garg.2012.countermodelsfromsequentcalculiinmultimodallogics

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Countermodels from Sequent Calculi in Multi-Modal Logics

Deepak Garg

MPI-SWS

Germany

Valerio Genovese

University of Luxembourg and

University of Torino, Italy

Sara Negri

University of Helsinki

Finland

January 06, 2012

Abstract

A novel countermodel-producing decision procedure that applies to several multi-modal log-ics, both intuitionistic and classical, is presented. Based on backwards search in labeled sequentcalculi, the procedure employs a novel termination condition and countermodel construction.

Using the procedure, it is argued that multi-modal variants of several classical and intuitionisticlogics including K, T, K4, S4 and their combinations with D are decidable and have the finitemodel property. At least in the intuitionistic multi-modal case, the decidability results are new.It is further shown that the countermodels produced by the procedure, starting from a set ofhypotheses and no goals, characterize the atomic formulas provable from the hypotheses.

1 Introduction

Modal logics are widely used in several fields of Computer Science and their decidability is a subjectof deep interest to the academic community. The subject has been investigated through varioustechniques, notably semantic filtrations [5, 8], semantic tableaux [4, 10, 11], and translation into

decidable fragments of first-order logic [2], yet many areas related to decidability of modal logicsremain open. Two such areas are: (1) Decidability of multi-modal intuitionistic logics, especiallywhen modalities interact with each other, and (2) Decision procedures based on sequent calculithat can be directly implemented. Both these areas are challenging. Decidability of intuitionisticmodal logics is challenging because standard techniques like semantic filtrations and tableaux havenot been studied extensively in the intuitionistic setting, whereas sequent calculi are difficult touse for decision procedures in modal logic because of a well-known problem of looping [10,16,25],which is exacerbated by the interaction between modalities and intuitionistic implication.

Spanning both these areas, we present a uniform decision procedure for several propositionalmulti-modal logics (both intuitionistic and classical), based on backwards search in labeled sequentcalculi. Our decision procedure is constructive, which means that for any given formula it eitherproduces a derivation which shows that the formula is true in all (Kripke) models or produces a

finite set of finite countermodels on all of which the formula is false.The decision procedure is also general; it applies to any intuitionistic modal logic without

possibility (diamond) modalities and any classical modal logic (even with possibility modalities),provided the logic satisfies a specific technical condition, namely the existence of what we calla suitable closure relation or SCR. As examples, we show that the classical and intuitionisticvariants of the following multi-modal logics are constructively decidable by our method: K (the

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basic normal modal logic), T (reflexive frames), K4 (transitive frames), S4 (reflexive and transitiveframes) and their combinations with D (serial frames). We further show that several interactionaxioms between modalities such as I ((A) (BA)) [9, 13], unit ( (A)) [7] andsubsumption ((A) (B)) result in decidable logics. Constructive decidability also impliesthe finite model property, so our results also show this property for all the logics listed earlier. For

multi-modal intuitionistic logics, not only our method, but also the decidability results are new.

Technical approach and challenges. We present our method separately for classical and in-tuitionistic logics due to minor technical differences between the two. Our method uses the labeledapproach to proof theory of modal logic as developed, among others, by [16, 23, 25], and morespecifically [16], to produce labeled sequent calculi with strong analyticity properties. We definea multi-modal intuitionistic (classical) logic MMI (MM) as the set of all formulas that are validin all intuitionistic (classical) Kripke frames satisfying stipulated conditions, represented as a set. Conditions in can be arbitrary, but are restricted in two ways: (1) They must be of theform x.((i=1,...,n xiRix

i) (xRx

)), where R, Ri range over the relations of a Kripke frame,and (2) The conditions must have a SCR, which is a relation over Kripke frames satisfying somestipulated properties, as discussed later in the paper. Briefly, the existence of the SCR implies thatframe relations can be deduced from the conditions only in some specific ways. We then builda non-terminating, standard labeled sequent calculus for the logic, which we refine in two steps toobtain a constructive decision procedure that, for a given sequent, either produces a proof of it, ora finite set of finite countermodels for it. Next, we build an extension of our method that worksfor any logic on which our original method works, extended with seriality. Finally, we prove aninteresting property of our decision procedure: The set of countermodels it produces for a givenhypotheses without a specific goal completely characterizes the atomic formulas that can be provedfrom the hypotheses. Thus the set of countermodels produced is, in a sense, complete. We call thisproperty comprehensiveness.

The first challenge for our work is to find a general termination condition for backwards searchin labeled sequent calculi. Our termination condition is based on containment of the sets of formulas

labeling worlds, which we show to be complete so long as the logic has an SCR. This idea is a non-trivial generalization of existing work on logic-specific termination conditions for many uni-modalclassical logics [6, 10, 16]. Finding an appropriate definition for SCRs, that is both sufficient toobtain termination and general enough, is the main technical challenge of our work and also itsmain technical contribution.

The second challenge in our work is to actually build the countermodel when we know thatbackwards proof search has unsuccessfully terminated. To this end, we observe that a straightfor-ward extension of the model inherent in the sequent at which search terminates (with a few morerelations) is actually a countermodel to the sequent. As far as we know, this construction is novel.

Contributions. In summary, our work makes the following contributions to the area of decid-

ability of modal logics:

It proves, by uniform method, the decidability of the necessitation-only, multi-modal intu-itionistic variants of the logics K, T, K4 and S4 and their combinations with the logic D. Ourdecision procedure produces countermodels, and also establishes the finite model propertyfor these logics. (The corresponding uni-modal intuitionistic logics are already known to bedecidable due to the work of Simpson[23] and Hasimoto [15].)

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We provide the first sequent calculus based constructive decision procedure for multi-modallogics. (Sequent calculus based decision procedures for specific uni-modal logics exist in boththe classical [16,24] and intuitionistic [23]settings.)

At a technical level, we provide a general method for forcing termination in labeled sequentcalculi for modal logics and a sufficient condition (the existence of SCRs) under which it works

without loss of completeness. We also present a simple method to extract countermodels whensearch terminates.

As far as we know, ours is the first decision procedure which produces a set of countermodelsthat is comprehensive in the sense described above.

Limitations. There are existing undecidability results for modal logics with frame conditionssuch as symmetry and transitivity [3]. Consequently, we cannot hope for a method that provesdecidability for all logics MMI or all logics MM. Nonetheless, there are some classes of frameconditions which are not known to immediately result in undecidability, but to which our methoddoes not apply (either because they do not fit our definition of or because they do not haveSCRs):

Due to an interaction with intuitionistic implication, our method cannot handle possibilitymodalities in intuitionistic logic, even though it works fine with them in classical logic. Thisis discussed in Section4.

