Content uploaded by Mohammad Ardeshir
Author content
All content in this area was uploaded by Mohammad Ardeshir on Jan 30, 2015
Content may be subject to copyright.
arXiv:1409.5699v1 [math.LO] 19 Sep 2014
The Σ1-Provability Logic of HA
Mohammad Ardeshir∗
, S. Mojtaba Mojtahedi†
Department of Mathematical Sciences,
Sharif University of Technology
September 24, 2014
Abstract
In this paper we introduce a modal theory Hσwhich is sound and complete for arithmetical
Σ1-substitutions in HA, in other words, we will show that Hσis the Σ1-provability logic of HA.
Moreover we will show that Hσis decidable. As a by-product of these results, we show that
HA +✷⊥has de Jongh property.
Contents
1 introduction 2
2 Definitions and conventions 4
3 Arithmetic 5
3.1 Some arithmetical preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3.2 q-Realizability and Leivant’s principle . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3.3 The extended Leivant’s Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.4 Interpretability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
4 Propositional modal logics 13
4.1 The NNIL formulae and related topics . . . . . . . . . . . . . . . . . . . . . . . . . . 14
4.1.1 The NNIL-algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
4.1.2 The TNNIL-algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
4.1.3 The TNNIL−-algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
4.2 The Box Translation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
4.3 Axiomatizing the TNNIL-algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
4.4 TNNIL-Conservativity of LC over LLe+.......................... 21
4.5 Kripke semantics for LC .................................. 23
5 Transforming Kripke models 25
5.1 Definition of the Solovay function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
5.2 Elementary properties of the Solovay function . . . . . . . . . . . . . . . . . . . . . . 28
5.3 Deciding the boxed formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
5.4 The Solovay function is a constant function . . . . . . . . . . . . . . . . . . . . . . . 32
5.5 Proof of the main theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
6 The Σ1-provability logic of HA 36
∗mardeshir@sharif.ir
†mojtahedy@gmail.com
1
1 introduction
As far as we know, there are at least two updated reliable sources [AB04], [BV06], for current
situation, historical background and motivations for provability logic. In order to our paper to be
self-contained, in this introduction, we extract a brief backgrounds of provability logic from the
mentioned sources for readers not much familiar to the subject.
provability Logic is a modal logic in which the modal operator ✷has intended meaning of
provability in some formal system. Unlike the other realms of modal logic, e.g. temporal logic,
epistemic logic and deontic logic, here in provability logic, we have a rational meaning for ✷A: “A
is provable in the system T”.
The notion of provability logic goes back essentially to K. G¨odel [G¨od33] in 1933, where he
intended to provide a semantics for Heyting’s formalization of intuitionistic logic IPC. He defined a
translation, or interpretation τfrom the propositional language to the modal language such that
IPC ⊢A⇐⇒ S4 ⊢τ(A).
The translation τ(A) adds a ✷before each sub-formula of A. The idea behind this translation is
hidden in the intuitionistic meaning of truth (the BHK interpretation): “The truth of a proposition
coincides with its provability”. Hence if one assume ✷Aas “provability of A”, then it is reasonable
to add a ✷behind each sub-formula and expect to have a correspondence between the intuitionistic
propositional calculus IPC and some classical modal logic.
On the other hand, by works of G¨odel in [G¨od31], for each arithmetical formula Aand recursively
axiomatizable theory T(like PA), we can formalize the statement “there exists a proof in Tfor A” by
a sentence of the language of arithmetic, i.e. ∃xProvT(x, pAq), where pAqis the code of A. Now the
question is whether we can find some modal propositional theory such that the operator captures
provability in classical mathematics. Let us restrict our attention to the part of mathematics known
as Peano Arithmetic, PA. Hence the question is to find some propositional modal theory T✷such
that:
T✷⊢A⇐⇒ ∀∗ PA ⊢A∗
By ( )∗, we mean a mapping from the modal language to the first-order language of arithmetic, such
that
•p∗is an arithmetical first-order sentence, for any atomic variable p, and (⊥)∗=⊥,
•(A◦B)∗=A∗◦B∗, for ◦ ∈ {∨,∧,→},
•(A)∗:= ∃xProvPA(x, pA∗q).
It turned out that S4 is not a right candidate for interpreting the notion of provability, since
¬⊥is a theorem of S4, contradicting G¨odel’s second incompleteness theorem (Peano Arthmetic,
PA does not prove its own consistency).
In 1976, R. Solovay [Sol76] proved that the right modal logic, in which the operator interprets
the notion of provability in PA is GL. This modal logic is well-known as the G¨odel-L¨ob logic, and
has the following axioms and rules:
•all tautologies of classical propositional logic,
•✷(A→B)→(✷A→✷B),
•✷A→✷✷A,
•L¨ob’s axiom (L): ✷(✷A→A)→✷A,
•Necessitation Rule: A/✷A,
•Modus ponens: (A, A →B)/B.
2
Theorem 1.1. (Solovay) For any sentence Ain the language of modal logic, GL ⊢Aif and only if
for all interpretations ( )∗,PA ⊢A∗.
There are many open problems which could be assumed as a generalization of the above theorem.
A list of such problems could be found in [BV06]. Also a live list of open problems could be found
in the homepage of Lev Beklemishev1.
The question of generalizing Solovay’s result from classical theories to intuitionistic ones, such as
the intuitionistic counterpart of PA, well-known as Heyting Arithmetic, HA, proved to be remarkably
difficult [AB04]. This problem was taken up by A. Visser, D. de Jongh and their students. The
problem of axiomatizing the provability logic of HA remains a major open problem since the end of
70s [AB04]. Precisely speaking, the problem of the provability logic of HA is as follows:
Find a modal theory Hsuch that: H⊢A⇐⇒ ∀∗ HA ⊢A∗
Note that in the above statement of the provability logic of HA, we have (✷A)∗:= ProvHA (pA∗q).
The following list contains important results about the provability logic of HA with arithmetical
nature:
•Friedman 1975. H0✷(A∨B)→(✷A∨✷B), [Fri75]
•Leivant 1975. H⊢✷(A∨B)→✷(✷
.A∨✷
.B), in which ✷
.Ais a shothand for A∧✷A, [Lei75]
•Visser 1981. H⊢✷¬¬✷A→✷✷Aand H⊢✷(¬¬✷A→✷A)→✷(✷A∨ ¬✷A), [Vis81,Vis82]
•Iemhoff 2001. Introduced a uniform axiomatization of all known axiom schemas of Hin an
extended language with a bimodal operator ✄. In her Ph.D. dissertation [Iem01], Iemhoff
raised a conjecture that implies directly that her axiom system, iPH, restricted to the normal
modal language, is equal to H, [Iem01]
•Visser 2002. Introduced a decision algorithm for H⊢A, for all Anot containing any atomic
variable. [Vis02]
In this paper we introduce an axiomatization of a modal logic Hσand prove the following result
which partially answers the question. We show that for any modal proposition A, which has no nested
implication to the left (TNNIL-formula), Ais in the Σ1-provability logic of HA, iff iGL+CP ⊢A, where
iGL is the intuitionistic G¨odel-L¨ob’s logic and CP is the completeness principle B→✷B(this theory
is named in this paper as LC). This in combination with the conservativity result of Theorem 4.23
and also some variant of Visser’s NNIL-algorithm in [Vis02], implies that the Σ1-provability logic of
HA is a decidable modal theory, that is called Hσhere. More precisely, we find a system Hσsuch
that
Hσ⊢A⇐⇒ ∀∗ HA ⊢A∗,
in which, ∗range over all of the substitutions that p∗is a Σ1-sentence for atomic variables p. It
is worth mentioning that a non-modal variant of all the axioms of Hσ, were already discovered by
Visser in [Vis81,Vis82,Vis02]. He also showed in [Vis02] that those variant of axioms of Hσare
sound for Σ1arithmetical interpretations in HA.
