Craig interpolation on the logic of knowledge

Craig Interpolation on the Logic of Knowledge

Everardo Bárcenas¹, José-de-Jesús Lavalle-Martínez², Guillermo Molero-Castillo^3,4, and Alejandro Velázquez-Mena¹

1 Universidad Nacional Autónoma de México [barcenas,mena]@fi-b.unam.mx

2 Benemérita Universidad Autónoma de Puebla jlavalle@cs.buap.mx

3 Consejo Nacional de Ciencia y Tecnología

4 Universidad Veracruzana ggmoleroca@conacyt.mx

Abstract. The Craig interpolation property is described as follows: if a formula φimplies another formulaψ, then there is a formulaβin the common language ofφandψ, such thatφalso impliesβ, as well asβimpliesψ. In this paper, we provide a constructive proof of the Craig interpolation property on the modal logic of knowledgeKm. The proof is based on the application of the Maehara technique on a tree-hypersequent calculus. We also show, as a consequence of the interpolation property, Beth definability and Robinson joint consistency .

1 Introduction

Philosophy and Artificial Intelligence have been the traditional study framework to rea- son about knowledge [12]. More recently, reasoning about knowledge has become of much importance in many areas of computer science, such as distributed systems, cryp- tography, natural language processing and databases [8]. In this paper, we consider as a formal framework to study knowledge, the basic modal logic of knowledge, also known as the multi-modal logicK_m. This logic has been shown to be mathematically well-founded [11]. As consequence of being a bisimulation invariant fragment of first order logic,Kmposses a nice balance of expressiveness and reasoning efficiency [3].

The interpolation property was first proved for classical first order logic by Craig [6].

Considering a formulaφimplies another formulaψ, the interpolation property consists in the existence of a formula β, called the interpolant, in the common language of φandβ, which is assumed to be non-empty, such thatφimpliesβ, andβ impliesψ.

Some logical consequences of interpolation are Beth definability [1] and Robinson joint consistency [22]. Applications of interpolation in computer science have been recently studied for formal verification [18], computational complexity [5], and knowledge rep- resentation [13, 4], among others.

In this paper, we give a constructive proof the Craig interpolation theorem for the modal logic of knowledgeKm. The proof implies a straightforward algorithm to com- pute interpolants.

1.1 Related work

Early studies about the interpolation property in modal logics are reported in [10, 16].

In [10], Gabbay proved interpolation for several mono-modal logics includingKand S4. Maksimova in [16] indentifies a close conection of amalgamability of modal logics containingS4, and proved that only a finite number modal logics conainingS4enjoys interpolation. Maksimova later proved in [15] interpolation of all normal modal log- ics via amalgamation. This result was extended for multi-modal logics in [14]. Marx proved interpolation in [17] for several modal logics with bisimulation. This work in- cludes interpolation proofs forK, fibered modal logics and the multi-modal logics of knowledge and belief. In all the above works, interpolation is proved by semantics meth- ods. Although these methods are quite general and can be applied to several logics, they not provide an explicit construction of interpolants. In the current paper, we provide a syntactic proof of interpolation for the multi-modal logicK_m. This proof includes an explicit construction of interpolants.

Syntactic interpolation proofs for modal logicsKB,KDB,K5andKD5are de- scribed in [20]. In this work, interpolation is proved by means of a cut-free complete sequent-like tableau deduction system. Constructive interpolation for modal logicsK andT is given in [2]. More precisely, a stronger form of interpolation, called uniform interpolation, is proved in this work. In uniform interpolation, interpolants are com- posed by the common language language of formulas in the implication, but restricted by a choice of propositional variables. The closest work to our paper is [9]. In this work, constructive interpolation is proved for the entire modal cube, composed by the logics resulting from any combination ofK,D,T,B,5and4. The proof technique used in this work is based on nested sequents. In our paper, we obtain a constructive interpo- lation proof for the multi-modal logicKm, using the Maehara technique on a cut-free complete tree-hypersequent calculus.

In [7], D’Agostino reports an extensive survey on interpolation for non-classical logics, including modal logics.