We do not know whether our method can handle label-producing conditions like densityor confluence. However, it can be easily extended to work with seriality, as discussed inSection3.6.

Further, like many other methods in this domain, an analysis of our proofs does not necessarilyproduce good upper bounds on the actual complexity of modal logics.

Organization. Since our constructive decision procedure and its correctness proof are almostidentical for intuitionistic and classical modal logics, in the main body of the paper, we present

our results only for the intuitionistic case. The case of classical modal logic is briefly discussed inSection5 and its details are deferred to Appendix B.

In Section2, we define the syntax, semantics and a standard, but non-terminating labeled se-quent calculus for the intuitionistic multi-modal logic MMI. We illustrate the known problem ofnon-termination due to looping in the sequent calculus in Section2.3. Section3 presents the maintechnical work. Starting from an informal introduction, we proceed to a description of the deci-sion procedure (Sections3.13.4), its comprehensiveness (Section3.5), its extension with seriality(Section3.6), and its instantiation to several known intuitionistic multi-modal logics (Section 3.7).We discuss limitations of our work and its relation to semantic filtrations in Section 4. Section5briefly lists modifications needed to adapt the method to classical logic. Related work is discussed

in Section6 and Section7concludes the paper.

2 MMI: Multi-Modal Intuitionistic Logic

We start by defining formally the family of intuitionistic multi-modal logics we consider in thispaper. The family is parametrized by a set of conditions on Kripke frames that must have a

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specific (standard) form, as described later in this section. The logic obtained by instantiating ourdefinition of syntax and semantics with a specific set is called MMI.

Syntax. Let I = {A , B , . . .} be a finite set of indices for modalities and p denote an atomicformula, drawn from a countable set of such formulas. Then, the syntax of formulas of the logic

MMI

is:

Formulas ,, ::= p | | | | | | A nec

Connectives,,, and have their usual meanings. Implication is interpreted intuition-istically. A nec is the necessitation modality of index A. This is commonly written A, but weprefer the more descriptive notation. Negation is not primitive, but may be defined in a standardway as = . We do not consider possibility modalities in the intuitionistic setting, sincethey are incompatible with our method (see Section 4).

2.1 Semantics

We provide Kripke (frame) semantics to formulas of MMI

and assume in our presentation thatthe reader has basic familiarity with this style of semantics. Unlike classical (multi-)modal logic,whose Kripke semantics are standard, several different Kripke semantics for intuitionistic modallogic have been proposed[1,23,26]. They differ in the number of relations used, how the relationsare related to each other and also how the modalities are interpreted. In what follows, we usesemantics that are closest in spirit to those of Wolter et al. [26]. This choice makes the technicaldevelopment easier.

Definition 2.1 (Kripke model). An intuitionistic model, Kripke model or, simply, model, M is atuple (W, , {NA}AI, h) where,

(W, ) is a preorder. Elements of W are called worlds and written x, y , z , w. Since is a

preorder, it satisfies the following conditions:x.(x x) (refl)

x,y,z.(((x y) (y z)) (x z)) (trans)

1 is also written .

Each NA is a binary relation on Wthat satisfies the condition ( NA) NA, i.e.,1

x,y,z.(((x y) (yNAz)) (xNAz)) (mon-N)

h assigns to each atom p the set of worlds h(p) W where p holds. We require h to bemonotone w.r.t. , i.e., ifx h(p) and x y then y h(p).

A model without the assignment, i.e., the tuple (W, , {NA}AI) is also called a frameand theconditions on relations above, e.g., (mon-N), are called frame conditions.

1Wolter et al. [26]require the stronger condition ( NA ) = NA, but our condition results in the same set ofvalid formulas.

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The frame conditions . In addition to the frame conditions (refl), (trans) and (mon-N), weallow a countable number of additional frame conditions as rules of the following form: x.((i=1,...,nxiRix

i) (xRx

)), where R1, . . . , Rn, R are from the set {NA | A I} {} and all variablesxi, x

i, x , x

are in x. A set of such additional frame conditions is denoted . MMI is the logicwhose valid formulas are exactly those that are valid (in the sense defined below) in frames that

satisfy al lconditions in .Definition 2.2 (Satisfaction). Given a model M= (W, , {NA}AI, h) and a world w W, wedefine the satisfaction relationM |=w : , read the world w satisfies formula in model M byinduction on as follows:

M |=w : p iffw h(p)

M |=w : (unconditionally)

M |=w : iffM |=w : and M |=w :

M |=w : iffM |=w : or M |=w :

M |=w : iff for every w such that w w and M |=w :, we have M |=w :.

M |=w : A nec iff for every w such that wNAw, we have M |=w :.

We say thatM |=w : if it is not the case thatM |=w : . In particular, for every M and everyw,M |=w : .

A formula is true in a model M, written M |=, if for every world w M, M |=w : . Aformula is valid in MMI, written |= , ifM |= for every model M satisfying all conditionsin .

Valid axioms. For the benefit of readers, we list below some valid axioms and admissible rules

that are common to all logics MMI

. If a rule/axiom is standard in literature, its common nameis mentioned to the extreme right. means ( ) ( ).

(All tautologies of intuitionistic propositional logic are valid in MMI )

|=

|=A nec(nec)

|= (A nec( )) ((A nec) (A nec)) (K)

|= ((A nec) (A nec)) (A nec( ))

The frame conditions can be used to force additional axioms in a standard way, which has

been explored in great detail in literature on correspondence theory[5]. For example,

The conditionx,y.((xNAy) (x y)) corresponds to the axiom (A nec), commonlycalled (unit) and of central importance in lax logic [7].

The condition x,y,z.(((xNAy) (yNAz)) (xNAz)) corresponds to the axiom (A nec) (A necA nec), commonly called (4).

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The following is a very important, fundamental property of Kripke models of all intuitionisticlogics, including ours. It is used to prove our sequent calculus (Section 2.2) sound with respect toour Kripke semantics.

Lemma 2.3 (Monotonicity). IfM |=w : andw w M, thenM |=w :.

Proof. By induction on .