Map of sections
Let us explain the content of sections and their interrelationship. All of the contents of this paper
are minimally chosen for one major goal: “The completeness of Hσfor arithmetical Σ1-substitutions,
i.e. Theorem 6.4”. In section 2, we give definitions of some elementary notions and also make some
conventions. In section 3, we gather all the required statements with arithmetical nature. Most
of the lemmas and definitions are for one purpose: proving a refinement of Leivant’s principle in
1http://www.mi.ras.ru/~bekl
3
Lemma 3.18 (or its simplified form in Theorem 3.14). This will be used in section 5. In section 4,
we collect all required notions with propositional nature. The most crucial fact we will show in this
section is that in Hσ, one could transform any modal proposition to another proposition with simpler
form (roughly speaking with no nested implications to the left). Then we show that the theory LC,
which consists of iGL plus the completeness principle and Hσprove the same TNNIL-propositions.
Moreover we show that LC is sound and complete for a special class of finite Kripke models. In
section 5, we show that one could transform a finite Kripke model of LC (with tree-frame) to a
first-order Kripke-model of HA. This transformation is such that there is a natural correspondence
between these two Kripke-models. Finally in section 6, we use the results of section 4 and section 5
to prove the completeness of Hσfor arithmetical Σ1-substitutions.
For better understanding of what is going on this paper, we propose to those readers who are
somehow familiar with intuitionistic arithmetic and intuitionistic modal logics, have a look at first
Example 6.1 and Example 6.2 at the beginning of the section 6.
2 Definitions and conventions
The propositional non-modal language, L0contains atomic variables, ∨,∧,→,⊥and the proposi-
tional modal language, L✷has an additional operator ✷. In this paper, the atomic propositions
(in modal or non-modal language) includes atomic variables and ⊥. For an arbitrary proposi-
tion A,Sub(A) is defined to be the set of all sub-formulae of A, including Aitself. We take
Sub(X) := SA∈XSub(A) for a set of propositions X. We use ✷
.Aas a shorthand for A∧✷A. The
logic IPC is intuitionistic propositional non-modal logic over usual propositional non-modal language.
The theory IPC✷is the same theory IPC in the extended language of propositional modal language,
i.e. its language is propositional modal language and its axioms and rules are the same as the one in
IPC . Because we have no axioms for ✷in IPC✷, it is obvious that ✷Afor each A, behaves exactly
like an atomic variable inside IPC✷. First-order intuitionistic logic is denoted IQC and the logic CQC
is its classical closure, i.e. IQC plus the principle of excluded middle. For a set of sentences and rules
Γ∪ {A}in the propositional non-modal, propositional modal or first-order language, Γ ⊢Ameans
that Ais derivable from Γ in the system IPC,IPC✷,IQC, respectively. For an arithmetical formula
A,pAqrepresents the G¨odel number of A. For an arbitrary arithmetical theory Twith a set of ∆0-
axioms, we have the ∆0-predicate ProofT(x, pAq), that is a formalization of “xis code of a proof
for Ain T”. We also have the provability predicate ProvT(pAq) := ∃xProofT(x, pAq). The set of
natural numbers is denoted by ω:= {0,1,2,...}.
Definition 2.1. Suppose Tis an recursively enumerable (r.e.) arithmetical theory and σis a
function from atomic variables to arithmetical sentences. We extend σto all modal propositions A,
inductively:
•σT(A) := σ(A)for atomic A,
•σTdistributes over ∧,∨,→,
•σT(✷A) := ProvT(pσT(A)q).
We call σaΣ1-substitution, if for every atomic A,σ(A)is a Σ1-formula.
Definition 2.2. The provability logic of a sufficiently strong theory, Tis defined to be a modal
propositional theory PL(T)such that P L(T)⊢Aiff for all arithmetical substitutions σ,T⊢σT(A).
If we restrict the substitutions to Σ1-substitutions, then the new modal theory is PLσ(T).
Lemma 2.3. Let A(p1,...,pn)be a non-modal proposition with pi6=pjfor all 0< i < j ≤n. Then
for all modal sentences B1,...,Bnwith Bi6=Bjfor 0< i < j ≤nwe have:
IPC ⊢Aiff IPC✷⊢A[p1|✷B1,...,pn|✷Bn]
4
Proof. By simple inductions on complexity of proofs in IPC and IPC✷.✷
We define NOI (No Outside Implication) as the set of modal propositions A, such that any occurrence
of →is in the scope of some ✷. To be able to state an extension of Leivant’s Principle (that is
adequate to axiomatize Σ1-provability logic of HA) we need a translation on the modal language
which we name Leivant’s translation. We define it recursively as follows:
•Al:= Afor atomic or boxed A,
•(A∧B)l:= Al∧Bl.
•(A∨B)l:= ✷
.Al∨✷
.Bl.
•(A→B)lis defined by cases: If A∈NOI, we define (A→B)l:= A→Bl, otherwise we define
(A→B)l:= A→B.
Definition 2.4. Minimal provability logic iGL, is same as G¨odel-L¨ob provability logic GL , without
the principle of excluded middle, i.e. it has the following axioms and rules:
•The theorems of IPC✷.
•✷(A→B)→(✷A→✷B),
•✷A→✷✷A,
•L¨ob’s axiom (L):✷(✷A→A)→✷A,
•Necessitation Rule: A/✷A,
•Modus ponens: (A, A →B)/B.
iK4 is iGL without L¨ob’s axiom. Note that we can get rid of the necessitation rule by adding ✷Ato
the axioms, for each axiom Ain the above list. We will use this fact later in this paper. We list the
following axiom schemas:
•the Completeness Principle: CP := A→✷A.
•Restricted Completeness Principle to atomic formulae: CPa:= p→✷p, for atomic p.
•Leivant’s Principle: Le := ✷(B∨C)→✷(✷B∨C). [Lei75]
•Extended Leivant’s Principle: Le+:= ✷A→✷Al.
We define theories LC := iGL +CP and LLe+:= iGL +Le++CPa. Note that in the presence of
CP and modus ponens, the necessitation rule is superfluous. Later we will find a Kripke semantics
for LC and also we will see that LC and LLe+proves the same formulae w.r.t. a class of restricted
complexity, to wit TNNIL.
3 Arithmetic
In this section, we gather some preliminaries from intuitionistic arithmetic. Mostly we will prove
some refinements of well-known theorems such as: Π2-conservativity of PA over HA, G¨odel’s diag-
nolization lemma and Σ1-completeness of HA. Most of these preliminaries will be used to prove a
refinement of Leivant’s principle ✷(A∨B)→✷(✷
.A∨✷
.B) in the technical Lemma 3.17. For better
understanding of what Lemma 3.17 is going to state, we suggest the reader first to see Theorem 3.14.
5
3.1 Some arithmetical preliminaries
The first-order language of arithmetic contains three functions (successor, addition and multiplica-
tion), one predicate symbol and a constant: (S, +, ., <, 0). First-order intuitionistic arithmetic (HA)
is the theory over IQC with the axioms:
Q1 S(x)6= 0,
Q2 S(x) = S(y)→x=y,
Q3 y= 0 ∨ ∃xS(x) = y,
Q4 x+ 0 = x,
Q5 x+S(y) = S(x+y),
Q6 x.0 = 0,
Q7 x.S(y) = (x.y) + x,
Q8 x < y ↔ ∃z(x+S(z) = y),
Ind: For each formula A(x):
Ind(A, x) := UC[A(0) ∧ ∀x(A(x)→A(S(x)))] → ∀xA(x)]
In which UC(B) is the universal closure of B.
Peano Arithmetic PA, has the same axioms of HA over CQC.
Notation 3.1. ¿From now on, when we are working in first-order language of arithmetic, for a
first-order sentence A,✷Aand ✷+Aare shorthand for ProvHA (pAq)and ProvPA (pAq), respectively.
