𝑘   𝑝𝑘𝑞
   ℕ>0        
             
           
    1 𝗈𝗋𝖽+   
      ℕ   ℕ>0   𝑛𝑛1
 𝑛 
  §
              
                
            
  
   𝜔12
𝜔12 22 𝜔      2 2 
1 1     
           
 𝗈𝗋𝖽+      
       ℕ  𝗈𝗋𝖽+𝑘   
             
           
       
   
∗
𝜔 𝜔
  𝜔∗𝜔
∗
𝑎,
𝑎𝑎∗
∗
     
    𝐿∶=
𝑏𝗈𝗋𝖽+(𝑎𝑎)𝗈𝗋𝖽
 𝗈𝗋𝖽+𝗈𝗋𝖽     
   𝐿        
𝒥            
       𝐿   
 1  𝗈𝗋𝖽+𝗈𝗋𝖽2 𝗈𝗋𝖽+  3 𝗈𝗋𝖽  
  𝐿 1       2 3
          ℤℤ   
         
1 2        
               ℕ
  ℕ  𝑙𝑘𝑙      
𝗈𝗋𝖽+ 𝐿        
 2 3

   
             
           
           
         
   𝒜        𝒫 
    
𝜔
grp𝒜𝜔+
grp𝒜+
grp𝒜𝜔
ord+
grp 𝒜     𝒮 𝒫   
       
ord
grp𝒜      ord+
grp 𝒜  
   
   +
grp𝒜 𝜔
grp𝒜   ord+
grp 𝒜 
     
 
    
   𝐴𝗈𝗋𝖽+ = 𝐴 ⋅ 𝐴𝗈𝗋𝖽
    
      
     
   
ord+
grp 𝒜𝒜ord
grp𝒜
    
∗
{𝒃𝒂𝝎𝒂,
𝒃𝒂𝝎}
{𝑏𝑎𝜔𝑎} {𝑏𝑎𝜔}
{𝒃𝒂,
𝒃𝒂𝒂}
{𝒂𝝎𝒂,
𝒂𝝎}∗
 {𝑎𝜔𝑎}∗{𝑎𝜔}∗
{𝒂,
𝒂𝒂}∗
∗
{𝑎},
{𝑎𝑎}∗
 

     
  ord+
grp (𝑆𝐿) 
    𝐿∶=
𝑏𝗈𝗋𝖽+(𝑎𝑎)𝗈𝗋𝖽
       
ord+
grp 𝐿    𝐿        𝒫𝐿 
 ord+
grp 𝐿         
   𝜔𝜔 𝜔𝜔   ord+
grp 𝐿 
   ord+
grp 𝐿   
             
           
              
                
     
         𝒜
 𝒫 
   𝒜  ord+
grp 𝒜ord+
grp 𝒜 
 ord+
grp 𝜔
grp𝒜ord+
grp 𝒜 
 ord+
grp 𝒜    ℒ ℛ    
 𝒜𝗈𝗋𝖽+ 𝒜⋅,𝜔  
     ord+
grp 𝐿      
   ord+
grp 𝐿ord+
grp 𝐿
   §            
          §
    
             
   
       𝗈𝗋𝖽+ 
    ord+
grp

        
    
   
    

        
 
     ord+
grp     
              
            
              
    
   ord+
grp 
     𝗈𝗋𝖽+   
           ℕ𝑘  
            

    ord+
grp
              
       
         
   
𝐴𝗈𝗋𝖽 → 𝑈    
  
𝐴𝗈𝗋𝖽 →ord
grp(𝑈)
   𝗈𝗋𝖽+  
         
𝗈𝗋𝖽+
ord+
grp   
  𝗈𝗋𝖽+
    
 
 
 
 
 
 