2.2 Seq-MMI: A Labeled Sequent Calculus for MMI

As a first step towards building a constructively complete decision procedure for logics MMI , webuild a sound, complete, cut-free sequent calculus for MMI. Following the work of Negri [16],our calculus is presented in what is known as labeled style of calculi for modal logics, whichmeans that the calculus proves formulas labeled with symbolic worlds. A labeled formula containsa symbol x, y , z , w, u denoting a world and a formula , written together as x : . A sequent in ourcalculus has the form ;M; , where

- is a finite set of world symbols appearing in the rest of the sequent. World symbols are

also called labels.

- M is a finite multi-set of relations between labels in . Relations have the formsx y andxNAy.

- is a finite multi-set of labeled formulas.

- is a finite multi-set of labeled formulas.

The intuition is that if ; M; is valid, then every model with a world set containing atleast , satisfying all relations in Mand all labeled formulas in must satisfy at least one labeledformula in . This is formalized in the following definition.

Definition 2.4 (Sequent satisfaction and validity). A model M and a mapping from elementsof to worlds of M satisfy a (possibly non-provable) sequent ; M; , written M, |=(;M; ), if one of the following holds:

- There is an xRy Mwith R {} {NA | A I}such that (x)R (y) M.

- There is an x: such that M |=(x) :.

- There is an x: such that M |=(x) :.

A model M satisfies a sequent ;M; , written M |= (;M; ), if for everymapping , we have M, |= (;M; ). Finally, a sequent ;M; is valid, written|= (;M; ), if for every modelMwe have M |= (;M; ).

Rules of the sequent calculus. The sequent calculus for MMI is shown in Figure1. Followingstandard approach in labeled calculi, the rules for each connective mimic the (Kripke) semanticdefinition of the connective. For example, in the rule (R), to prove x : in the conclusion,we prove x : and x : in the premises. The rules (R) and (necR) introduce fresh worldsinto , consistent with the semantic definition (Definition 2.2). As illustrated in Section 2.3, it

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Axiom Rules

;M, x y; , x: p y : p, init

Logical Rules

;M; x: , R

;M; , x: L

;M; x: , x: , ;M; x: , x: ,

;M; x: , R

;M; , x: , x: , x:

;M; , x: L

;M; x: , x: , x: ,

;M; x: , R

;M; , x: , x: ;M; , x: , x:

;M; , x: L

, y;M, x y; , y: y: , x: ,

;M; x: , R

;M, x y; , x: y : , ;M, x y; , x: , y:

;M, x y; , x: L

, y;M, xNAy; y: , x: A nec ,

;M;

x: A nec ,

necR;M, xNAy; , x: A nec , y:

;M, xNAy; , x: A nec

necL

Frame Rules

, x;M, x x;

, x;M; refl

;M, x y, y z, x z ;

;M, x y, y z; trans

;M, x y,yNAz,xNAz;

;M, x y,yNAz;

mon-N

(x.((i(xiRix

i)) (xRx))) xiRix

i M ;M,xRx;

;M;

Figure 1: Seq-MMI: A labeled sequent calculus for MMI

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is primarily because of these rules that a backwards derivation in the sequent calculus may notterminate trivially; this motivates most of our technical development.

We say that (;M; ) if ;M; has a proof in the calculus. The sequentcalculus is both sound and complete with respect to the semantics.

Theorem 2.5 (Soundness). If(;M; ), then |= (;M; ).

Proof. Fix an M. It is easily proved by induction on the given derivation of ;M; thatfor every mapping ,M, |= (;M; ).

The converse of Theorem2.5,completeness, also holds. It can be proved using a Henkin-styleargument, but we do not need this result in the rest of our development so we do not present theproof here. For those logics MMI to which our method applies, completeness is also a consequenceof the completeness of our decision procedure, which we prove later. The following property iscritical to the design and correctness of our constructively complete decision procedure.

Theorem 2.6 (Weak subformula property). If a formula appears in any proof tree (possiblyinfinite) obtained by applying the rules of Figure1 backwards starting from a concluding sequent

;M; , then is a subformula of some formula in either or.

Proof. By induction on the distance (in the proof tree) of the occurrence of from the conclusion;M; .

2.3 Non-Termination in the Sequent Calculus

The main challenge in constructing a decision procedure based on the sequent calculus of Figure1is that backwards search in it can loop forever, due to unbounded creation of new worlds in therules (R) and (necR). We illustrate this through an example. Suppose = {x,y,z.(((xNAy)(yNAz)) (xNAz))}, i.e., the relation NA is transitive. Then the calculus admits the followinginfinite derivation. The omitted steps marked () are the same as the steps between the two (L)

rules with the difference that x1 is used in place ofx0 in all rules. To keep the derivation concise,we drop those formulas from M, and that are no longer needed after each backwards step.

...x0, x1, x2; x0NAx2; x0 : A nec((A nec ) )

x2 :

x0, x1, x2; x0NAx1, x1NAx2; x0 : A nec((A nec) ) x2 :

..

.()

..

.

. . .

x0, x1; x0NAx1, x1 x1; x0 : A nec ((A nec) ) x1 :

L

x0, x1; x0NAx1; x0 : A nec((A nec) ), x1 : (A nec ) x1 :

refl

x0, x1; x0NAx1; x0 : A nec((A nec) ) x1 :

necL

x0; ; x0 : A nec((A nec) ), x0 : (A nec ) x0 : A nec necR

. . .

x0; x0 x0; x0 : A nec((A nec) ), x0 : (A nec ) x0 :

L

x0; x0NAx0, x0 x0; x0 : A nec ((A nec) ) x0 :

necL

Note that the sequent at the top of the derivation is identical to the premise of the rule marked with a fresh world x2 replacing x1, so this derivation loops forever. The problem is caused by an

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interaction of the rules (L), (necR) and our frame condition that keeps creating new worlds.Although we do not illustrate them here, similar loops can also be created using the rule (R) inplace of (necR).

The question then is: How do we detect such loops during backwards proof search to obtain adecision procedure? In the rest of this paper we present a general technique that not only detects

such loops in a wide variety of logics, but also produces Kripke countermodels witnessing thenon-validity of the end-sequent when such loops are detected.