Let iΣ1be the theory HA, where the induction principle is restricted to Σ1-formulae. We also define
the theories HAxto be the theory with axioms of HA, in which the induction principle is restricted
to formulas satisfying one of the following conditions:
•formulas of the form (A→B)→Bin which Aand Bare Σ1.
•formulas with G¨odel number less than x.
We can define similar concept for PAx. Note that classically, formulas of the form (A→B)→B
in which Aand Bare Σ1, are equivalent to the Σ1-formula A∨Band hence PA0is the well-known
theory IΣ1. We also define ✷xAand ✷+
xAto be ProvHAx(pAq)and ProvPAx(pAq), respectively.
We recall that a function fon ω:= {0,1,2,...}is recursive iff there exists some Σ1-formula Af(¯x, y)
such that N|=Af(¯x, y) iff f( ¯x) = y. It is called to be provably (in T) total, iff T⊢ ∀¯x∃yAf(¯x, y).
It is well known that all primitive recursive functions are provably total in IΣ1with a ∆0-formula
as defining formula. So we may use primitive recursive function symbols in the language of arithmetic
with their defining axioms (as far as we work in IΣ1).
Lemma 3.2. Let A,Bbe Σ1-formulae such that PA ⊢A→B. Then HA ⊢A→B.
Proof. This is an immediate consequence of Π2-conservativity of PA over HA. [TvD88](3.3.4). ✷
Lemma 3.3. For any ∆0-formula A(¯x), we have HA0⊢ ∀¯x(A(¯x)∨ ¬A(¯x)).
Proof. This is well-known in the literature. ✷
The G¨odel-Gentzen translation associates a formula Agto any formula Ain a first-order language,
and is defined inductively by the following items:
6
•Ag:= A, for atomic A,
•(A∧B)g:= Ag∧Bg,
•(A∨B)g:= ¬(¬Ag∧ ¬Bg),
•(A→B)g:= Ag→Bg,
•(∀xA)g:= ∀xAg,
•(∃xA)g:= ¬ ∀x¬Ag.
The Friedman translation associates a formula AC, for an arbitrary formula C, to any formula
Ain a first-order language. Roughly speaking, ACis the result of adding Cas a disjunct to all
atomic sub-formulas of A. To define AC, we assume that free variables of Cdo not appear as bound
variables of A. It is obvious that we can always take care of this detail by renaming bound variables
of Ato fresh variables.
•AC:= A∨C, for atomic A,
•(A∧B)C:= AC∧BC,
•(A∨B)C:= AC∨BC,
•(A→B)C:= AC→BC,
•(∀xA)C:= ∀xAC,
•(∃xA)C:= ∃xAC.
As shown in [TvD88], we have the following properties for G¨odel-Gentzen and Friedman trans-
lations:
•For each Σ1-formula Ain the language of arithmetic, HA ⊢Ag↔ ¬¬Aand HA ⊢AC↔(A∨C).
•For any Ain the language of arithmetic, CQC ⊢Aimplies IQC ⊢Ag.
•HA0is closed under Friedman’s translation with respect to Σ1-formulas. i.e. for any Σ1-formula
Band any A,HA0⊢Aimplies HA0⊢AB. Actually in [TvD88], this property is proved for
HA instead of HA0, but this case is very similar to that one.
We have the following variant of Lemma 3.2.
Lemma 3.4. For any Σ1-formula A,PA0⊢Aimplies HA0⊢A. Hence for any Π2-sentence A,
PA0⊢Aimplies HA0⊢A.
Proof. First observe that PA0⊢Bimplies HA0⊢Bg, by induction on proof of Bin PA0. We refer
the reader to [TvD88] for a detailed proof of this fact for PA and HA instead of PA0and HA0.
It should only be noted that for any instance Bof induction over Σ1formulae, by definition and
properties of G¨odel-Gentzen translation, Bgbelongs to the axioms of HA0, since, for Aand Bin
Σ1, the formula (A→B)→Btranslates, modulo provable equivalence, to ¬¬ (A∨B). Hence, we
have HA0⊢ ¬¬A, and, thus, HA0⊢(¬¬A)A. This implies HA0⊢A, as desired. ✷
Consider the mapping:
F:n7→ A(Sn(0)) := A(
ntimes
z}|{
S...S(0))
Let Gbe the function that assigns to nthe G¨odel number of F(n). We use pA( ˙x)qas a term for G.
We may omit the dot over variables when no confusion is likely.
7
Lemma 3.5. For every formula A(x, x1...,xn)with free variables exactly as shown, there exists a
formula B(x1,...,xn)such that
HA0⊢B(x1,...,xn)↔A(pB( ˙x1,..., ˙xn)q, x1,...,xn)
Moreover, if the formula Ais ∆0, then Bis also ∆0.
Proof. It is easy to see that the usual proof of the Fixed Point lemma holds in this setting. ✷
The following lemma, states the Σ1-completeness of HA0.
Lemma 3.6. HA0proves all true Σ1sentences. Moreover this argument is formalizable and provable
in HA0, i.e. for every Σ1-formula A(x1,...,xk)we have HA0⊢A(x1,...,xk)→✷0A( ˙x1,..., ˙xk).
Proof. It is a well-known fact that any true (in standard model N) Σ1-sentence is provable in iΣ1.
Moreover this argument is constructive and formalizable in iΣ1.✷
Lemma 3.7. For every formula A, we have PA ⊢ ∀x✷+(✷+
xA→A)and HA ⊢ ∀x✷(✷xA→A).
Proof. The case of PA is well known. For the case HA, see [Smo73b] or Theorem 8.1 in [Vis02]. ✷
Coding of finite sequences
We use some fixed method for encoding of finite sequences and use hx1,...,xnias the code of the
finite sequence (x1,...,xn). We assume here that the encoding is a one-one correspondence between
natural numbers and the assiciated finite sequences. For details on coding of finite sequences, we
refer the reader to [Smo85], Chapter 0.
Let x=hx0, x1,...,xniand y=hy0, y1,...,ymi. The following notations are used in this paper:
•lth(x) is defined as the length of the sequence with the code x, i.e. here lth(x) := n+ 1,
•x∗y:= hx0,...,xn, y0,...,ymi,
•(x)iis defined (if i < lth(x)) as the i-th element in the sequence with the code x, i.e. here
(x)i:= xi. If also i≥lth(x), we define (x)i:= 0,
•ˆxis defined as the final element of the sequence with the code x, i.e. here ˆx:= (x)lth(x)˙
−1,
•xis an initial segment of y(x⊆iy) if lth(x)≤lth(y) and for all j < lth(x), we have (x)j= (y)j.
Kripke models of HA
A first-order Kripke model for HA is a triple K= (K, <, M) such that:
•The frame of K, i.e. (K, <), is a non-empty partially ordered set,
•Mis a function from Kto the first-order classical structures for the language of the arithmetic,
i.e. M(α) is a first-order classical structure, for each α∈K,
•For any α≤β∈K,M(α) is an elementary (weak) substructure of M(β).
For any α∈Kand first-order formula A∈ Lα(the language of arithmetic augmented with constant
symbols ¯afor each a∈ |M(α)|), we define K, α A(or simply αA, if no confusion is likely)
inductively as follows:
•For atomic A,αAiff M(α)|=A. Note that in the structure M(α), ¯ais interpreted as a,
•If Ais conjunction, disjunction or implication, αAas in modal propositional case (see
subsection 4.5),
8
•If A=∀xB,αAiff for all β≥αand each b∈ |M(β)|, we have βB[x:¯
b].
It is well-known in the literature that HA is complete for first-order Kripke models.
Lemma 3.8. Let K= (K, <, M)be a Kripke model of HA and Abe an arbitrary Σ1-formula. Then
for each α∈K, we have αAiff M(α)|=A.
Proof. Use induction on the complexity of Ato show that for each β∈K, we have βAiff
M(β)|=B. In the inductive step for →and ∀, use Lemma 3.3.✷
3.2 q-Realizability and Leivant’s principle
A variant of realizability introduced by Kleene, is q-realizability (see [TvD88]) which is defined
inductively for arithmetical formula Aas follows:
•xqA:= Afor atomic A.