 
𝜔 𝜔𝜔
𝜔𝜔𝜔
𝜔 𝜔𝜔
𝜔𝜔𝜔
      
   𝐿
    
     𝗈𝗋𝖽+ 𝐿      −1
−1−1−1−1𝜔−1𝜔−1 𝜔  −1𝜔 
           
    𝐿ord+
grp 𝐿        
 𝗈𝗋𝖽+ ord+
grp 𝐿      
−1
−1
    
−1+∗+
      
−1𝜔𝜔  
            𝗈𝗋𝖽+ +
      
  
               
            
           
    §
           §    
            
        § 
            𝒜𝗈𝗋𝖽+

ord+
grp 𝒜 𝒜𝒫     𝒜 ord+
grp 𝒜 
  𝒜             
   𝑘      §   
  §      𝒜     
   𝒜        
             
              
         §  §
     
       
      
      
  
ord+
grp (𝒜) ⋅ 𝑎 ⊊ ord+
grp (𝒜)
    
    
 
             
       
            

+ord+
grp 
           
 
        
𝜔   §
      
     
       𝜔 
      𝜔
grp
            
   

      
    
∗∗
    
𝜔1/∼
    1       
            
   1        
           
    
      
         𝒞   
            
         
    𝒞      
 lim𝑛∞ 𝑛    𝜔    lim𝑛∞ 𝑛
      𝒞𝜔
   2 lim𝑛→∞ 𝑛 lim𝑛→∞𝑘𝑛  ℕ
   3          
              
      𝜔𝜔
  
      max      
 𝜔 𝑥+1
2    max𝜔    
   3
max
max max
 𝒥      𝒥   
            
𝒥         

  
    
           
         
     𝒜𝒫     
𝒜++
grp𝒜     𝒜+
             
  +
grp𝒜 𝒜
Y  +
grp𝓐  ⟨𝒜⟩⋅    
+
grp(𝒜)
     𝒜++
grp𝒜
    +
grp𝒜 𝒜⋅ 
Y  𝓐   𝒜     𝒜⋅  
  𝒜⋅      𝑛   
 𝒢 𝒜⋅  𝑘𝒜+ 

𝑘 
𝒢
         𝑘   
 𝑘     𝒢   𝑘     
     𝒜+
Y  +
grp𝓐  𝓐   
    𝒜  +
grp𝒜 +
grp𝒜   
  +
grp𝒜 +
grp𝒜  +
grp𝒜  
YY   +
grp𝒜+
grp𝒜  𝒜     
𝑤 ∈ 𝒜∗   
   𝑎   
   𝑎  
   
 ℬ𝒜
   𝒜∗ℬ∗+ℬ+∗∗    ℬ   
  𝒜        𝑎 +
+
grp ℬℬ++
grpℬ    𝑎    +
ℬ  ℬ+
  +
grp𝒜+
grp𝒜    
+
grp+
grp+
grpℬ+
grp𝒜+
grp𝒜
        
+
grp+
grpℬ++
grp+
grp+
grpℬ
    𝑎ℬ   +
grp +
grpℬ+  
𝒜++
grp𝒜   ℬ𝑎,𝑖 ℬ,𝑖𝑖<𝑘𝑎𝒜∗ℬ∗+ℬ+∗∗
      ̂𝜋
      
  
   
  
 ∗
grp(−)   +
grp(−)
 𝑤ℬ𝑤𝑎 (𝑤𝑎,𝑖𝑤ℬ,𝑖 )𝑖<𝑘
       
   
 ℬℬ𝑎𝑎,𝑖ℬℬ,𝑖𝑖<𝑘𝑎𝑎
    𝑎ℬ     
         
 ∗
grpℬ∗
grp+
grp+
grpℬ∗
grp∗
grp𝒜

       +
grp𝒜     
         
YY   +
grp𝒜  +
grp𝒜    𝒜  
    
YY   +
grp𝒜         
   
|+
grp(𝒜)| ≥ 2   
     
+
grp(𝒜)     
 
       
   
     
 
    
      