3 Constructively Complete Decision Procedure for MMI

In this section we present our general constructively complete (countermodel producing) decisionprocedure for several intuitionistic multi-modal logics of the form MMI. Since any application ofa rule other than (R) and (necR) is unnecessary in a backward proof search when the labeledformulas introduced in the premise(s) already exist in and , the use of all rules other than(R) and (necR) can be bounded easily. So, the main technical challenge is to be able to detectloops, such as the one illustrated in Section 2.3. Although the end-result of our technique, i.e.,

the decision procedure itself is quite straightforward, building up to it requires some non-standardmachinery, which we motivate here by presenting an informal outline of our method.To control the use of rules (R) and (necR) in backwards proof search, we generalize a technique

from existing work on tableau calculi for classical uni-modal logics [10]. The technique preventsloops by checking for containment of formulas that label a world in those labeling another. We startby observing that in any sequent ;M; obtained during backwards proof search startingfrom a single goal formula, all worlds in M lie on a rooted, directed tree, whose edges are relationsin M that were introduced by the rules (R) and (necR) in earlier steps of the search. We callthe reflexive-transitive closure of this tree . Next, we define a function Sfor(;M; ; , x) thatlists, approximately, all formulas labeled by x in and (the exact definition of Sfor depends on, and is one of our key technical contributions). This function satisfies a very important, criticalproperty, whose proof requires induction on : If there is a world y (y = x) such that y xand Sfor(;M; ; , x) Sfor(;M; ; , y), then it isuselessto apply any of the rules (R) and(necR) on any principal formula labeled by x in the sequent ;M; in backwards proofsearch. It only remains to show that this condition forces termination. This follows from the factthat Sfor(;M; ; , x) increases monotonically for each x in a backwards proof search and the factthat the number of possible values of Sfor is finite, which, in turn, is a consequence of the weaksubformula property (Theorem2.6).

We further show that if no rule applies backwards to a sequent ;M; (after imposingour termination checks), then we can obtain a countermodel to the sequent ;M; byadding an edge x y whenever Sfor(;M; ; , x) Sfor(;M; ; , y). This forms the basis ofour countermodel extraction. (As explained in Section6,this method of extracting countermodelsis motivated by Negris proof of completeness of labeled sequent calculi for uni-modal logic with

respect to their Kripke semantics [17]. In that proof, Negri shows how to extract countermodelsfrom failed branches of a non-terminating labeled sequent calculus. However, our construction ofthe countermodel is different.)

The definition of the function Sfor depends on the conditions that define the logic. In ourformal development, we define a suitable Sfor for every logic for which there exists what we calla suitable closure relation (SCR). Technically, a SCR is a family of relations on frames, which

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satisfies some stipulated properties. Our entire method applies to any logic MMI for whose a SCR exists. This begs the question of how general the existence of a SCR is. As we show inSection 3.7, several (multi-) modal logics with reflexivity, transitivity, and modality interactionconditions, including the multi-modal logics K, S4, T, and I [9,13]have SCRs. We further show inSection3.6that our method can be extended to any logic with a SCR plus the seriality condition on

its accessibility relations. This is, in an informal sense, the most we could hope for from a generaldecision procedure for multi-modal logics because there are negative results for decidability of multi-modal logics with other classes of frame conditions like symmetry (see Section 4). On the negativeside, our method does not directly generalize to handle possibility modalities in intuitionistic logic(Section 4). However, the method does work for possibility modalities in classical logic, wherepossibility can be defined as the DeMorgan dual of necessity (Section 5).

Our technical presentation consists of the following steps:

We define the term SCR for . Its existence is the only condition that must hold for ourmethod to apply to the logic MMI.

We define the function Sfor using SCRs. We also define a predicate on sequents, whose

elements (sequents) are called saturated histories. Roughly, a saturated history is a sequentwhich satisfies all the termination conditions listed above, i.e., applying any rule other than(R) and (necR) backwards on the sequent does not add any new labeled formulas, and theapplication of these two rules is blocked by the containment condition on Sfor. We prove, byconstruction, the key property of our entire method: If a sequent is a saturated history, thenit has a finite countermodel.

We define an intermediate sequent calculus Seq-MMICM with judgments ;M; CM

S. Here, Sis a (possibly empty) finite set of finite countermodels. The correctness propertyof this calculus is: If ;M; CM {} has a proof, then (;M;

) and if;M; CM Shas a proof for S={}, then every M Ssatisfies: M |= (;M;

). Backwards search in this calculus does not necessarily terminate because the calculus

allows every rule of, and, in addition, it has a new rule that produces a countermodelwhen ;M; ; is a saturated history. Consequently, in itself, the calculus is not a decisionprocedure. However, we find the calculus a useful intermediate step to prove many properties.

We observe that a specific strategy for proof search in Seq-MMI CM terminates. This strategyis presented as a sequent calculus Seq-MMIT, which is a countermodel producing decisionprocedure.

In the next four subsections, we present the technical details of each of these steps. In particular,the last of these subsections, Section3.4, describes our decision procedure. In Section3.5we stateand prove a comprehensiveness property of countermodels generated by our procedure.

3.1 Suitable Closure Relations (SCRs)

We start our technical presentation by defining suitable closure relations (SCRs) for frame condi-tions . Our constructive decision procedure applies to any logic MMI whose has a SCR. Call aframe M closed if it is closed under the conditions (refl), (trans), (mon-N) and. Given a frame M,let M denote its closure obtained by closing the frame under the conditions (refl), (trans), (mon-N)and , obtained as the least fixed point of the application of these rules. Informally, a SCR is

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a family of relations (R(A))AI that, given any closed frame M and any extension M of it with

additional edges of the form, characterizes all relations NA in M in terms of the relations in Mand the difference M M.

Definition 3.1(Suitable closure relation (SCR)). A family of binary relations (R(A))AI is calleda suitable closure relation or SCR for if the following hold:

0. Each R(A) is definable in first-order logic in terms of the relations {NA | A I}.

1. For a finite frameMand x, y M, it can be decided whether x(R(A))y or not.

2. ((R(A)) NA) NA can be derived from the frame conditions (refl), (trans), (mon-N) and.

3. For anyclosed frame M and any C {x y | x, y M}, ifxNAy M C, then x((R(A) C) NA )y, where all relations R(A) and NA on the right are in M.

4. For any closed frame M and anyC {x y | x, y M}, ifx y M C, thenx( C)y,where all relations on the right are in M.

Observe that condition (4) depends on , not on the choice of R(A), but we include it here foruniformity.

SCRs for many different are listed in Section3.7,but we describe one in the following examplefor illustration.