•xq(A1∧A2) := j1(x)qA1∧j2(x)qA2,
•xq(A1∨A2) := (j1(x) = 0 →j2(x)qA1)∧(j1(x)6= 0 →j2(x)qA2),
•xq(A1→A2) := ∀y(yqA1→ ∃u(Txyu ∧U(u)qA2)) ∧(A1→A2),
•xq∃yA(y) := j1(x)qA(j2(x)),
•xq∀yA(y) := ∀y∃u(Txyu ∧U(u)qA(y))
In above definition j1,j2are inverses for a one-to-one onto, pairing function, j, such that x=
j(j1(x),j2(x)). Also Txyu is Kleene’s predicate formalizing “uis a computation for the Turing
Machine with code xwith input y”, and Uis the result extractor function, i.e. if uis a computation
for a Turing Machine, then U(u) is its output.
Lemma 3.9. For any formula Awe have HA0⊢xqA→A.
Proof. See [TvD88]. ✷
In the following, {x}is partial recursive function of Turing Machine with code x. The notation
{x}y↓means that “The function {x}is defined on input y”, or equivalently “The Turing machine
with code xhalts with input y”. It is well known that {x}y↓is a Σ1sentence. We use terms which
contain some Kleene’s bracket notation. In that case, we use t↓to mean that all the brackets in t
are defined (terminate).
One immediate consequence of q-realizability, is Church’s Rule for HA:
Lemma 3.10. For every formula A(x, y), if HA ⊢ ∀x∃y A(x, y), then there exists some n∈ωsuch
that HA ⊢ ∀x({n}(x)↓ ∧ A(x, {n}(x))).
Proof. See [TvD88]. ✷
It is easy to observe that “HA ⊢A” implies “there exists some nsuch that HA ⊢nqA”([TvD88]).
The point of the following lemma is that we can actually compute this nfrom the (code of) proof
of Ain HA and moreover we can formalize this argument in HA:
Lemma 3.11. Suppose that A(x1,...,xm)is an arithmetical formula with free variables as shown.
Then, there exists a provably (in HA)total recursive function fsuch that:
HA ⊢✷xA( ˙x1,..., ˙xm)→ ∃z✷f(x)({˙z}h ˙x1,..., ˙xmi↓ ∧ { ˙z}h ˙x1,..., ˙xmiqA( ˙x1,..., ˙xm))
9
Proof. The proof is very similar to the proof of the soundness part of [TvD88, Theorem 4.10]. First
define f(n) in this way:
f(n) := max({pBq,xq|pBq< n, x is a free variable of B} ∪ {n})
in which, Bq,x := {t(u)}hxi↓ ∧{t(u)}hxiqB,u6=xand t(u) is a primitive recursive function that
will be defined later in the proof. Let’s fix some sequence of numbers m. With induction on the
complexity of the proof HAn⊢A(m), we show that (by A(m), we mean A[x:m])
HA ⊢“HAn⊢A(m)” →“ there exists some number ksuch that HAf(n)⊢kqA(m)”
We only treat the case that Ais an instance of induction schema. All the other cases are trivial and
left to reader. Assume that pBq< n and
A(m) = (B[x: 0] ∧ ∀x(B→B[x:S(x)])) → ∀xB
We should find some ksuch that HAf(n)⊢kq[(B(0) ∧ ∀x(B(x)→B(x+ 1)) → ∀xB]. By definition
of q-realizability, we have:
kqA(m) =
C
z}| {
∀u[uq(B(0) ∧ ∀x(B(x)→B(x+ 1)) →({k}(u)↓ ∧{k}(u)q∀xB)] ∧A(m)
Since f(n)≥n, we have HAf(n)⊢A(m). Hence it remains only to show that HAf(n)⊢C. Define
the primitive recursive function t(u) in the following way. For any given u,t(u) is the code of the
Turing Machine that fulfills the following conditions:
({t(u)}h0i=j1(u)
{t(u)}(x+ 1) = {{j2(u)}hxi}h{t(u)}hxii
Finally, let kbe the code of the Turing Machine that computes the prmitive recursive function t.
Now it is not defiicult to observe that, by induction on Bq,x , one could deduce Cin HA0, and hence
HAf(n)⊢C. This implies HAf(n)⊢A(m), as desired. ✷
Lemma 3.12. For every sentence A, there exists some provably (in HA) total recursive function hA
such that HA ⊢ ∀x✷hA(x)(✷˙xA→A).
Proof. By Lemma 3.7 we have HA ⊢ ∀x∃y✷y(✷˙xA→A). Now we have the desired result by use of
Lemma 3.10.✷
Lemma 3.13. Suppose that A(x1,...,xm)is a Σ1-formula with variables as shown. Then there
exists some nA∈N, such that
HA ⊢A(x1,...,xm)→({nA}hx1,...,xmi↓ ∧ {nA}hx1,...,xmiqA(x1,...,xm))
Proof. This theorem for r-realizability instead of q-realizability is proved in [TvD88](Proposition
4.4.5). The proof for q-realizability is quite similar and we leave it to the reader. ✷
It is well-known that the disjunction property holds for IPC and HA, however it is also shown
that in case of HA , the proof is not formalizable in HA, i.e. HA 0✷(A∨B)→(✷A∨✷B). But this
is not the end of story! Daniel Leivant in his PhD dissertation [Lei75] showed that HA ⊢✷(A∨B)→
✷(A∨✷B). Albert Visser in an unpublished paper showed that we can extend Leivant’s principle
to the following version. For every Σ1-sentence A,HA ⊢✷(A→(B∨C)) →✷(A→(✷B∨C)).
In the following lemma, we will show that we can find (constructively) from the code xof the proof
of A→(B∨C), some f(x) such that ✷(A→(✷f(x)B∨C)) holds. Although the statement of
this theorem would not be used later in this paper, we bring it here for better understanding of its
generalization in a more technical lemma, i.e. Lemma 3.17.
10
Theorem 3.14. For arbitrary sentences A, B, C such that A∈Σ1, there exists a provably (in HA)
total recursive function fsuch that
HA ⊢✷x(A→(B∨C)) →✷f(x)(A→(✷f(x)B∨C))
Proof. First observe that, by Lemma 3.13, there exists some finite number nA∈Nsuch that HA ⊢
A→({nA}hi↓ ∧ {nA}hi qA). We set t0:= {nA}hi. Hence there exists some n0∈Nsuch that
(1) HA ⊢✷n0(A→(t0↓ ∧ t0qA))
We work inside HA. Assume ✷x(A→(B∨C)). By Lemma 3.11, there exists some zsuch
that ✷g0(x)({˙z}hi ↓ ∧ { ˙z}hi q(A→(B∨C))), in which g0is the recursive function provided by
Lemma 3.11. We define t1:= {˙z}hi and hence we have ✷g0(x)t1↓. If we set g1(y) := g0(y) + n0, by
use of Equation 1, we can deduce ✷g1(x)(A→(t0↓ ∧ {t1}(t0)q(B∨C))). We set t2:= {t1}(t0).
Then, by definition of q-realizability, we have:
✷g1(x)(A→(t2↓ ∧ (j1(t2) = 0 →j2(t2)qB)∧(j1(t2)6= 0 →j2(t2)qC))).
Let B′:= (j1(t2) = 0) →j2(t2)qBand C′:= (j1(t2)6= 0) →j2(t2)qCThen we have
✷g1(x)(A→B′) and, hence, by Σ1-completeness (Lemma 3.6), we can deduce ✷0✷g1(x)(A→B′),
that again by use of Lemma 3.6, implies ✷0(A→✷g1(x)B′). Thus we have
✷g1(x)(A→(t2↓ ∧ ✷g1(x)B′∧C′))
Again by Lemma 3.6 and Lemma 3.9,✷g1(x)(A→(t2↓∧(j1(t2) = 0 →✷g1(x)B)∧(j1(t2)6= 0 →C))).