   𝒜
   
    
  
  
 𝒜++
grp𝒜
             
         𝒜+  
  𝒜+⋅       
    +   

 𝑈
𝑈  
 
𝒫(𝑈)
+
 + + +       𝑈  
      𝒫+𝒫 𝒫   
  𝒫          
          
           +
grp
      +  +
grp  +
grp 
       
  ++
grp   + 
      
  +   
       +   
     
            
       𝛼     1   
    𝜔          
    01𝑛−1 0
                    ℤℤ
 
𝑖𝜔
𝑖−1 𝑖−1 𝜔
𝑖−1 𝑖−1 𝑖𝜔
𝑖−2 𝑖 𝜔𝑘−1
𝑖−𝑘 𝑖 𝜔𝑛−1
𝑖𝑖𝜔𝑛−1+1
𝑖
  𝜔
𝑖(𝜔𝑛−1+1)𝜔
𝑖𝜔𝑛
𝑖
 𝑖+1  𝑖  0 1   𝑛−1          
          
    
       

           
    𝜆   
              
   𝜆         
               𝜔
       𝜆ℛ𝜔ℛ  𝜔𝜋ℛ    
𝜔2𝜔         ℕ>0  𝜔𝑛𝜔
          𝜔   𝜆 𝜔 ⋅𝑝  
  ℕ    𝜔 ⋅𝑝   ℋ𝜔    ℋ𝜔  
𝜔        
        𝛼   
      §
    ℕ        
 𝜔𝑛     𝜔𝑛      
    𝜔𝑚𝜔𝑛⋅    𝜔𝑚𝜔𝑛⋅𝑘 
   ℕ    𝜔𝑚     
  𝜔𝑛      𝜔𝑚𝑚<𝑛    
  
    
        §     
             
   
    
  
          
              
           
    𝜔 ⋅ 3
   n       n
               
       
  n        1
  n       
   n    𝜔
           
 
           
†     1
† n
 †
           
              
 
            ℕ  𝑘+2
 n 𝑘n

         ℕ     ℕ>0  
𝑛            
 
      ℕ       
𝑛+1    𝑛+1  n  𝑛    
          𝑛   
    1   † n 𝑛 𝑛+1
   
     𝑛 1     
  𝑛 <𝑥  𝑛1
    
       
 ord+
grp 𝒜ord+
grp 𝒜  𝒜 
 ord+
grp 𝜔
grp𝒜ord+
grp 𝒜 
 ord+
grp 𝒜    ℒ ℛ      
𝒜𝗈𝗋𝖽+ ord+
grp 𝒜 
     
 ord+
grp (𝒜)
             𝒥
 ord+
grp 𝒜         ℛ  𝒜   
𝒥  ℛ  𝒜       
   𝒥     ℛ
           𝜔𝒥𝜔   
ord+
grp 𝒜  𝒥   𝒥  𝒥    
𝒥𝒥𝜔𝒥  
     ℋ      ℒ  
 ℛ          ℒ 
    ℒ      
     ℛ        
               
       𝜄𝜄<𝜅      
       𝜄𝜄<𝜅𝜃   
         ord+
grp 𝒜  
 𝗈𝗋𝖽+          ord+
grp 𝒜
    ord+
grp      ℋ   
     𝒜    ord+
grp 𝒜  ord+
grp    
ord+
grp 𝒜   ℒ ℛ
     𝜋
    
  𝐽   
  
      𝒜𝗈𝗋𝖽+ ord+
grp 𝒜  
  ord+
grp 𝒜        ord+
grp 𝒜   
                
       ord+
grp 𝒜      ℒ𝜔
                 
         

    
            
              
     
      
 