Example 3.2 (SCR for transitivity). Let trans(A) be the frame condition x,y,z.(((xNAy) (yNAz)) (xNAz)) and let = {trans(A)| A I}. Then, the relation R(A) =NA is a SCRfor . To prove this, we verify each of the conditions (0)(4) in the definition of SCR. Conditions(0) and (1) are trivially true. (2) is equivalent to ((NA )

NA) NA, which follows from theframe conditions (mon-N) and . To prove (3), suppose that xNAy M C. Then, because the

only way to derive a relation NAis to use either (mon-N) or trans(A), it follows that in M C, wehave x((NA )

NA)y. So, we also have x((NA C) NA)y, where all and NA relations

are in M, i.e.,x((R(A)C) NA)y. Finally, since is reflexive, we have: x((R(A)C) NA )y,

as required. The proof of (4) is similar to that of (3).

3.2 Saturated Histories

Our method applies only to those logics MMI whose has a SCR, so in the sequel we fix a setof frame conditions and assume there is a SCR (R(A))AI for this . Although we present thetechnical material generically with respect to an abstract and SCR, we strongly advise the readerto choose a specific and its SCR (for instance, those from Example 3.2), and instantiate ourdefinitions and theorems on them.

A history is a tuple ;M; ; (equivalently, a sequent ;M; ) such that all labels inM, and occur in . Let T() and F() be two uninterpreted unary relations. Informally, weread T() as should be true and F() as should be false. Given a history ;M;; andx , the signed formulas ofx, written Sfor(;M; ; , x) are defined as follows:

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Sfor(;M; ; , x) ={T() | x: }{F() | x: }

{T(A nec) | y. y(R(A))x Mand y: A nec }{T( )| y. y x M and y: }

{T(p) | y. y x M and y: p }

When ,M, , are clear from context, we abbreviate Sfor(; M; ; , x) to Sfor(x). We saythat x y iff Sfor(x) Sfor(y).

We call a frame M tree-likeif it can be derived from a finite tree of the relations and NAand(possibly partial) closure by frame rules. This tree is called the underlying tree ofM and we saythat x y (in M) iff there is a directed path from x to y in the tree underlying M.

The key definition in our method is that of a saturated history. Intuitively, this definitioncharacterizes those histories ;M; ; for which we can directly define a countermodel for thesequent ;M; . (The definition of this countermodel is given immediately after the definitionof a saturated history.)2

Definition 3.3 (Saturated history). A history ;M; ; is called saturated if the following hold:

1. M is tree-like and saturated with respect to the conditions (refl), (trans), (mon-N) and. (Inparticular, because Mis tree-like, it has a relation defined on it.)

2. Ifx: p , then there is no y such that x y Mand y: p .

3. There is no x such that x: .

4. There is no x such that x: .

5. Ifx: , then x: and x: .

6. Ifx: , then either x: or x: .7. Ifx: , then either x: or x: .

8. Ifx: , then x: and x: .

9. Ifx: and x y M, then either y: or y: .

10. Ifx: , then either:

(a) There is ay such that x y M, y: and y: or

(b) There is ay such that y=x, y x and x y.

11. Ifx: A nec and xNA

y M, then y: .

12. Ifx: A nec , then either:

(a) There is ay such that xNAy M and y: or


2Saturated histories are labeled generalizations of Hintikka sets.

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Definition 3.4 (Countermodel of a saturated history). For a saturated history ; M;;, thecountermodelof the history, CM(;M; ; ) is defined as follows:

- The worlds of CM(;M; ; ) are exactly those in .

- The relations of CM(;M; ; ) areM C, where C={x y | x y}.

- h(p) ={x | y. (y x M) (y: p )}.

It is not obvious that CM(;M; ; ) is a model. It does satisfy all frame conditions. However,we must also show monotonicity: If x y CM(;M;;) and x h(p), then y h(p). Thefollowing lemma states that this is the case.

Lemma 3.5. If;M; ; is a saturated history, then CM(;M;;)has a monotonic valuationh, i.e., x h(p) andx y CM(;M;;)implyy h(p).

Proof. LetCbe the set {x y | x y}. Suppose thatx y CM(;M; ; ), i.e.,x y M Cand x h(p). From the latter, there is a z such that z x Mand z: p . From the definitionof SCR, clause (4) it follows that x( C)y, where all the relations are in M. Hence, we have

a chain x = x0( C)x1 . . . ( C)xn= y where all relations are in M. We induct onito showthat T(p) Sfor(xi). The result then follows immediately from T(p) Sfor(xn) = Sfor(y). SeeAppendixA,LemmaA.1 for details.

The next Lemma states the central property of our method. In particular, the Lemma imme-diately implies that if ;M; ; is a saturated history, then CM(;M; ; ) is a countermodel tothe sequent ;M; .

Lemma 3.6. The following hold for any saturated history;M; ; :

A. IfT() Sfor(;M; ; , x), then CM(;M;;)|=x :

B. IfF() Sfor(;M; ; , x), then CM(;M;;)|=x : Proof. We prove both properties simultaneously by lexicographic induction, first on, and then onthe partial (tree-like) order ofM. (Note that we cannot induct on either M or the relation inCM(;M; ; ), because both of these may potentially be cyclic.) Since the context ;M; ; isfixed here, we abbreviate Sfor(;M; ; , x) to Sfor(x). We show only one interesting case here;for the remaining cases, see AppendixA, LemmaA.2.

Case. Proof of (A), = A nec . We are given that T(A nec ) Sfor(x). We need toshow that CM(;M;;) |= x : A nec , i.e., for any y such that xNAy in the model, we haveCM(;M;;) |= y : . Pick any y such that xNAy in the model. Because of the definition ofSCR, clause (3), we have x((R(A) C) NA )y, where the relations R(A) andNA are in M. So

there are x0, . . . , xn, y such that x = x0(R(A) C)x1 . . . (R(A) C)xnNAy y. We now prove,by induction on i, that T(A nec) Sfor(xi) for each i.

For i= 0, x0 = x and we are given that T(A nec) Sfor(x).

For the inductive case, assume that T(A nec) Sfor(xi) for some i. We show that T(A nec) Sfor(xi+1) by case analyzing the relation xi(R(A) C)xi+1.

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xi(R(A))xi+1 M: By the i.h., T(A nec ) Sfor(xi) so there is some z such thatz(R(A))xi M and z : A nec . Clearly, we have z(R(A))

xi+1 M, so T(A nec) Sfor(xi+1).

(xi, xi+1) C: Because of the definition of C, Sfor(xi+1) Sfor(xi). Thus, T(A nec) Sfor(xi) immediately implies T(A nec) Sfor(xi+1).