1.2 Outline

We first introduce the multi-modal logic of knowledgeKmin Section 2. In Section 3, we describe a complete cut-free tree-hypersequent calculus forKm. Then, in Section 4, by means of Maehara technique, we extract interpolants from tree-hypersequent proofs ofK_mimplications. In Section 5, as a consequence of interpolation, we also prove Beth definability and Robinson joint consistency. Finally, in Section 6, we give a summary of the article and briefly argue further research perspectives.

2 Logic of Knowledge

We assume a basic modal language: a non-empty set of propositions PROP; and a non- empty finite set of modalities MOD.

The set of formulas is inductively defined by the following grammar.

φ:=p| ¬φ|φ∧φ|2mφ

wherepis a proposition andmis a modality.

Notation:

>:=p∨ ¬p ⊥:=¬>

φ∨ψ:=¬(¬φ∧ ¬ψ) φ→ψ:=¬φ∨ψ 3mφ:=¬2m¬φ

A Kripke structure is a tupleM= (W, R, V)where:

– Wis a non-empty set calleddomain;

– Ris a finite set of binary relationsR^m:W ×W, for every modalitym; and – V :PROP7→2^W is valuation function mapping propositions to domain subsets.

Given a Kripke structureM= (W, R, V), formulas are interpreted as follows:

[[p]]^M={w∈V(p)}

[[¬φ]]^M=W\[[φ]]^M [[φ∧ψ]]^M= [[φ]]^M∩[[ψ]]^M

[[2mφ]]^M={w| ∀w⁰∈W :if(w, w⁰)∈R^m, thenw⁰∈[[φ]]^Mo

We may also writeM, w |= φinstead ofw ∈ [[φ]]^M,M |= φwhen for everywin M, we have thatM, w |=φ, in which case we sayMis a model ofφ. If any Kripke structure is a model ofφ, we write|=φ.

Definition 1 (Hilbert derivation system).We define the derivation systemH by the following schemas and rules, for eachm∈MOD:

A1 φ→(ψ→φ)

A2 (φ→(ψ→β))→((φ→ψ)→(φ→β)) A3 (¬φ→ ¬ψ)→(ψ→φ)

A4 2m(φ→ψ)→(2mφ→2mψ) R1

φ→ψ φ

ψ R2

φ 2mφ

We say a formula φ_n is derivable from H, written `H φ_n, if there is a sequence φ₁, φ₂, . . . , φ_n, such that for eachi∈ {1, . . . , n}:

– φiis either an instance, up to substitution, of a schema inH, or

– there is (are)j < i(andk < i) such thatφ_iandφ_j (andφ_k) are instances of the conclusion and premises, resp, of a rule inH.

Consider for instance the following derivation of2m(φ∧ψ)→2mφ:

1. (φ∧ψ)→φ, which by notation is an instance of A₁,¬φ∨ψ∨φ;

2. 2m((φ∧ψ)→φ), from1by R2;

3. 2m((φ∧ψ)→φ)→(2m(φ∧ψ)→2mφ), from A4; and 4. 2m(φ∧ψ)→2mφ, from2and3byR₁.

Theorem 1 (Correctness [11]).For any formulaφ,`H φ, if and only if,|=φ.

3 Tree-hypersequents

A sequent is an expressionΓ ` ∆, whereΓ and∆are formula multisets, non-empty and finite. Intuitively, a sequentΓ `∆is interpreted, in terms of logical symbols, as an implication, where the antecedent is composed by the disjunction of formulas inΓ, and the consequent is the conjunction of formulas in∆. We then the define the following interpretation function:

(φ₁, . . . , φ_n`ψ₁, . . . , ψ_k)^I :=

n

^

i=1

φ_i →

k

_

j=1

ψ_j

wherenandkare some positive integers.

In sequents, we often writeφ, Γ orΓ, φinstead of{φ} ∪Γ, also` ∆instead of

> `∆, andΓ `in place ofΓ ` ⊥.

Tree hypersequents expressions are inductively defined by the following grammar:

T :=S[ST] ST :=∅ |m:T, ST

whereS is a sequent and mis a modality. We extend the interpretation function of sequents for tree hypersequents as follows:

(S[ST])^I :=S^I∨(ST)^I (∅)^I :=⊥

(m:T, ST)^I :=2mT^I∨(ST)^I

When clear from context, we often write tree instead of tree hypersequent. It is usually writtenSinstead ofS[∅], also ifSisΓ `∆, we writeφ, SandS, φinstead of φ, Γ `∆andΓ `φ, ∆, respectively.