  𝒜𝒫      𝒜𝜔
𝜔
grp𝒜     𝒜𝜔
     
    𝒜 +
grp𝒜
Y  𝓐 𝒜𝜔       
   𝒜𝜔 
Y  𝓐       
    𝒜  +
grp𝒜 +
grp𝒜    
𝒜  +
grp𝒜 +
grp𝒜   +
grp𝒜  
YY   +
grp𝒜 +
grp𝒜  𝒜 ℬ𝒜      𝒜𝜔 
𝐿𝑎 𝐿ℬ   
    {𝑎} 
ℬ
    𝒜𝜔 𝑎ℬ 𝑎ℬ  𝑎ℬ∗+ℬ+∗𝜔
ℬℬ𝜔ℬ∗+ℬ+∗+ℬ𝜔 𝑎ℬℬ∗+ℬ+𝜔      
      𝒜𝜔𝜔
grp𝒜     𝑎ℬ
 𝑎ℬ              
           
  
𝒜∗∗
grp𝒜   𝒜∗
𝑎 ++
grp  𝑎  +
ℬℬ++
grpℬ ℬ  ℬ+
  +
grp𝒜+
grp𝒜   +
grp+
grp+
grpℬ
+
grp𝒜           
 
+
grp𝒜+
grpℬ𝜔𝜔
grp+
grp𝒜+
grpℬ
        +
grp𝒜 +
grpℬ𝜔   

𝐿𝑎ℬ𝑎ℬ𝜔
grp
ℬ𝑎,𝑖ℬ,𝑖𝑖<𝜔  ℬ𝑎𝑎,𝑖ℬℬ,𝑖𝑖<𝜔
    𝐿𝑎 𝐿ℬ      
   ℬ       
          
    
YY   +
grp𝒜   +
grp𝒜    𝒜 
   
YY   +
grp𝒜       
𝜔    𝜔
grp𝒜   𝜔    


             
              
             
              
               
            § 
  𝒫𝗈𝗋𝖽+ 𝒫       
  𝒫
    𝒜𝒫       
𝒜𝗈𝗋𝖽+ ord+
grp 𝒜     𝒜𝗈𝗋𝖽+
     𝒜 ord+
grp 𝒜
Y  𝓐   𝒜𝗈𝗋𝖽+    1    
𝒜𝗈𝗋𝖽+    1  𝒜𝜋 𝒮    ℕ   ℕ>0  𝑛
     𝒢     𝑛  
𝑛 𝒢    𝑛 𝑛  ℕ  
𝑚 𝜌     𝑎𝑖= 0̂𝜋(𝜔𝑚−1𝑎𝑖)
    
ord
grp(𝒜)
𝑚𝑚𝑚−1 𝑚−1 10 𝑚 𝑚−1𝑚−1 1 0
 𝑚1 𝑚−10ℕ      
         
Y  𝓐      ord+
grp 𝒜ord+
grp 𝒜
  𝒜   ord+
grp 𝜔
grp𝒜ord+
grp 𝒜   𝒜𝗈𝗋𝖽+ 𝒜⋅,𝜔
 
YY   ord+
grp 𝒜ord+
grp 𝒜  𝒜 ℬ𝒜
 𝒜𝗈𝗋𝖽 ℬ𝗈𝗋𝖽𝗈𝗋𝖽+ℬ𝗈𝗋𝖽+𝗈𝗋𝖽𝗈𝗋𝖽    
ord+
grp ord+
grp ord+
grp ℬord+
grp 𝒜
               §
YY   ord+
grp 𝜔
grp𝒜ord+
grp 𝒜     𝒜𝗈𝗋𝖽 = (𝒜𝜔)𝗈𝗋𝖽𝒜∗
     