Since we just proved that T(A nec ) Sfor(xi), it follows, in particular, that T(A nec ) Sfor(xn). Consequently, there is somez

such that z(R(A))xn M and z : A nec . Then,

we also have (within M) that: z(R(A))xnNAy. So, by clause (2) of the definition of SCR, zNAy

M. Hence, by clause (11) of the definition of saturated history, we must havey : . Therefore,T() Sfor(y) and by the i.h., CM(; M;;) |= y : . Since y y CM(;M;;), byLemma2.3, CM(;M;;)|=y : .

Corollary 3.7(Existence of countermodel). If;M; ; is a saturated history, then CM(;M;;)|=(;M; ).

Proof. Lemma3.6immediately implies that CM(;M;;), |= (;M; ), where :

is the identity substitution.

3.3 Seq-MMICM

: Countermodels for MMI

Having defined a saturated history, i.e., a sequent for which a countermodel exists (Corollary 3.7), wenow define a sequent calculus Seq-MMICM, written

CM, which uses this fact to emit countermodels

from unprovable sequents. Although this calculus is not a decision procedure, we find it a usefulstep in proving several results, in particular, the results of Section 3.5.

Sequents of Seq-MMICM have the form ;M; CM S, where S is a finite set of finite

models. We write (; M; CM S) if ;M; CM Shas a proof. The meaning of

;M; CM Sdepends on S. If (;M; CM {}), then (;M;

) and if(; M; CM S) withS={}, then every model M Sis a countermodel to ;M;

in the sense of (the converse of) Definition 2.4.The rules of the sequent calculus Seq-MMICM are shown in Figure 2. First, every rule in

the ordinary sequent calculus (Figure 1) is modified to have in the conclusion the union of the(counter)models in the premises. This is sound because the rules of the sequent calculus areinvertible (i.e., the conclusion of each rule holds iff the premises hold). Second, there is a new rule(CM) that produces the countermodel CM(;M; ;) when ;M; ; is a saturated history.

We emphasize again that this calculus is not necessarily a decision procedure because it includesall rules of and hence admits all of the latters infinite backwards derivations as well.

Theorem 3.8 (Soundness 1). If(;M; CM {}), then(; M; ).

Proof. By induction on the given derivation of ; M; CM {}. Note that the case of

rule (CM) does not apply because the set of countermodels in it is non-empty. The proof isstraightforward because the rules ofCM mimic those of

.

Theorem 3.9 (Soundness 2). If (;M; CM S), then for every modelM S, M |=(;M; ).

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Axiom Rules

;M; ; is a saturated history

;M; CM

{CM(;M;;)}CM

;M, x y; , x: p CM

y : p, {}init

Logical Rules

;M; CM

x : , {}R

;M; , x: CM

{}L

;M; CM

x : , x: , S1 ;M; CM

x : , x: , S2

;M; CM

x : , S1, S2R

;M; , x: , x: , x: CM

S

;M; , x:

CM S

L;M;

CMx : , x: , x: , S

;M;

CMx : , S

R

;M; , x: , x: CM

S1 ;M; , x: , x: CM

S2

;M; , x: CM

S1, S2L

, y;M, x y; , y: CM

y : , x: , S

;M; CM

x : , SR

;M, x y; , x: CM

y : , S1 ;M, x y; , x: , y: CM

S2

;M, x y; , x: CM

S1, S2L

, y;M, xNAy; CM

y : , x: A nec , S

;M; CMx : A nec , S

necR;M, xNAy; , x: A nec , y:

CM

S

;M, xNAy; , x: A nec CM S

necL

Frame Rules

, x;M, x x; CM

S

, x;M; CM

Srefl

;M, x y, y z, x z ; CM

S

;M, x y, y z ; CM

Strans

;M, x y,yNAz,xNAz; CM

S

;M, x y,yNAz; CM

Smon-N

(x.((i(xiRix

i)) (xRx))) xiRix

i M ;M,xRx;

CM S

;M; CM

S

Figure 2: Seq-MMICM: Countermodel producing sequent calculus for MMI

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Proof. By induction on the given derivation of ;M; CM S and case analysis of its lastrule. The rules (init), (L), and (R) are vacuous because they have empty S. For all other rules,except (CM), we simply observe that contexts in all major premises are a superset of correspondingcontexts in the conclusion and hence we can trivially conclude by induction on one of the premises.The case of rule (CM) is shown below:

Case. ;M; ; is a saturated history;M; CM {CM(;M;;)}

CM

HereM = CM(;M; ; ). So, the result follows by Corollary3.7.

3.4 Seq-MMIT

: Termination and Countermodel Extraction for MMI

Next, we describe a particular backwards proof search strategy in Seq-MMICM that always ter-minates without losing completeness, thus obtaining a countermodel producing decision proce-dure for MMI. This strategy is described as a calculus Seq-MMIT, with sequents of the form;M; T S. Operationally, the rules of the calculus can be interpreted backwards asa decision procedure with inputs , M, , and and output S. For a given , M, , and ,

(;M;

) is provable iffS= {}, else every model in Sis a countermodel to the sequent.The rules of the calculus Seq-MMITare shown in Figure3. Each rule in the calculus correspondsto a rule of the same name in Seq-MMICM (Figure 2). The only significant difference betweenthe two calculi is that the premise of the rule (CM) in Seq-MMI CM requires that ;M;; bea saturated history, but the rule (CM) applies in Seq-MMIT only when no other rule applies.To ensure that no other rule applies implies that ; M; ; is a saturated history, we spreadthe negations of the conditions (1) and (5)(12) from the definition of saturated history to rulesother than (CM) as pre-conditions, called applicability conditions. Conditions (2), (3) and (4)obviously hold when the rules (init), (R) and (L) do not apply, respectively. Hence, when norule other than (CM) applies, all 12 conditions of the definition of saturated history must hold, so;M; ; must be a saturated history. The conditions are spread to the obvious rules; for example,the negation of condition (5) is applied to the rule (R). In Figure 3, applicability conditions

are highlighted using boxes . It only remains to show that the calculus with these applicabilityconditions does not admit infinite backwards derivations. This follows from a counting argument,as in the proof of Theorem 3.12.

Lemma 3.10 (Correctness of CM). Let, M, and be such thatM is tree-like and no ruleexcept (CM) applies backwards to ;M; T . . .. Then, ;M; ; is a saturated history.

Proof. We verify all conditions in the definition of a saturated history. Each condition correspondsto the negation of premises of one of the rules of Figure 3.

Lemma 3.11(Tree-like M). LetM be tree-like. Then, theM in any sequent;M; T . . .

appearing in a backwards search starting from;M; T . . . is tree-like.