We writeThSiwhen a sequentSoccurs in a treeT, more precisely:

– S[ST]hSi;

– S⁰[ST]hSi, provided thatS⁰is different thanSandSThSi; and – (m:T⁰, ST)hSi, when eitherT⁰hSiorSThSi.

We extend the occurring relationThT⁰ibetween trees as expected:

– TandT⁰are the same;

– (S[ST])hT⁰i, whenSThT⁰i;

– (m:T⁰, ST⁰)hT⁰i; and

– (m:T⁰⁰, ST⁰)hT⁰i, provided thatT⁰⁰is different thanT⁰andST⁰hT⁰i.

We also distinguish whenm:T⁰occurs in a treeT, writtenThm:T⁰i:

– (S[ST])hm:T⁰i, whenSThm:T⁰i;

– (m:T⁰, ST⁰)hm:T⁰i; and

– (m⁰ :T⁰⁰, ST⁰)hT⁰i, provided that eithermis different thanm⁰ orT⁰⁰is different thanT⁰, andST⁰hm:T⁰i.

We say a sequentS occurs, under a modalitym, in a finite sequence of tree hyperse- quentsm₁:T₁, m₂:T₂, . . . , m_k :T_k, when there is anisuch thatm_iismandT_ihas the formS[ST]. Moreover, we often writeS[m:S⁰]instead ofS[ST], provided S⁰ occurs underminST.

Definition 2 (Tree-hypersequents derivation system).The inference system for tree hypersequentsGis defined as follows.

– Initial tree hypersequents:

Thp, S, pi – Propositional rules:

ThS, φi Th¬φ, Si ¬L

Thφ, Si ThS,¬φi ¬R Thφ, ψ, Si

Thφ∧ψ, Si ∧L

ThS, φi ThS, ψi ThS, φ∧ψi ∧R – Modal rules:

Th2mφ, S[m:φ, S⁰]i 2mL Th2mφ, S[m:S⁰]i

ThS[m:`φ, ST]i 2mR ThS,2mφ[ST]i

We now define the concept of derivation tree:

– any rule (up to subtitution) ofGis a derivation tree;

– ^T_T⁰ and ^T0 _T^T00 are derivation trees, provided thatT⁰ andT⁰⁰are derivation trees, and

– ^T

0 0

T and ^T

0 0 T₀⁰⁰

T are rules inGandT₀⁰ andT₀⁰⁰ are the lowest tree hypersequents occurring inT⁰andT⁰⁰.

If all the branches of a derivation tree, where T is the lowest tree hypersequent, are finite and ends with an initial tree hypersequent, then we say the derivation tree is a proof tree, or simply a proof, ofT, or thatTis derivable inG, and we write`GT.

Consider now for instance the following proof ofA₄:

Thφ`ψ, φi

Thφ, ψ`ψi Thφ`ψ,¬ψi ¬R Thφ`ψ, φ∧ ¬ψi ∧R

2m¬(φ∧ ¬ψ),2mφ`[m:¬(φ∧ ¬ψ), φ`ψ] ¬L 2mL 2m¬(φ∧ ¬ψ),2mφ`[m:φ`ψ]

2mL 2m¬(φ∧ ¬ψ),2mφ`[m:`ψ]

2mR 2m¬(φ∧ ¬ψ),2mφ`2mψ

2m¬(φ∧ ¬ψ),2mφ,¬2mψ` ¬L 2m¬(φ∧ ¬ψ)∧2mφ∧ ¬2mψ` ∧L

` ¬(2m¬(φ∧ ¬ψ)∧2mφ∧ ¬2mψ) ¬R

Theorem 2 ([21, 19]).For any sequentS,`GS, if and only if,`H S.

Corollary 1. For any sequentS,`G S, if and only if,|=S^I.

4 Interpolation

We define the set of non-logical symbolsSym(φ)of a formulaφas follows:

– Sym(p) ={p};

– Sym(¬φ) =Sym(φ);

– Sym(φ∧ψ) =Sym(φ)∪Sym(ψ); and – Sym(2mφ) ={m} ∪Sym(φ).

The set of non-logical symbols of a (multi-)set of formulas is defined as expected.

