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FO[<]-uniformity
Abstract
Uniformity notions more restrictive than the usual FO[<, +, *]-uniformity = FO[<, Bit]-uniformity are introduced. It is shown that the general framework exhibited by Barrington et al. still holds if the fan-in of the gates in the corresponding circuits is considered
... Roy and Straubing asked whether MOD q + FO[<, +] also has a circuit characterization. This was answered by the first and third authors who proved, using a new encoding for circuit connections, that ExtFO[<, X] is meaningfully captured by FO[<, X]-uniform circuits, for any reasonable set X of numerical predicates [BL06]. ...
... -uniformity is defined here as in [BL06] using unary shuffled encoding. For purposes of comparison, we also define a FO[+1]-uniform bin world, where "<" is replaced with the weaker "+1" and circuit parameters are expressed in binary notation (following [BCGR92]). ...
... Definition 2 (Unary shuffled encoding [BL06]). The unary shuffled nencoding of a sequence α 1 , . . . ...
The effect of severely tightening the uniformity of Boolean circuit families is investigated. The impact on NC 1 and its subclasses is shown to depend on the characterization chosen for the class, while classes such as P appear to be more robust. Tightly uniform subclasses of NC 1 whose separation may be within reach of current techniques emerge.
... The following viewpoint has emerged [5,3,6] over two decades: when a circuitbased language class is characterized using first-order descriptive complexity, the circuit uniformity conditions spring up in the logic in the form of restrictions on the set of numerical predicates allowed. ...
... Barrington, Immerman and Straubing [5] have shown that Q[arb] equals AC 0 [Q], that is, non-uniform AC 0 with Q gates. Behle and Lange [6] have shown that Q[<, ...
... (Another prominent example that refutes (2) is the "Wotschke language" W = {(a n b) n : n ≥ 0}, again a co-context-free language [25]. Similar to the case of the Immerman language we observe that W ∈ FO+MAJ[<, +], hence W does not refute (6) [15,6]. Let us call this latter class sTC 0 (for small TC 0 or strict TC 0 ). ...
Imposing an extensional uniformity condition on a non-uniform circuit complexity class C means simply intersecting C with a uniform class L. By contrast, the usual intensional uniformity conditions require that a resource-bounded machine be able to exhibit the circuits in the circuit family defining C. We say that (C,L) has the "Uniformity Duality Property" if the extensionally uniform class C \cap L can be captured intensionally by means of adding so-called "L-numerical predicates" to the first-order descriptive complexity apparatus describing the connection language of the circuit family defining C. This paper exhibits positive instances and negative instances of the Uniformity Duality Property.
... Are all neutral letter languages of FO [ARB] in FO[≤]? Note that this echoes the above intuition on uniformity, since the numerical predicates correspond precisely to the allowed power to compute the circuit for a given input length [11]. The intuition on the logic side is even more compelling: if a letter can be introduced anywhere without impacting membership, then the only meaningful relation that can relate positions is the linear order. ...
... [10]. • Numerical predicates correspond in a precise sense [11] to the computing power allowed to construct circuit families for a language. Is there a natural way to present FO[≤, FIN]-uniform circuits? ...
First-order logic (FO) over words is shown to be equiexpressive with FO equipped with a restricted set of numerical predicates, namely the order, a binary predicate MSB, and the finite-degree predicates: FO[Arb] = FO[<, MSB, Fin]. The Crane Beach Property (CBP), introduced more than a decade ago, is true of a logic if all the expressible languages admitting a neutral letter are regular. Although it is known that FO[Arb] does not have the CBP, it is shown here that the (strong form of the) CBP holds for both FO[<, Fin] and FO[<, MSB]. Thus FO[<, Fin] exhibits a form of locality and the CBP, and can still express a wide variety of languages, while being one simple predicate away from the expressive power of FO[Arb]. The counting ability of FO[<, Fin] is studied as an application.
... Is GROUP[<, +] different from FO + MOD[<, +] ? Behle and Lange [7] gave a notion of interpreting L S [<, +] as highly uniform circuit classes. Our results therefore can be summarized as: every FO[<, +] uniform constant depth polynomial size circuit with gates that compute a product in S and that recognizes a language with a neutral letter can be made FO[<]uniform. ...
... Behle and Lange [7] give a notion of interpreting L S [<, +] as highly uniform circuit classes. As a consequence we can interpret the following results as a separation of the corresponding circuit classes. ...
We consider first-order logic with monoidal quantifiers over words. We show
that all languages with a neutral letter, definable using the addition
numerical predicate are also definable with the order predicate as the only
numerical predicate. Let S be a subset of monoids.