Proof. By backwards analysis of each rule observing that the M in the premises of each rule istree-like if that in the conclusion is.

Note that the underlying tree ofM in any sequent of a backward proof search starting froma single formula consists of exactly those edges that are introduced in one of the rules (R) and(necR).

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Axiom Rules

No other rule applies

;M; T

{CM(;M; ;)}CM

;M, x y; , x: p T

y : p, {}init

Logical Rules

;M; T

x : , {}R

;M; , x: T

{}L

x: and x : ;M; T

x : , x: , S1 ;M; T

x : , x: , S2

;M; T

x : , S1, S2R

x: or x : ;M; , x: , x: , x: T

S

;M; , x: T

SL

x: or x : ;M; T

x : , x: , x: , S

;M; T

x : , SR

x: and x : ;M; , x: , x: T

S1 ;M; , x: , x: T

S2

;M; , x: T

S1, S2L

y .(x y M) (y : or y : )

y .(y x) (x= y or x y) , y;M, x y; , y: T

y : , x: , S

;M; T

x : , SR

y: and y : ;M, x y; , x: T

y : , S1 ;M, x y; , x: , y : T

S2

;M, x y; , x: T

S1, S2L

y .(xNAy M) y :

y .(y x) (x= y or x y) , y;M, xNAy; T

y : , x: A nec, S

;M; T

x : A nec, SnecR

y : ;M, xNAy; , x: A nec, y: T

S

;M, xNAy; , x: A nec T

SnecL

Frame Rules

x x M , x;M, x x; T

S

, x;M; T

Srefl

x z M ;M, x y, y z , x z ; T

S

;M, x y, y z ; T

Strans

xNAz M ;M, x y,yNAz,xNAz; T

S

;M, x y,yNAz; T

Smon-N

(x.((i (xiRix

i)) (xRx

))) xiRix

i M xRx

M ;M, xRx

; T S

;M; T

S

Figure 3: Seq-MMIT: Terminating, countermodel producing sequent calculus for MMI. Appli-

cability conditions are written in boxes . Wherever mentioned, the relation is the equivalencerelation of the contexts ;M; ; in the conclusion of the rule. Similarly, is the order of theunderlying tree ofM.

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Theorem 3.12 (Termination). The following hold:

1. Any backwards search in Seq-MMITstarting from a sequent;M; T withM tree-like

terminates.

2. For any;M; ; withMtree-like, there is anSsuch that(;M; T S) and such

anScan be finitely computed.

Proof. Proof of (1): Suppose, for the sake of contradiction, that there is an infinite backward proofstarting from ;M; T . . .. Since the proof is finitely branching (every rule has a boundednumber of premises), it must have an infinite path. Observe that , are monotonic backwards, sothe applicability conditions in the rules prevent application of the same rule on the same principallabeled formulamore than once in any branch. Since there are only a finite number of formulas thatcan appear in any search (weak subformula property, Theorem2.6), it follows that in the infinitepath there must be an infinite number of labels. Let Tbe the underlying tree of this entire path(i.e., the underlying tree of the union ofM for each sequent on this path). Since the tree is finitelybranching (because we cannot apply rules (R) and (necR) to the same label infinitely often), itmust have an infinite path. Let this path be x0x1 . . .. Let Si be the value of Sfor(xi) when

either of the rules (R) and (necR) is applied to createxi+1. Note that for i < j, SiSj , becauseifSi Sj, then at the time that xj+1 is created, Sfor(xi) Si Sj = Sfor(xj), so the applicationof the rules (R) and (necR) onxj would be blocked, so xj+1could not have been created. Hence,for i < j, Si Sj. Call this fact (A). (The reader may note that the deduction Sfor(xi) Si twosentences ago relies on the fact that Sfor(x) increases monotonically as we move backwards in aderivation.)

If is the set of all subformulas of the original sequent we start from, then by Theorem 2.6,each Si {T() | } {F() | }. Note that the right hand side is a finite set, so itssubsets form a finite partial order under set inclusion. Call this partial order P. Since P is finite,it has a finite number of chains and since the sequence S1, S2, . . . is infinite, at least one infinitesubsequenceR ofS1, S2, . . .must contain elements from only a single chain in P. Consider any two

elementsSi, Sj P with i < j. SinceP is a chain, we must have either Si Sj or Si Sj. Theformer is ruled out fact (A). So Si Sj. Hence, we have S1 S2 S3 . . ., so the chain P containsan infinite ascending sequence, which is a contradiction because P is finite.

Proof of (2): Follows immediately from (1), Lemma3.11, and the observation that all applicabilityconditions are finitely computable. The latter follows from condition (1) of the definition of SCRs.

Note that Theorem3.12(2) does not stipulate that the Sbe unique. Indeed, depending on theorder in which the rules of the calculus T are applied to a given sequent, S may be different.However, the fact that at least one such Sexists and can be computed is enough to get decidabilityfor MMI.

Lemma 3.13 (Simulation). IfM is tree-like and (;M; T S), then (;M; CM

S).

Proof. By induction on the given derivation of ; M; T S. The case of rule (CM) followsfrom Lemma3.10. The rest of the cases are immediate from the i.h. The only fact to take care ofis that the tree-like property holds for each i.h. application. This follows from Lemma3.11.

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Theorem 3.14 (Decidability). For a tree-likeM, suppose thatS is such that(;M; T S) (such anSmust exist and can be computed using Theorem3.12). Then:

1. IfS={}, then|= (;M; ).

2. IfS={}, then every modelM inSis a countermodel to the sequent, i.e., M |= (;M;

).

Proof. By Lemma3.13, we have that (;M; CM S). Now, (1) follows from Theorems3.8and2.5and (2) follows from Theorem3.9.

Corollary 3.15 (Decidability and finite model property). If a SCR exists for , then MMI isdecidable, has the finite model property and has a constructive decision procedure.

Proof. Immediate from Theorem3.14.

3.5 Comprehensiveness of Seq-MMICM

and Seq-MMIT

Countermodels

Countermodels generated by Seq-MMICM (and Seq-MMIT) have an interesting property: If

(;M; CM S), then (;M; CM x : p, {}) if and only ifM S. M |= x : p.Thus, if we can produce a set of countermodelsSby running without an actual goal (likex : p), thenthe set of atoms that are actually true are exactly those that are in the intersection of the valuationof all models in the set S. Further, because the result applies to derivations in Seq-MMICM, italso applies to derivations in Seq-MMIT due to Lemma3.13and the latter can be used to actuallyproduce the set S. We call this result comprehensiveness and prove it below.