     
𝜔⋅𝛼+𝑛 𝛼 < 𝜔1 𝑛 ∈ ℕ
𝒜𝗈𝗋𝖽
𝒜𝜔𝗈𝗋𝖽𝒜∗           
   
∗𝒜∗∗
grp𝒜 ∗  𝒜∗
𝜔𝒜𝜔𝜔
grp𝒜 𝜔  𝒜𝜔
  ord+
grp 𝜔
grp𝒜  ord+
grp 𝒜    
   𝜔
grp𝒜𝗈𝗋𝖽+ ord+
grp 𝜔
grp𝒜
    𝜔
grp𝒜𝗈𝗋𝖽+    
𝒜𝗈𝗋𝖽+ ord+
grp 𝒜
𝜄𝜄<𝜅∗ 𝜔𝜄𝜄<𝜅 ∗∗

          
  ∗ 𝜔           
    𝒜𝗈𝗋𝖽  𝒜𝜔𝗈𝗋𝖽𝒜∗   
     
YY           

            
         

    
  
    ℤℤℤℤ        +
   𝑎mod 𝑏mod     
−1

          
          
    ℤℤ    
      𝒥ℋ   ℤℤℤℤ




   +
grp        
  +
grp       
     +
grp        
          
  +
grp  ℋ      
 ℋ        
       ℋ  
  
𝑎2
∗,
𝑎3
     
   𝑎(𝑎𝑎)+
   𝐿 +      +
grp𝐿
   23 23      

      
       
1  
1𝑛       
2𝑛  𝑛

2𝑛         
 

𝑛  
𝑛+1     


𝑛
𝑛+1
  
          1   𝜔
  𝜔  𝜔       
𝜔 𝑛  ℕ     
  12           𝜔 
  1[0,𝜔𝜔[ 2 [0,𝜔𝜔[    𝗈𝗋𝖽+   
   1 2      

[0,𝜔𝜔[ 𝜋⋅,𝜔 ⋅,𝜔 [0,𝜔𝜔[
    
  
 1𝗈𝗋𝖽+  2𝗈𝗋𝖽+ 𝗈𝗋𝖽 𝗈𝗋𝖽 𝗈𝗋𝖽𝗈𝗋𝖽∗
∗(𝑏𝑎)𝜔
∗
∗(𝑎𝑏)𝜔
∗
     
    𝐿2
   21 2   𝜔  𝜔𝜔
      2     
     

        2
             
                
 2        𝜔 𝜔 
              
    
              
    
              
    
𝜔            
         
𝜔            
        


        
   
          
        
        
         
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          
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         
          
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         
        
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          
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        
       
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
          
         
       doi:
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           
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         doi:
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         
  
           
       
    doi:10.1007/BFb0030294
         
    https://www.irif.fr/~jep/PDF/MPRI/MPRI.
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         
       https://hal.
archives-ouvertes.fr/hal-00112831
          
         doi:10.
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           
         
         
         
       
      
 doi:10.4230/LIPIcs.FSTTCS.2018.47
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         
        
   http://www.jstor.org/stable/1995086
          
       
       https://hal.
archives-ouvertes.fr/hal-00160985
         
         https:
//tel.archives-ouvertes.fr/tel-00720658
        
         
      doi:10.1016/
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          
    http://www.jstor.org/stable/
1971037
          