Since atomic formulae are decidable in HA, so for any atomic formulae D, there exists some finite
n2such that in HAn2we have decidability of D. Let HAn2+t2↓decide j1(t2) = 0. If we set
f(x) := g1(x) + n2, we can deduce ✷f(x)(A→(✷f(x)B∨C)), as desired. ✷
3.3 The extended Leivant’s Principle
In this section, we study properties of the extended Leivant’s principle, Le+. We prove that for any
Σ1- substitution σ,HA ⊢σHA (LLe+).
Define a translation qσ(A, x) recursively for a modal proposition Aand a Σ1-substitution σ, as
follows:
•qσ(A, x) := σHA (A), if Ais atomic or boxed,
•qσ(A∧B, x) := qσ(A, j1(x)) ∧qσ(B, j2(x)),
•qσ(A∨B, x) := (j1(x) = 0 →qσ(A, j2(x))) ∧(j1(x)6= 0 →qσ(B , j2(x))),
•if A=B→Cand B∈NOI, we define qσ(B→C, x) := σHA (B)→({x}(nB)↓ ∧qσ(C, {x}(nB))),
in which nBis as in Lemma 3.13. If B6∈ NOI, then define qσ(A, x) := σHA (A).
Lemma 3.15. Let Abe a modal proposition and tbe a term in first-order language of arithmetic
which possibly contain Kleene’s brackets. Then
•HA0⊢xqσHA (A)→qσ(A, x),
•HA0⊢(t↓ ∧qσ(A, t)) →σHA (A),
Proof. Proof of both parts are by induction on the complexity of A.✷
For the next lemma, we need some auxiliary notation σl(A, x). Informally speaking, σl(A, x)
is going to be σHA (Al) with one difference. The new added boxes in Alshould be interpreted as
provability in HAx. More precisely, we define it inductively as the following.
11
•Ais atomic or boxed. σl(A, x) := σHA (A),
•A=B∧C. then σl(A, x) := σl(B, x)∧σl(C, x),
•A=B∨C. then σl(A, x) := ✷
.xσl(B, x)∨✷
.xσl(C, x), in which ✷
.xDis defined as D∧✷xD,
•A=B→C. Like the definition of Al, we define σl(A, x) by cases. If A∈NOI, then we define
σl(A, x) := σHA (B)→σl(C, x), otherwise we define σl(A, x) := σHA (A).
Lemma 3.16. Let Abe a modal proposition. Then
•HA0⊢(x≤y∧σl(A, x)) →σl(A, y),
•HA0⊢σl(A, x)→σHA (Al),
•HA0⊢σl(A, x)→σHA (A).
Proof. Use induction on A.✷
Lemma 3.17. Let Abe a modal proposition, Dbe any Σ1-sentence and tbe a term in first-order
language of arithmetic which possibly contain Kleene’s brackets. Then there exists a provably total
recursive function fsuch that
HA ⊢✷x(D→(t↓ ∧qσ(A, t)) →✷f(x)(D→σl(A, f (x)))
Proof. We use induction on A. For simplicity of notations, we assume here that tis a normal term.
One can build the general case easily.
Atomic, Boxed or conjunction: Trivial.
Disjunction. Let A=B∨C. Then by definition of qσ, we have
HA ⊢✷x(D→qσ(B∨C, t)) →[✷x((D∧j1(t) = 0) →qσ(B, j2(t)))∧✷x((D∧j1(t)6= 0) →qσ(C, j2(t))]
Hence by the induction hypothesis, there exists functions gand hsuch that
HA ⊢✷x(D→qσ(B∨C, t)) →
✷g(x)((D∧j1(t) = 0) →σl(B, g(x))) ∧✷h(x)((D∧j1(t)6= 0) →σl(C, h(x)))
Let f(x) be the maximum of g(x) and h(x). Finally, one can use Σ1-completeness of HA0(Lemma 3.6)
and Lemma 3.16 to derive
HA ⊢✷x(D→qσ(B∨C, t)) →✷f(x)(D→(✷
.f(x)σl(B, f (x)) ∨✷
.f(x)σl(C, f (x))))
Implication. Assume that A=B→C. If B6∈ NOI, by Lemma 3.15, we are done. So assume that
B∈NOI. By definition of qσ, there exists some term t1such that
HA ⊢✷x[D→qσ(B→C, t)] →✷x[(D∧σHA (B)) →(t1↓ ∧qσ(C, t1))]
Since B∈NOI,σHA (B) is a Σ1-formula. Hence by the induction hypothesis, there exists some
function fsuch that
HA ⊢✷x(D→qσ(A, t)) →✷f(x)((D∧σHA (B)) →σl(C, f(x)))
This by definition of σl(B→C, f (x)), implies the desired result. ✷
Lemma 3.18. For any Σ1-substitution σand modal proposition A, there exists some provably total
recursive function gsuch that HA ⊢✷xσHA (A)→✷g(x)σl(A, g(x)).
12
Proof. Work inside HA. Assume ✷xσHA (A). By Lemma 3.11, there exists some ysuch that
✷f0(x)(t↓ ∧ tqσHA (A))
in which t:= {y}hi and f0is a provably total recursive function as stated in Lemma 3.11. Hence by
the first item of Lemma 3.15,✷f0(x)(t↓ ∧qσ(A, t)). Hence by Lemma 3.17, we have the function f
such that ✷f(f0(x))σl(A, f (f0(x)). ✷
Theorem 3.19. For any Σ1-substitution σ, we have HA ⊢σHA (Le+).
Proof. Let Abe a modal proposition. We must show HA ⊢✷σHA (A)→✷σHA (Al). Now the desired
result may be deduced by Lemma 3.18 and the second item of Lemma 3.16.✷
Although there are other ways of proving the above theorem (see [Vis02] or [Iem01]), we need
its major preliminary lemma (i.e. Lemma 3.18) in the proof of the completeness theorem. Specially,
we use Lemma 3.18 in the proof of Lemma 5.11.
3.4 Interpretability
Let Tand Sbe two first-order theories. Informally speaking, we say that Tinterprets S(T✄S)
if there exists a translation from the language of Sto the language of Tsuch that Tproves the
translation of all of the theorems of S. For a formal definition see [Vis98]. It is well-known that for
recursive theories T , S containing PA, the assertion T✄Sis formalizable in first-order language of
arithmetic. For two arithmetical sentence A, B, we use the notation A✄Bto mean that PA +A
interprets PA +B. The following theorem due to Orey, first appeared in [Fef60].
Theorem 3.20. For recursive theories Tand Scontaining PA, we have:
PA ⊢(T✄S)↔ ∀x✷TCon(Sx),
in which Sxis the restriction of the theory Sto axioms with G¨odel number ≤xand Con(U) :=
¬✷U⊥.
Proof. See [Fef60]. p.80 or [Ber90]. ✷
Convention. ¿From above theorem, one can easily observe that PA ⊢(A✄B)↔ ∀x✷+(A→ ¬✷+
x¬B).
So from now on, A✄Bmeans its Π2-equivalent ∀x✷+(A→ ¬✷+
x¬B), even when we are working
in weaker theories like HA. We remind the reader that ✷+stands for provability in PA.
4 Propositional modal logics
In this section, we collect all the required notions with propositional flavour. This section is mostly
devoted to provide an axiomatic system for the Σ1-provability logic of HA, i.e. Hσ, and stating
some of its essential properties that we need them later in the proof of soundness (Theorem 6.3)
or completeness (Theorem 6.4) of Hσfor arithmetical Σ1-substitutions. The following are some of
important results that will be used in the proof of completeness theorem.
•In subsection 4.3, it is shown that the axiomatic system Hσis capable of simplifying any modal
proposition to an equivalent TNNIL−proposition (Corollary 4.18). This fact is useful for proof of
the completeness theorem (Theorem 6.4).