Let LS be the logic closed under quantification over the monoids in S and N
be the class of neutral letter languages. Then we show that: LS[<,+] cap N =
LS[<] Our result can be interpreted as the Crane Beach conjecture to hold for
the logic LS[<,+]. As a corollary of our result we get the result of Roy and
Straubing that FO+MOD[<,+] collapses to FO+MOD[<].
For cyclic groups, we answer an open question of Roy and Straubing, proving
that MOD[<,+] collapses to MOD[<]. Our result also shows that multiplication is
necessary for Barrington's theorem to hold.
All these results can be viewed as separation results for very uniform
circuit classes. For example we separate FO[<,+]-uniform CC0 from
FO[<,+]-uniform ACC0.
... Results showing that some logic classes correspond to some circuit classes abound in the literature, in particular in the corpus of Lange [4,5,10,13,14,16]. The usual pattern is that the quantifier part (say, FO, FO 2 , or more exotic logics with majority quantifiers) corresponds to the allowed circuit gates, while the numerical predicates (say, +1, reg, arb, or multiplication) correspond to the allowed computing power to wire the circuit, the so-called uniformity of the circuit class. ...
In this paper, the regular languages of wire linear AC0are characterized as the languages expressible in the two-variable fragment of first-order logic with regular predicates, FO2[reg]. Additionally, they are characterized as the languages recognized by the algebraic class QLDA. The class is shown to be decidable and examples of languages in and outside of it are presented.
... Roy and Straubing [14] (and also Krebs and Sreejith [8]), showed that there are regular languages not definable in an extension with modulo counting quantifiers, FOMOD[<, +]. All of the work above can be interpreted as an attempt to separate "highly uniform" versions of circuits classes (see Behle and Lange [2]). ...
We show that first order logic (FO) and first order logic extended with modulo counting quantifiers (FOMOD) over purely functional vocabularies which extend addition, satisfy the Crane beach property (CBP) if the logic satisfies a normal form (called positional normal form). This not only shows why logics over the addition vocabulary have the CBP but also gives new CBP results, for example for the vocabulary which extends addition with the exponentiation function. The above results can also be viewed from the perspective of circuit complexity. Showing the existence of regular languages not definable in FOMOD[<, +, *] is equivalent to the separation of the circuit complexity classes ACC0 and NC1 . Our theorem shows that a weaker logic , namely, FOMOD[<,+,2^x] cannot define all regular languages.
... The classes FO[Arb] and FO + MOD[Arb] correspond exactly to nonuniform AC 0 and ACC 0 respectively. Restrictions on the set of allowed numerical predicates translate in many natural cases into uniformity restrictions on the circuits [3,7]. Most notably, FO[+, * ] and FO + MOD[+, * ] correspond to the dlogtime-uniform versions of AC 0 and ACC 0 . ...
A letter e ∈ Σ is said to be neutral for a language L if it can be inserted and deleted at will in a word without affecting member-ship in L. The Crane-Beach Conjecture, which was recently disproved, stated that any language containing a neutral letter and definable in FO is in fact FO[<] definable and is thus a regular, star-free language. More generally, we say that a logic or a computational model has the Crane Beach property if the only languages with neutral letter that it can define/compute are regular. We develop an algebraic point of view on the Crane Beach properties us-ing the program over monoid formalism which has proved of importance in circuit complexity. Using recent communication complexity results we establish a number of Crane Beach results for programs over specific classes of monoids. These can be viewed as Crane Beach theorems for classes of bounded-width branching programs. We also apply this to a standard extension of FO using modular-counting quantifiers and show that the boolean closure of this logic's Σ1 fragment has the CBP.
... In the present paper we extend this to formulas with ordinary and modular quantifiers over the numerical predicate x+y = z. This can be viewed as proving the separation between ACC and NC 1 in a highly uniform setting (recently, a circuit interpretation of this logic was given by Behle and Lange [9]). ...
We consider an extension of first-order logic by modular quantifiers of a fixed modulus q. Drawing on collapse results from finite model theory and techniques of finite semigroup theory, we show that if the only available numerical predicate is addition, then sentences in this logic cannot define the set of bit strings in which the number of 1’s is divisible by a prime p that does not divide q. More generally, we completely characterize the regular languages definable in this logic. The corresponding statement, with addition replaced by arbitrary numerical predicates, is equivalent to the conjectured separation of the circuit complexity class ACC from NC 1 . Thus our theorem can be viewed as proving a highly uniform version of the conjecture.
... Building on work exposed in Libkin's book [Lib04], Roy and Straubing further showed that if p is a prime that does not divide q then the language Mod p is not expressible in FO + MOD q [+] where the q subscript indicates that only quantifiers counting modulo q are used [RS06]. In later work, Behle and Lange [BL06] translated the restriction of N to {+} into a uniformity restriction on circuits. Lautemann et al. [LMSV01], Schweikardt [Sch05] and Lange [Lan04] all provided further evidence of the fairly weak expressive power of addition even in the case where FO is augmented by so-called counting quantifiers or majority quantifiers. ...