      
   doi:10.4153/CMB-2018-014-8
          
        
  doi:10.1142/S0218196793000287
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We develop an algebraic notion of recognizability for languages of words indexed by countable linear orderings. We prove that this notion is effectively equivalent to definability in monadic second-order (MSO) logic. We also provide three logical applications. First, we establish the first known collapse result for the quantifier alternation of MSO logic over countable linear orderings. Second, we solve an open problem posed by Gurevich and Rabinovich, concerning the MSO-definability of sets of rational numbers using the reals in the background. Third, we establish the MSO-definability of the set of yields induced by an MSO-definable set of trees, confirming a conjecture posed by Bruyère, Carton, and Sénizergues.
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Pointlike sets for varieties determined by groups
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For a variety of finite groups H\mathbf H, let H‾\overline{\mathbf H} denote the variety of finite semigroups all of whose subgroups lie in H\mathbf H. We give a characterization of the subsets of a finite semigroup that are pointlike with respect to H‾\overline{\mathbf H}. Our characterization is effective whenever H\mathbf H has a decidable membership problem. In particular, the separation problem for H‾\overline{\mathbf H}-languages is decidable for any decidable variety of finite groups H\mathbf H. This generalizes Henckell's theorem on decidability of aperiodic pointlikes.
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Merge Decompositions, Two-sided Krohn-Rhodes, and Aperiodic Pointlikes
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This paper provides short proofs of two fundamental theorems of finite semigroup theory whose previous proofs were significantly longer, namely the two-sided Krohn-Rhodes decomposition theorem and Henckell's aperiodic pointlike theorem, using a new algebraic technique that we call the merge decomposition. A prototypical application of this technique decomposes a semigroup T into a two-sided semidirect product whose components are built from two subsemigroups T1,T2T_1,T_2, which together generate T, and the subsemigroup generated by their setwise product T1T2T_1T_2. In this sense we decompose T by merging the subsemigroups T1T_1 and T2T_2. More generally, our technique merges semigroup homomorphisms from free semigroups.
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Limited Set quantifiers over Countable Linear Orderings
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In this paper, we study several sublogics of monadic second-order logic over countable linear orderings, such as first-order logic, first-order logic on cuts, weak monadic second-order logic, weak monadic second-order logic with cuts, as well as fragments of monadic second-order logic in which sets have to be well ordered or scattered. We give decidable algebraic characterizations of all these logics and compare their respective expressive power.
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Recognisable Languages over Monads
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The principle behind algebraic language theory for various kinds of structures, such as words or trees, is to use a compositional function from the structures into a finite set. To talk about compositionality, one needs some way of composing structures into bigger structures. It so happens that category theory has an abstract concept for this, namely a monad. The goal of this paper is to propose monads as a unifying framework for discussing existing algebras and designing new algebras.
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Infinite Words: Automata, Semigroups, Logic and Games
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  • Jan 2004
Automata theory arose as an interdisciplinary field, with roots in several scientific domains such as pure mathematics, electronics and computer science. This diversity appears in the material presented in this book which covers topics related to computer science, algebra, logic, topology and game theory. The elementary theory of automata allows both the specification and the verification of simple properties of finite sequences of symbols. The possible practical applications include lexical analysis, text processing and sofware verification. There are at least two possible extensions of this theory. The theory of formal series is one of them. Words are replaced by functions associating to each word some numerical value. This value can be an integer counting the number of paths labeled by this word in an automaton or the integer represented by this word in some basis. It can also be a real number corresponding to some probability of the word. The other possible extension is the subject of this book: Finite sequences of symbols are replaced by infinite sequences. The motivation for this generalization originates in the early work of Richard Büchi in the sixties. Working on weak logical theories of the integers, he was lead to consider the monadic second order theory of the successor function on the integers. He was able to prove the decidability of this theory. He actually showed that all properties of the integers expressible in this logic can also be defined in terms of finite automata. Later on, Robert McNaughton proved the equivalence of deterministic and non-deterministic automata, a natural extension of the corresponding result for finite words. This difficult result had been conjectured by David Muller while working at questions related to oscillating circuits. Many other results have appeared since then and the theory has known an important increase of interest motivated by applications to problems in computer science. The notion of an infinite sequence is of interest to model the behavior