•In subsection 4.4, the TNNIL-conservativity of a stronger theory LC over Hσ(Theorem 4.23) is
proved. This conservativity plays an important role in the proof of completeness theorem. As far
as working with TNNIL-formulas, we get rid of all those complicated axioms of Hσand just use
the more handful theory LC.
13
•In subsection 4.5, we will prove the finite model property for the theory LC (Theorem 4.25). With
the aid of our main theorem in next section (Theorem 5.1), such finite counter-models are used
to be transformed to a first-order counter-models of HA.
4.1 The NNIL formulae and related topics
The class of No Nested Implications to the Left,NNIL formulae in a propositional language was in-
troduced in [VvBdJRdL95], and more explored in [Vis02]. The crucial result of [Vis02] is providing
an algorithm that as input, receives a non-modal proposition Aand returns its best NNIL approxi-
mation A∗from below, i.e., IPC ⊢A∗→Aand for all NNIL formula Bsuch that IPC ⊢B→A, we
have IPC ⊢B→A∗. Also for all Σ1-substitutions σ, we have HA ⊢σHA (✷A↔✷A∗) [Vis02].
•In subsubsection 4.1.1, we state Visser’s NNIL-algorithm for computing A∗, and some of its useful
properties.
•In subsubsection 4.1.2, we explain the extension of this algorithm to the modal language (the
TNNIL-algorithm), which computes A+and is essentially the same as the NNIL-algorithm with
this extra rule: treat inside ✷as a fresh proposition, i.o.w. in the inductive definition of the
algorithm (✷A)+:= ✷A+. Then we prove some useful properties of TNNIL-algorithm: Lemma 4.6
and Corollary 4.7. The best feature of TNNIL-algorithm is that for all Σ1-substitutions σ, we have
HA ⊢σHA (✷A↔✷A+) (first part of Corollary 4.7).
•In subsubsection 4.1.3, we define another algorithm TNNIL−for computing A−, which is essentially
the same as the TNNIL-algorithm, with this minor difference: Only treat those sub-formulae which
are boxed and leave the others. With this minor change, we even have a better feature for A−,
i.e., for all Σ1-substitutions σ, we have HA ⊢σHA (A↔A−) (Lemma 4.9).
Now we define the class NNIL of modal propositions precisely by NNIL := {A|ρA ≤1}, in which
the complexity measure ρ, is defined inductively as follows:
•ρ(✷A) = ρ(p) = ρ(⊥) = ρ(⊤) = 0, for an arbitrary atomic variables pand modal proposition A,
•ρ(A∧B) = ρ(A∨B) = max(ρA, ρB),
•ρ(A→B) = max(ρA + 1, ρB),
In the following, we define an special complexity measure o(.) on modal propositions. We need this
measure for termination of the NNIL-algorithm.
Definition 4.1. Let Dbe a modal proposition. Let
•I(D) := {E∈Sub(D)|Eis an implication that is not in the scope of a ✷}.
•i(D) := max{|I(E)| | E∈I(D)}, where |X|is the number of elements of X.
•cD:= the number of occurrences of logical connectives which are not in the scope of a ✷.
•dD:= the maximum number of nested boxes. To be more precise,
–dD:= 0 for atomic D,
–dD:= max{dD1,dD2}, where D=D1◦D2and ◦ ∈ {∧,∨,→},
–d✷D:= dD+ 1,
•oD:= (dD, iD, cD).
We order the measures oDlexicographically, i.e., (d, i, c)<(d′, i′, c′)iff d < d′or d=d′, i < i′or
d=d′, i =i′, c < c′.
14
For the definition of NNIL-algorithm, we use the bracket notation [A]Bof [Vis02]. We also use a
variant of this notation, [A]′B, which implicitly were used in [Vis02]:
Definition 4.2. For any two modal propositions Aand B, we define [A]Band [A]′Bby induction
on the complexity of B:
•[A]B= [A]′B=B, for atomic or boxed B,
•[A](B1◦B2) = [A](B1)◦[A](B2),[A]′(B1◦B2) = [A]′(B1)◦[A]′(B2)for ◦ ∈ {∨,∧},
•[A](B1→B2) = A→(B1→B2),[A]′(B1→B2) = (A′∧B1)→B2, in which A′=A[B1→B2|B2],
i.e., replace each occurrence of B1→B2in Aby B2,
4.1.1 The NNIL-algorithm
For each modal proposition A, the proposition A∗is produced by induction on complexity measure
oAas follows: [Vis02]
1. Ais atomic or boxed, take A∗:= A.
2. A=B∧C, take A∗:= B∗∧C∗.
3. A=B∨C, take A∗:= B∗∨C∗.
4. A=B→C, we have several sub-cases. In the following, an occurrence of Ein Dis called
an outer occurrence, if Eis neither in the scope of an implication nor in the scope of a boxed
formula.
(a) Ccontains an outer occurrence of a conjunction. In this case, there is some formula J(q)
such that
•qis a propositional variable not occurring in A.
•qis outer in Jand occurs exactly once.
•C=J[q|(D∧E)].
Now set C1:= J[q|D], C2:= J[q|E] and A1:= B→C1, A2:= B→C2and finally, define
A∗:= A∗
1∧A∗
2.
(b) Bcontains an outer occurrence of a disjunction. In this case, there is some formula J(q)
such that
•qis a propositional variable not occurring in A.
•qis outer in Jand occurs exactly once.
•B=J[q|(D∨E)].
Now set B1:= J[q|D], B2:= J[q|E] and A1:= B1→C, A2:= B2→Cand finally, define
A∗:= A∗
1∧A∗
2.
(c) B=VXand C=WYand X, Y are sets of implications or atoms. We have several
sub-cases:
i. Xcontains atomic variables or boxed formula B. We set D:= V(X\ {B}) and take
A∗:= p∗→(D→C)∗.
ii. Xcontains ⊤. Define D:= V(X\ {⊤}) and take A∗:= (D→C)∗.
iii. Xcontains ⊥. Take A∗:= ⊤.
15
iv. Xcontains only implications. For any D=E→F∈X, let
B↓D:= ^((X\ {D})∪ {F}).
Let Z:= {E|E→F∈X} ∪ {C}and A0:= [B]Z:= W{[B]E|E∈Z}. Now if
oA0<oA, we take
A∗:= ^{((B↓D)→C)∗|D∈X} ∧ A∗
0,
otherwise, first set A1:= [B]′Zand then take
A∗:= ^{((B↓D)→C)∗|D∈X} ∧ A∗
1
Remark 4.3. In fact in [Vis02], the NNIL-algorithm is only for non-modal propositions. But one
could compute the best NNIL-approximation for modal propositions, with the original algorithm of
[Vis02] in the following way. Let Abe a given modal proposition. Let B1,...,Bnbe all boxed
sub-formulae of Awhich are not in the scope of any other boxes. Let A′(p1,...,pn) be the unique
non-modal proposition such that {pi}1≤i≤nare fresh atomic variables not occurring in Aand A=
A′[p1|B1,...,pn|Bn]. Let γ(A) := (A′)∗[p1|B1,...,pn|Bn]. Then it is easy to observe that IPC✷⊢
γ(A)↔A∗.
The above defined algorithm is not deterministic, however by the following theorem we know that
A∗is unique up to IPC✷equivalence. Notation A⊲IPC✷,NNIL B(A,NNIL-preserves B) from [Vis02],
means that for each NNIL modal proposition C, if IPC✷⊢C→A, then IPC✷⊢C→B, in which
A, B are modal propositions.
Theorem 4.4. For each modal proposition A,
1. NNIL algorithm with input Aterminates and the output formula A∗, is an NNIL proposition
such that IPC✷⊢A∗→A.
2. IPC✷⊢A∗→Biff A⊲IPC✷,NNIL B.
3. A∗is the best NNIL approximation of Afrom below i.e. IPC✷⊢A∗→Aand for each NNIL
proposition B, with IPC✷⊢B→A, we have IPC✷⊢B→A∗.