The study of finite automata and regular languages is a privileged meeting
point of algebra and logic. Since the work of Buchi, regular languages have
been classified according to their descriptive complexity, i.e. the type of
logical formalism required to define them. The algebraic point of view on
automata is an essential complement of this classification: by providing
alternative, algebraic characterizations for the classes, it often yields the
only opportunity for the design of algorithms that decide expressibility in
some logical fragment.
We survey the existing results relating the expressibility of regular
languages in logical fragments of MSO[S] with algebraic properties of their
minimal automata. In particular, we show that many of the best known results in
this area share the same underlying mechanics and rely on a very strong
relation between logical substitutions and block-products of pseudovarieties of
monoid. We also explain the impact of these connections on circuit complexity
theory.
We introduce a new framework for a descriptive complexity approach to arithmetic computations. We define a hierarchy of classes based on the idea of counting assignments to free function variables in first-order formulae. We completely determine the inclusion structure and show that and appear as classes of this hierarchy. In this way, we unconditionally place properly in a strict hierarchy of arithmetic classes within . Furthermore, we show that some of our classes admit efficient approximation in the sense of FPRAS. We compare our classes with a hierarchy within defined in a model-theoretic way by Saluja et al and argue that our approach is better suited to study arithmetic circuit classes such as which can be descriptively characterized as a class in our framework.
Complexity theory and the theory of regular languages both belong to the branch of computer science where the use of resources in computing is the main focus. However, they operate at different levels. While complexity theory seeks to classify computational problems by resource use, such as space and time, regular language theory remains at the very base of this hierarchy and is concerned with classes of computational problems for which membership is (potentially) decidable.
Logical formulas are naturally decomposed into their subformulas and circuits into their layers. How are these decompositions expressed in a purely language-theoretical setting? We address that question, and in doing so, introduce a product directly on languages that parallels formula composition. This framework makes an essential use of languages of higher-dimensional words, called hyperwords, of arbitrary dimensions. It is shown here that the product thus introduced is associative over classes of languages closed under the product itself; this translates back to extra freedom in the way formulas and circuits can be decomposed.
Imposing an extensional uniformity condition on a non-uniform circuit complexity class means simply intersecting with a uniform class . By contrast, the usual intensional uniformity conditions require that a resource-bounded machine be able to exhibit the circuits in the circuit family defining . We say that has the Uniformity Duality Property if the extensionally uniform class can be captured intensionally by means of adding so-called
-numerical predicates to the first-order descriptive complexity apparatus describing the connection language of the circuit family defining .
This paper exhibits positive instances and negative instances of the Uniformity Duality Property.
The characterization of the class of FO[+]-definable languages by some generating or recognizing device is still an open problem. We prove that, restricted to word bounded languages, this class coincides with the class of semilinear languages. We also study the closure properties of the classes of languages definable in FO[+1], FO[<], FO[+] and FOC[+] under the main classical operations.
Following recent works connecting two-variable logic to circuits and monoids, we establish, for numerical predicate sets
satisfying a certain closure property, a one-to-one correspondence between FO[<,\ensuremath{\mathfrak{P}}]-uniform linear circuits, two-variable formulae with \ensuremath{\mathfrak{P}} predicates, and weak block products of monoids. In particular, we consider the case of linear TC0, majority quantifiers, and finitely typed monoids. This correspondence will hold for any numerical predicate set which is FO[ < ]-closed and whose predicates do not depend on the input length.
Motivated by the open question whether \mbox{TC{$^0$}}=\mbox{NC{$^1$}} we consider the case of linear size TC0. We use the connections between circuits, logic, and algebra, in particular the characterization of \mbox{TC{$^0$}} in terms of finitely typed monoids. Applying algebraic methods we show that the word problem for finite non-solvable groups cannot be described by a FO+MOD+MAJ[REG] formula using only two variables. This implies a separation result of FO[REG]-uniform linear TC0 from linear NC1.
We study probabilistic complexity classes and questions of derandomisation from a logical point of view. For each logic L we introduce a new logic BPL, bounded error probabilistic
L, which is defined from L in a similar way as the complexity class BPP, bounded error probabilistic polynomial time, is defined from P.
Our main focus lies on questions of derandomisation, and we prove that there is a query which is definable in BPFO, the probabilistic version of first-order logic, but not in C
, finite variable infinitary logic with counting. This implies that many of the standard logics of finite model theory, like transitive closure logic and fixed-point logic, both with and without counting, cannot be derandomised. We prove similar results for ordered structures and structures with an addition relation, showing that certain uniform variants of AC0 (bounded-depth polynomial sized circuits) cannot be derandomised. These results are in contrast to the general belief that most standard complexity classes can be derandomised.