of systems which are supposed to work endless, as operating systems for example. This book presents a comprehensive treatment of all aspects of this theory. It gathers for the first time the basic results with the advanced ones. Actually, if several surveys have appeared on infinite words, this book is the first manuel devoted to this topic. All proofs are given in detail, with a few, duly mentionned, exceptions. The book is intended for researchers or advanced students in mathematics or computer science. No particular background is required to read it, except for a standard mathematical culture. The dependence between chapters is not too strong, making it possible to read some chapters independently from other ones. The book can be used to lecture and the authors have used the manuscript for several years for graduate courses in computer science. It is unlikely to cover the whole content in one single course, but a selection with emphasis either on topology, or on logic, or on automata and semigroups, is possible. The book is organized as follows. The first chapter contains the definitions of rational expressions, Büchi and Muller automata and recognizable sets. It covers the necessary elements of the theory of automata on finite words such as Kleene's theorem. A proof of McNaughton theorem is given, using Safra's determinization algorithm. Although this construction is rather involved, we have chosen to place it at the very beginning of the book because it is straightforward. Other proofs of McNaughton's theorem are given later on. In the second chapter, we shift to a more algebraic point of view. The key idea of this chapter is to give a purely algebraic definition of recognizable sets. This point of view will be adopted quite often in the sequel. Our main tools are finite semigroups and their counterpart for infinite words, called omega-semigroups. The third chapter introduces the topological aspects of the theory. It is really an excursion in the field known as descriptive set theory, situated at the border between analysis and logic. We show that the main notions introduced so far have a natural translation in terms of topology. The fourth chapter is devoted to games. These games are two player mathematical games which are used as a tool to prove some results on infinite words and automata. For instance, in this chapter, games are used to prove the Büchi-Landweber theorem. Some particular games, such as Wadge games or Fraïssé-Ehrenfeucht games, are used in further chapters. In the fifth chapter, we present a classification of recognizable sets of infinite words known as the Wagner hierarchy. It emphasizes once again the importance of finite semigroups in this theory. Chapters 6 and 7 present the theory of varieties for infinite words. It is an extension of the so-called Eilenberg variety theory, which associates sets of finite words and families of finite semigroups. The families of semigroups are actually varieties of finite semigroups. The extension to infinite words leads to the notion of varieties of omega-semigroups. The classical result of Schützenberger on star-free sets and aperiodic semigroups founds its generalization with an appropriate notion of aperiodic omega-semigroup. Logic comes in with Chapter 8. The main point is that there is a close connexion between the concepts of automata theory and those of logic, as it was the case with topology. Thus, recognizability is equivalent with monadic second order definability while aperiodicity is equivalent to first order definability. The last two chapters deal with two natural extensions of infinite words. The first one is concerned with two-sided infinite words, for which all notions generalize in a natural way. The second one deals with infinite trees. This case is important because of its role in the applications. The situation is very different with trees instead of words and, in particular, Büchi and Muller automata are no longer equivalent. The main result is Rabin's theorem which states the equivalence between recognizability by tree automata and monadic second order definability.
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An algebraic theory for regular languages of finite and infinite words
Article
  • Nov 2011
  • INT J ALGEBR COMPUT
An algebraic approach to the theory of regular languages of finite and infinite words (∞-languages) is presented. It extends the algebraic theory of regular languages of finite words, which is based on finite semigroups. Their role is taken over by a structure called right binoid. A variety theorem is proved: there is a one-to-one correspondence between varieties of ∞-languages and pseudovarieties of right binoids. The class of locally threshold testable languages and several natural subclasses (such as the class of locally testable languages) as well as classes of the Borel hierarchy over the Cantor space (restricted to regular languages) are investigated as examples for varieties of ∞-languages. The corresponding pseudovarieties of right binoids are characterized and in some cases defining equations are derived. The connections with the algebraic description and classification of regular languages of infinite words in terms of finite semigroups are pointed out.
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Pointlike sets: the finest aperiodic cover of a finite semigroup
Article
  • Nov 1988
  • J PURE APPL ALGEBRA
The research in this paper is motivated by the open question: “Is the complexity of a finite semigroup S decidable?” Following the lead of the Presentation Lemma (Rhodes), we describe the finest cover on S that can be computed using an aperiodic semigroup and give an explicit relation. The central idea of the proof is that an aperiodic computation can be described by a new ‘blow-up operator’ Hω. The proof also relies on the Rhodes expansion of S and on Zeiger coding.
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