4. IPC✷⊢A1→A2implies IPC✷⊢A∗
1→A∗
2.
5. IPC✷⊢A↔Bimplies IPC✷⊢A∗↔B∗.
6. For each Σ1-substitution σ,HA ⊢✷σHA (A)↔✷σHA (A∗).
Proof. 1. Direct consequence of [Vis02, Theorem 7.1]. First assume A′[p1,...,pn] be as in
Remark 4.3. By [Vis02, Theorem 7.1], we have IPC ⊢(A′)∗→A′, and hence by Lemma 2.3,
IPC✷⊢(A′)∗[p1|B1,...,pn|Bn]→A.
2. Direct consequence of [Vis02, Thorem. 7.2]. First suppose that IPC✷⊢A∗→B. Let
A′, B′be non-modal propositions as defined in Remark 4.3, i.e, A=A′[p1|C1,...,pn|Cn], B =
B′[p1|C1,...,pn|Cn]. Then by Lemma 2.3,IPC ⊢(A′)∗→B′. Now by [Vis02, Theorem 7.2],
we have A′⊲IPC,NNIL B′, and then by Lemma 2.3,A⊲IPC✷,NNIL B.
3. Suppose IPC✷⊢B→Aand Bis NNIL. Since IPC✷⊢A∗→A∗, from part 2 above, we get
A⊲IPC✷,NNIL A∗. By IPC✷⊢B→Aand B∈NNIL, we have IPC✷⊢B→A∗.
4. Suppose that IPC✷⊢A1→A2. By part 1, IPC✷⊢A∗
1→A2and hence by part 3,
IPC✷⊢A∗
1→A∗
2.
16
5. Direct consequence of part 4.
6. First suppose that Ais a non-modal proposition. Combining Theorem 10.2 and Corol-
lary 7.2 from [Vis02], implies that IPC ⊢A∗→Biff A|∼HA
HA,ΣB, in which A|∼HA
HA,ΣB
means that for each Σ1-substitution σ, we have HA ⊢✷σ(A)→✷σ(B). This implies that
HA ⊢✷(σ(A)) ↔✷σ(A∗). Now for a modal proposition A, suppose that A′(p1,...,pn) and
B1,...,Bnbe such that A=A′[p1|B1,...,pn|Bn] in which A′is a non-modal proposition and
p1,...,pnare fresh atomic variables (not occurred in A). Let σ′be the substitution defined by
σ′(pi) := σ(Bi), for each 1 ≤i≤n, and for any other atomic variable q,σ′(q) = σ(q). Clearly,
σ′is again a Σ1-substitution and hence we have HA ⊢✷(σ′(A′)) ↔✷σ′((A′)∗). This implies
HA ⊢✷(σ(A)) ↔✷σ(A∗).
✷
4.1.2 The TNNIL-algorithm
Definition 4.5. TNNIL (Thoroughly NNIL) is the smallest class of propositions such that
•TNNIL contains all atomic propositions,
•if A, B ∈TNNIL, then A∨B, A ∧B, ✷A∈TNNIL,
•if all →occurring in Aare contained in the scope of a ✷(or equivalently A∈NOI) and
A, B ∈TNNIL, then A→B∈TNNIL.
Finally we define TNNIL−as the set of all the propositions like A(✷B1,...,✷Bn), such that A(p1,...,pn)
is an arbitrary non-modal proposition and B1,...,Bn∈TNNIL.
Here we define A+as TNNIL-formula approximating A. The ma jor difference between A+and
A∗is that IPC✷⊢A+→Amay not hold any more. Informally speaking, to find A+, we first
compute A∗and then replace all outer boxed formula ✷Bin Aby ✷B+. To be more accurate,
we define A+by induction on dA. Suppose that for all Bwith dB < dA, we have defined B+.
Suppose that A′(p1,...,pn) and ✷B1,...,✷Bnsuch that A=A′[p1|✷B1,...,pn|✷Bn] where A′is
a non-modal proposition and p1,...,pnare fresh atomic variables (not occurred in A). It is clear
that dBi<dAand then we can define A+:= (A′)∗[p1|✷B+
1,...,pn|✷B+
n].
Lemma 4.6. For every modal proposition B,
•If iGL ⊢Bthen iGL ⊢B+.
•If iK4 ⊢Bthen iK4 ⊢B+.
Proof. We prove first part by induction on complexity of proof iGL ⊢B. Proof of the second part is
similar to the first one.
•Bis an axiom.
–Bis L¨ob’s axiom, i.e., B=✷(✷C→C)→✷C. Then B+=✷(✷C+→C+)→✷C+,
that is valid also in iGL.
–B=✷C→✷✷C. Then B+=✷C+→✷✷C+, that is valid in iGL.
–B= (✷(C→D)∧✷C)→✷D. Then B+= (✷(C→D)+∧✷C+)→✷D+. On the
other hand, IPC✷⊢(C∧(C→D)) →Dand hence IPC✷⊢(C∧(C→D))∗→D∗,
by Theorem 4.4(4). Now we can infer IPC✷⊢(C+∧(C→D)+)→D+, by definition
of TNNIL-algorithm and Lemma 2.3. Finally, by the necessitation rule in iGL, we have
iGL ⊢(✷C+∧✷(C→D)+)→✷D+.
•Bis a theorem of IPC✷. Then IPC✷⊢B+, by Theorem 4.4(5) and Lemma 2.3.
17
•B=✷Cand Bis derived by applying the necessitation rule. Let iGL ⊢C. By induction
hypothesis, iGL ⊢C+and then iGL ⊢✷C+.
•Bis derived by modus ponens. Let iGL ⊢Aand iGL ⊢A→B. ¿From these, we have
iGL ⊢A+∧(A→B)+and then (A∧(A→B))+inside iGL. Since IPC✷⊢(A∧(A→B)) →B,
then IPC✷⊢(A∧(A→B))∗→B∗. Then by Lemma 2.3,IPC✷⊢(A∧(A→B))+→B+and
hence iGL ⊢B+as desired.
✷
Corollary 4.7. For any modal proposition A,
1. for all Σ1-substitution σwe have HA ⊢✷σHA (A)↔✷σHA (A+)and hence HA ⊢σHA (A)iff
HA ⊢σHA (A+).
2. iGL ⊢A1→A2implies iGL ⊢A+
1→A+
2, and iK4 ⊢A1→A2implies iK4 ⊢A+
1→A+
2.
3. iGL ⊢A1↔A2implies iGL ⊢A+
1↔A+
2, and iK4 ⊢A1↔A2implies iK4 ⊢A+
1↔A+
2.
Proof. The first assertion can be deduced simply by induction on dAand using Theorem 4.4(6).
To prove the second part, first note that by Theorem 4.4(4), if IPC✷⊢A1→A2, then IPC✷⊢
A∗
1→A∗
2. By Lemma 2.3, we can replace each outer occurrence of boxed formulae by arbitrary
propositions, in particular, by their TNNIL approximations. Then by definition of A+, we have
IPC✷⊢A+
1→A+
2. Now suppose that iGL ⊢A1→A2(iK4 ⊢A1→A2). This implies IPC✷+X⊢
A1→A2, for some Xwith iGL ⊢X(iK4 ⊢X). So IPC✷⊢(X∧A1)→(X∧A2). Then
IPC✷⊢(X∧A1)+→(X∧A2)+, and hence by TNNIL-algorithm, IPC✷⊢(X+∧A+
1)→(X+∧A+
2).
This implies IPC✷+X+⊢A+
1→A+
2and by Lemma 4.6,iGL ⊢A+
1→A+
2(iK4 ⊢A+
1→A+
2).
Proof of the third part is direct consequence of the second part. ✷
4.1.3 The TNNIL−-algorithm
Corollary 4.8. There exists a TNNIL−-algorithm such that for any modal proposition A, it halts
and produces a proposition A−∈TNNIL−such that IPC✷⊢A+→A−.