Finally, we note that BPIFP+C, the probabilistic version of fixed-point logic with counting, captures the complexity class BPP, even on unordered structures.
A language L over an alphabet A is said to have a neutral letter
if there is a letter e∈A such that inserting or deleting e's from
any word in A* does not change its membership (or non-membership) in L.
The presence of a neutral letter affects the definability of a language
in first-order logic. It was conjectured that it renders all numerical
predicates apart from the order predicate useless, i.e., that if a
language L with a neutral letter is not definable in first-order logic
with linear order then it is not definable in first-order. Logic with
any set 𝒩 of numerical predicates. We investigate this conjecture
in detail, showing that it fails already for 𝒩={+, *}, or possibly
stronger for any set 𝒩 that allows counting up to the m times
iterated logarithm, 1g(m), for any constant m. On the
positive side, we prove the conjecture for the case of all monadic
numerical predicates, for 𝒩={+}, for the fragment BC(Σ) of
first-order logic, and for binary alphabets
We consider an extension of first-order logic by modular quantifiers of a fixed modulus q. Drawing on collapse results from finite model theory and techniques of finite semigroup theory, we show that if the only
available numerical predicate is addition, then sentences in this logic cannot define the set of bit strings in which the
number of 1’s is divisible by a prime p that does not divide q. More generally, we completely characterize the regular languages definable in this logic. The corresponding statement, with
addition replaced by arbitrary numerical predicates, is equivalent to the conjectured separation of the circuit complexity
class ACC from NC
1. Thus our theorem can be viewed as proving a highly uniform version of the conjecture.
In order to study circuit complexity classes within NC¹ in a uniform setting, we need a uniformity condition which is more restrictive than those in common use. Two such conditions, stricter than NC¹ uniformity, have appeared in recent research: Immerman's families of circuits defined by first-order formulas and a uniformity corresponding to Buss' deterministic log-time reductions. We show that these two notions are equivalent, leading to a natural notion of uniformity for low-level circuit complexity classes. We show that recent results on the structure of NC¹ still hold true in this very uniform setting. Finally, we investigate a parallel notion of uniformity, still more restrictive, based on the regular languages. Here we give characterizations of subclasses of the regular languages based on their logical expressibility, extending recent work of Straubing, Thérien, and Thomas. A preliminary version of this work appeared in “Structure of Complexity Theory: Third Annual Conference” pp. 47–59, IEEE Comput. Soc., Washington, DC, 1988.
Part 1 Mathematical preliminaries: words and languages automata and regular languages semigroups and homomorphisms. Part 2 Formal languages and formal logic: examples definitions. Part 3 Finite automata: monadic second-order sentences and regular languages regular numerical predicates infinite words and decidable theories. Part 4 Model-theoretic games: the Ehrenfeucht-Fraisse game application to FO [decreasing] application to FO [+1]. Part 5 Finite semigroups: the syntactic monoid calculation of the syntactic monoid application to FO [decreasing] semidirect products categories and path conditions pseudovarieties. Part 6 First-order logic: characterization of FO [decreasing] a hierarchy in FO [decreasing] another characterization of FO [+1] sentences with regular numerical predicates. Part 7 Modular quantifiers: definition and examples languages in (FO + MOD(P))[decreasing] languages in (FO + MOD)[+1] languages in (FO + MOD)[Reg] summary. Part 8 Circuit complexity: examples of circuits circuits and circuit complexity classes lower bounds. Part 9 Regular languages and circuit complexity: regular languages in NC1 formulas with arbitrary numerical predicates regular languages and non-regular numerical predicates special cases of the central conjecture. Appendices: proof of the Krohn-Rhodes theorem proofs of the category theorems.
Building upon the known generalized-quantifier-based first-order characterization of LOGCFL, we lay the groundwork for a deeper investigation. Specifically, we examine subclasses of LOGCFL arising from varying the arity and nesting of groupoidal quantifiers in first-order logic with linear order. Our work extends the elaborate theory relating monoidal quantifiers to NC1 and its subclasses. In the absence of arithmetical predicates for plus and times (equivalently, in the absence of the BIT predicate), we resolve the main issues: we show in particular that no single outermost unary groupoidal quantifier with FO can capture all the context-free languages, and we obtain the surprising result that a variant of Greibach's hardest context-free language is LOGCFL-complete under quantifier-free reductions without arithmetic. We then prove that FO with unary groupoidal quantifiers is strictly more expressive with predicates for plus and times than without. Considering a particular groupoidal quantifier, we prove that first-order logic with the “majority of pairs” quantifier is strictly more expressive than first-order with majority of individuals. As a technical tool of independent interest, we define the notion of an aperiodic nondeterministic finite automaton and prove that FO translations are precisely the mappings computed by single-valued aperiodic nondeterministic finite transducers.