For technical convenience, we consider an equivalent extensionG⁰of the derivation systemG, where formulas>are consideredper se(not as notation). All rules inGare also inG⁰. Additionally, the initial sequentThS,>iis also included inG⁰.

Lemma 1 (Maehara’s Lemma).LetThΓ `∆ibe derivable inG, and letΓ1, Γ2and

∆1, ∆2 be partitions ofΓ and∆, respectively. Then there is a formulaβ, called the interpolant, such that ThΓ1`∆1, βi and Thβ, Γ2`∆2i are derivable in G⁰, and Sym(β)⊆(Sym(Γ1)∪Sym(∆1))∩(Sym(Γ2)∪Sym(∆2)).

Proof. By induction on the height of the proof tree.

The base case isThp, Γ `∆, pi. The interpolantβis then defined according to the occurrence of propositionspin partitions:

Thp, Γ₁`∆<sub>1</sub>, p,¬>i Th¬>, Γ<sub>2</sub>`∆₂i ThΓ1`∆1,>i Th>, p, Γ2`p, ∆2i Thp, Γ1`∆1, pi Thp, Γ2`∆2, pi ThΓ1`∆<sub>1</sub>, p,¬pi Th¬p, p, Γ2`∆₂i Induction step. Assume the last inference is the following:

ThΓ `∆, φi ThΓ `∆, ψi ThΓ `∆, φ∧ψi

By induction hypothesis, there are interpolants β₁ and β₂ for the upper tree hyper- sequents. There are two possible cases according the occurrence of φ andψ in the respective partitions.

ThΓ1`∆1, φ, β1i Thβ1, Γ2`∆2i ThΓ1`∆1, ψ, β2i Thβ2, Γ2`∆2i ThΓ1`∆1, β1i Thβ1, Γ2`∆2, φi ThΓ1`∆1, β2i Thβ2, Γ2`∆2, ψi

Depending on the occurrence ofφ∧ψin the partitions, we then construct the interpolant φas follows:

ThΓ1`∆1, φ∧ψ, β1∨β2i Thβ1∨β2, Γ2`∆2i ThΓ1`∆1, β1∧β2i Thβ1∧β<sub>2</sub>, Γ<sub>2</sub>`∆₂, φ∧ψi In the induction step, now consider the last inference is

Th2mφ, Γ `∆[m:φ, Γ<sup>0</sup>`∆⁰]i Th2mφ, Γ `∆[m:Γ<sup>0</sup>`∆⁰]i

By induction, there is an interpolantβ for the upper tree hypersequent. By the occur- rence of2mφin partitions, we distinguish two cases:

Th2mφ, Γ1`∆1, β[m:φ, Γ<sup>0</sup>`∆⁰]i Thβ, Γ2`∆<sub>2</sub>[m:φ, Γ<sup>0</sup>`∆⁰]i ThΓ1`∆1, β[m:φ, Γ<sup>0</sup>`∆⁰]i Thβ,2mφ, Γ₂`∆<sub>2</sub>[m:φ, Γ<sup>0</sup>`∆⁰]i

We then construct the following interpolants:

Th2mφ, Γ1`∆1,2mφ∧β[m:Γ<sup>0</sup>`∆⁰]i Th2mφ∧β, Γ₂`∆<sub>2</sub>[m:Γ<sup>0</sup>`∆⁰]i ThΓ1`∆1,¬2mφ∨β[m:Γ<sup>0</sup>`∆⁰]i Th¬2mφ∨β,2mφ, Γ₂`∆<sub>2</sub>[m:Γ<sup>0</sup>`∆⁰]i

Consider now the last inference is the following:

ThΓ `∆[m:`φ, ST]i ThΓ `∆,2mφ[ST]i We obtain the following interpolantβby induction:

ThΓ1`∆<sub>1</sub>, β[m:`φ, ST]i Thβ, Γ2`∆2[m:`φ, ST]i

There are then two cases depending on the occurrence of2mφin partitions:

ThΓ1`∆1,2mφ,¬2mφ∧β[ST]i Th¬2mφ∧β, Γ2`∆2[ST]i ThΓ1`∆<sub>1</sub>,2mφ∨β[ST]i Th2mφ∨β, Γ2`∆2,2mφ[ST]i

Theorem 3 (Craig Interpolation).For any two formulasφandψ, if|=φ→ψ, then there is a formulaβ, such that|= φ → β,|= β → ψand Sym(β) ⊆ Sym(φ)∩ Sym(β), provided that there is a propositionpsuch thatp∈Sym(φ)∩Sym(β).