Proof. Let A:= B(✷C1,...,✷Cn), and B(p1,...,pn) is non-modal. Clearly such Bexists. Then
define A−:= B(✷C+
1,...,✷C+
n). Now definition of A+in Corollary 4.7 implies A+= (A−)∗and
hence Theorem 4.4.1 implies that A−has desired property. ✷
Lemma 4.9. For each modal proposition Aand Σ1-substitution σ,HA ⊢σHA A↔σHA A−.
Proof. Use definition of (.)−and Corollary 4.7.1. ✷
Remark 4.10. Note that LC ⊢A↔Bdoes not imply LC ⊢A+↔B+. A counterexample is
A:= ¬¬pand B:= ¬✷
.(¬p). We have A+=A∗=pand LC ⊢B+↔(✷¬p→p). Now one can use
Kripke models to show LC 0¬¬p→(✷¬p→p).
Remark 4.11. In the algorithm produced for NNIL, Let’s change step (1) in this way (and use new
symbol (.)†instead of (.)∗)
1. A†:= A, for atomic Aand (✷B)†:= ✷B†,
Then the new algorithm also halts, and for any modal proposition A, we have iK4 ⊢A†↔A+.
18
4.2 The Box Translation
The following definition of the box-translation, is essentially from ([Vis82, Def.4.1]). The box-
translation extends the well-known G¨odel-McKinsey-Tarski translation. In this subsection, we prove
that iGL is closed under box-translation (Proposition 4.16).
Definition 4.12. For every proposition Ain the modal propositional language, we associate a
proposition A✷, called the box-translation of A, defined inductively as follows:
•A✷:= A∧✷A, for atomic A,
•(A◦B)✷:= A✷◦B✷, for ◦ ∈ {∨,∧},
•(A→B)✷:= (A✷→B✷)∧✷(A✷→B✷),
•(✷A)✷:= ✷(A✷).
Lemma 4.13. For any modal proposition A, we have iK4 ⊢A✷→✷A✷.
Proof. Easy induction over the complexity of A.✷
In the following lemma we state some properties of ✷
..
Lemma 4.14. For each modal sentences A,Band C, the following propositions are provable in
iK4.
1. ✷ ✷
.A↔✷A↔✷
.✷A.
2. ✷
.A✷↔A✷.
Proof. The first part is easily deduced in iK4. For the second one deduce from Lemma 4.13.✷
Following Visser’s definition of the notion of a base in arithmetical theories [Vis82], we define
Definition 4.15. We say that a modal theory Tis closed under box-translation if for every propo-
sition A,T⊢Aimplies T⊢A✷.
Proposition 4.16. The theory iGL is closed under box-translation.
Proof. The proof can be carried out in three steps:
1. For any proposition Afirst we show that IPC✷⊢Aimplies iK4 ⊢A✷. This can be done by a
routine induction on the length of the proof in IPC. Note that for any axiom Aof IPC, we have
iK4 ⊢A✷. As for the rule of modus ponens, suppose that IPC✷⊢Aand IPC✷⊢A→B. By
induction hypothesis, then iK4 ⊢A✷and iK4 ⊢(A✷→B✷)∧✷(A✷→B✷) and so iK4 ⊢B✷.
2. Next observe that
(✷A→✷✷A)✷=✷A✷→✷✷A✷
and also
iK4 ⊢[(✷(A→B)∧✷A)→✷B]✷↔(✷(A✷→B✷)∧✷A✷)→✷B✷
3. Observe that the box translation of an instance of L¨ob’s axiom L, is also an instance of L.✷
19
4.3 Axiomatizing the TNNIL-algorithm
In this subsection we present axioms which we need for the TNNIL−-algorithm (.)−. More precisely,
we will find some axiom set Xsuch that X⊢A−↔A.
To do that, we use some relation ◮between modal propositions. A variant of this relation for
non-modal case first came in [Vis02]. The relation ◮is defined to be the smallest relation on modal
propositional sentences satisfying:
A1. If iK4 ⊢A→B, then A◮B,
A2. If A◮Band B◮C, then A◮C,
A3. If C◮Aand C◮B, then C◮A∧B,
A4. If A◮B, then ✷A◮✷B,
B1. If A◮Cand B◮C, then A∨B◮C,
B2. Let Xbe a set of implications, B:= VXand A:= B→C. Also assume that Z:= {E|E→F∈X} ∪ {C}.
Then A◮[B]Z,
B3. If A◮B, then for any atomic or boxed Cwe have C→A◮C→B.
A◮◭ Bmeans A◮Band B◮A. Let us define the theory
Hσ:= LLe++CPa+{✷A→✷B|A◮B}
in which CPais the Completeness Principle restricted to atomic propositions.
Notation. In the rest of the paper, we use A≡Bas a shorthand for iK4 ⊢A↔B.
The following theorem, shows that A1-A4 and B1-B3, axiomatize the TNNIL algorithm:
Theorem 4.17. For any modal proposition A, we have A◮◭ A+.
Proof. We prove the desired result by induction on o(A). Suppose we have the desired result for
each proposition Bwith o(B)<o(A). We treat Aby the following cases.
1. (A1) Ais atomic. Then A+=A, by definition, and result holds trivially.
2. (A1-A4, B1) A=✷B, A =B∧C, A =B∨C. All these cases hold by induction hypothesis.
In boxed case, we use of induction hypothesis and A4. In conjunction, we use of A1-A3 and in
disjunction we use A1,A2 and B1.
3. A=B→C. There are several sub-cases. similar to definition of NNIL-algorithm, an occurrence
of a sub-formula Bof Ais said to be an outer occurrence in A, if it is neither in the scope of a ✷
nor in the scope of →.
(a) (A1-A3) Ccontains an outer occurrence of a conjunction. We can treat this case using
induction hypothesis and TNNIL-algorithm.
(b) (A1-A3) Bcontains an outer occurrence of a disjunction. We can treat this case by induction
hypothesis and TNNIL-algorithm.
(c) B=VXand C=WY, where Xand Yare sets of implications, atoms and boxed formulae.
We have several sub-cases:
i. (A1-A3, B3) Xcontains atomic variables. Let pbe an atomic variable in X. Set
D:= V(X\ {p}). Then A+≡p→(D→C)+. On the other hand, we have by
induction hypothesis and A1,A2 and B3, that p→(D→C)+◮◭ p→(D→C).
Finally by A1 and A2 we have A+◮◭ A.
20
ii. (A1-A3, B3) Xcontains boxed formula. Similar to the previous case.
iii. (A1, A2) Xcontains ⊤or ⊥. Trivial.
iv. (A1-A3, B2) Xcontains only implications. This case needs the axiom B2 and it seems
to be the interesting case.
By the definition of A+, depending on o([B]Z)<o(A) or not, respectively we have:
A+≡^n(B↓D→C)+|D∈Xo∧([B]Z)+or
A+≡^n(B↓D→C)+|D∈Xo∧([B]′Z)+
We only treat the first case, i.e. when o([B]Z)<o(A). The other case is similar and left
to the reader. The reader should note that in the other case we have o([B]′Z)<o(A)
and hence one could apply induction hypothesis over [B]′Z. Then by induction hypoth-
esis, A1-A3 and B3 we have:
A+◮◭ ^{B↓D→C|D∈X} ∧ [B]Z
We show that for each E∈Z,
(2) iK4 ⊢^{(B↓D)→C|D∈X} ∧ [B]E→A
If E=C, we are done by IPC✷⊢[B]C→(B→C). So suppose some E→F∈X.
We reason in iK4. Assume V{(B↓D→C|D∈X}, [B]Eand B. We want to derive
C. We have (V(X\ {E→F})∧F)→C, [B]Eand B. From Band [B]E, we derive
E. Also from B, we derive E→F, and so F. Hence we have V(X\ {E→F})∧F,
which implies C, as desired.
Now Equation 2 implies iK4 ⊢(V{(B↓D→C|D∈X} ∧ [B]Z)→A