Proof. Assume|=φ→ψ, then by Corollary 1,φ`ψis derivable inG. By Lemma 1, there is a formula β, such that φ ` β and β ` φ are derivable in G⁰. Let p ∈ Sym(φ)∩Sym(β). Now, letβ⁰ be obtained fromβ by replacing>by¬(p∧ ¬p).

It is straightforward thatφ ` β andβ ` φare derivable inG, and hence (by Corol- lary 1)|=φ→βand|=β→ψ.

5 Definability and Consistency

Definition 3 (Implicit definability).Letφ(p, p₁, . . . , p_k)be a formula, wherep, p₁, . . . , p_k are propositions occurring in it. We sayφ(p, p₁, . . . , p_k)definespimplicitly if

|= (φ(p, p1, . . . , pk)∧φ(p⁰, p1, . . . , pk))→(p↔p⁰)

wherep6=p⁰.

Definition 4 (Explicit definability).Letφ(p, p₁, . . . , p_k)be a formula, wherep, p₁, . . . , p_k are propositions occurring in it. We sayφ(p, p1, . . . , pk)definespexplicitly, when

|=φ(p, p1, . . . , pk)→(p↔ψ)

whereSym(ψ)⊆Sym(φ(p, p1, . . . , pk))\ {p}.

Theorem 4 (Beth Definability).Letφ(p, p1, . . . , pk)be a formula, wherep, p1, . . . , pk

are propositions occurring in it. Ifφ(p, p1, . . . , pk)definespimplicitly, thenφ(p, p1, . . . , pk) definespexplicitly.

Proof. From the implicit definability assumption, it is easy to see that

|= (φ(p, p₁, . . . , p_k)∧p)→(φ(p⁰, p₁, . . . , p_k)→p⁰)

By the Craig Interpolation Theorem 3, we then obtain

|= (φ(p, p₁, . . . , p_k)∧p)→ψ

|=ψ→(φ(p⁰, p1, . . . , pk)→p⁰)

whereSym(ψ)⊆Sym(φ(p, p₁, . . . , p_k))\ {p}.

Before definining the notion of consistency, we need a precise description of some concepts. An axiom system is a finite set of formulas. An axiom sequence is a (possibly empty) subset of an axiom system. We say a sequentS is derivable (provable) inG from an axiom systemA, if there is an axiom sequenceA⁰ofA, such that`_GA⁰, S.

Definition 5 (Consistency).An axiom system is inconsistent if the empty sequent is derivable from it. We say an axiom system is consistent if it is not inconsistent.

Theorem 5 (Robinson Joint Consistency).Consider two consistent axiom systemsA1

andA2, if for any formulaφ, such thatSym(φ)⊆Sym(A1)∩Sym(A2), it is not the case that bothφand¬φare derivable fromA₁andA₂(orA₂andA₁), respectively, thenA₁∪A₂is consistent.

Proof. We prove the contrapositive. IfA₁∪A₂ is not consistent, then there are two axiom sequences A⁰₁ andA⁰₂ ofA₁ andA₂, resp., such thatA₁, A₂ `are derivable inG. Recall eachA<sub>1</sub> andA<sub>2</sub> is consistent, then not empty. By Lemma 1, there is an interpolantφ, whereSym(φ)⊆Sym(A<sub>1</sub>)∩Sym(A<sub>2</sub>), such thatA<sub>1</sub>`φandφ, A₂` (henceA<sub>2</sub> ` ¬φ) are both derivable inG⁰. As in the proof of Theorem 3, it is straight forward that bothA1 ` φandA2 ` ¬φare also derivable inGby replaceing all the occurrences of>inφby¬(p∧ ¬p)for ap∈Sym(A1)∪Sym(A2).

6 Conclusions

In this paper, we describe a constructive proof of the Craig interpolation property. The proof is based on the Maehara technique on a complete cut-free tree-hypersequent cal- culus. An interpolant algorithm can easily be inferred from the proof. A complexity analysis of this algorithm is prospected as further research. We are also interested in constructive interpolation proofs for other more expressive modal logics, such asK_m with converse, CTL and theµ-calculus.

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