Theorem 3.16 (Comprehensiveness). Suppose(;M; CM S). Then(;M; CM

x: p, {}) iffM S. M |=x : p.

Proof. See AppendixA, TheoremA.5.

Corollary 3.17 (Comprehensiveness in Seq-MMIT). Suppose (;M;

T S). Then,

(; M; x: p, ) iffM S. M |=x : p.

Proof. From (; M; T S) we derive (; M; CM S) using Lemma3.13. The

result then follows from Theorem3.16.

3.6 Adding Seriality

In this section we show that ifhas a SCR, then our method applies not only to the logic MMI

(Corollary3.15), but also to the logic which, in addition, forces serialitywith respect to some ofits relations NA. Seriality for index A is the condition x.y.(xNAy). This corresponds to theaxiom(A nec), also called D in literature[5]. Note that seriality does not fit our definition of

because frame conditions in cannot contain existentials, so it cannot be handled in the methoddescribed so far. Consequently, we must modify our method slightly to include seriality as a framecondition. The only new challenge is to control creation of worlds due to the seriality conditionduring backwards search; for this we use an approach similar to that for controlling the use of rules(R) and (necR). Proofs do not change significantly.

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Suppose we wish to make relations NA forA J Iserial. We first add the following rule toour sequent calculus Seq-MMI (Figure1) for every A J:

, x , y;M, xNAy;

, x;M; D

Next, we change clause (1) of the definition of saturated history notto require closure under thisnew frame condition, which would cause infinite models, but instead new conditions based on :

(1) M is tree-like and saturated with respect to the conditions (refl), (trans), (mon-N)and . In addition, at least one of the following must hold for each x and eachindex A J:

(a) There is ay such that xNAy M, or


With this new clause (1), we can show by induction on that CM(;M; ; ) is closed underseriality for A J, hence it is a model of our (modified) logic. Next, we add the following rule

for every A J to the terminating calculus Seq-MMIT and a corresponding rule without theapplicability conditions to Seq-MMICM.

y . (xNAy M) y .(y x) (y= x or x y) , x , y;M, xNAy; T

S

, x;M; T

SD

With these changes, our entire development works with only minor changes to the proofs (interest-ingly, the proof of Lemma3.6does not change).

Theorem 3.18 (Constructive decidability with seriality). Let D contain seriality conditions forsome set of indices and let the frame conditions have a SCR. Then the logic MMI,D is construc-tively decidable by our method.

3.7 Constructive Decidability for Common Intuitionistic Logics

In this section, we list some common sets of frame conditions with their SCRs, thus showing thatthe intuitionistic logics corresponding to each of them is constructively decidable by our method.Unfortunately, SCRs are not modular: We cannot combine the SCRs for frame conditions 1 and2 to get a SCR for a modal logic 1 2. As a result, we must explicitly construct a SCR forevery modal logic of interest.

Figure4lists some common intuitionistic logics, their frame conditions (), the correspondingaxioms they admit and the corresponding SCRs. In cases where the name of the logic is notcommon, we have cited the source of the logic. We note two things: (1) This list is not exhaustive,

but merely representative, and (2) Our method also applies to any of these logics combined withseriality from Section3.6due to Theorem3.18.

Theorem 3.19 (Decidability of Common Logics). The intuitionistic logics shown in Figure4 havethe SCRs also shown in that figure. Consequently, all these logics (and their combination with theseriality condition from Section3.6) are constructively decidable by our method.

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Logic Frame conditions Additional Axioms SCR

K {} R(A) = ()T A,x. xNAx (A nec ) R(A) = ()K4 A,x,y,z.((xNAy) (yNAz)) (xNAz) (A nec ) (A necA nec ) R(A) = (NA )S4 Conditions of K4 and T Axioms of K4 and T R(A) = (NA )I[9,13]

A,B,x,y,z.((xNBy) (yNAz)) (xNAz) (A nec ) (B necA nec) R(A) = ((BINB) )

unit[7] A,x,y. (xNAy) (x y) (A nec) R(A) = ()

x,y.(xNAy) (xNBy) (B nec) (A nec ) R(A) = ()

Figure 4: SCRs for some multi-modal intuitionistic logics. All these logics are constructivelydecidable by our method.

4 Discussion

This section discusses some loose ends: The connection between our technique and the techniqueof semantic filtrations, and some broad limitations of our work.

Relation to semantic filtration. Semantic filtrations [5] are a technique for establishing thefinite model property of modal logics. The key idea is to show the existence of an accessibilityrelation (called a filtration) on the finite model obtained by collapsing worlds of any model thatsatisfy the same set of formulas. The relation must satisfy some specific conditions. Often the ac-cessibility relation constructed has a definition similar to our SCR relations R(A) and, superficially,the two techniques may look similar. However, a careful examination reveals differences. Primarily,filtrations are semantic techniques that manipulate Kripke models whereas our method is purelysyntactic and SCRs only work with sets of formulas generated during a specific backwards search.A consequence of this difference is that, for any of the logics considered in this paper, we have notbeen able to find a suitable filtration on the obvious model whose worlds are equivalence classes of . In particular, it seems extremely difficult to satisfy what is known as the back condition,

which is required of a filtration. We also note that there is no standard definition of filtrations forintuitionistic modal logics. We know of only one work in this domain, and that work is also limitedto the uni-modal case[15] (the author of the paper notes in the conclusion that generalizing to themulti-modal case is not trivial).

Complexity bounds. Even though our method proves the finite model property and provides aconstructive decision procedure for a wide-variety of multi-modal logics, even in the intuitionisticcase, it does not provide tight upper bounds on the complexity of the logics. For example, based onthe result of [15], we expect that the intuitionistic modal logic K is decidable in doubly exponentialtime, but an analysis of Theorem3.12yields a bound that is at best quadruply exponential. Thisinability to produce accurate complexity bounds may be partly attributed to the methods general-

ity (it also applies to multi-modal logics) and partly to the fact that it constructs a comprehensiveset of countermodels (Section3.5). We do not know of any existing complexity bound for producingsuch comprehensive sets of countermodels.

Intuitionistic possibility modalities. Our method does not work when intuitionistic possibilitymodalities A pos (commonly written A) are included in the logic. The reason for this is subtle,

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but primarily stems from the fact that the binary relation PAused to define the possibility modalitymust satisfy ( PA) PA (as compared to ( NA) NA for necessitation) [26]. Consequently,in the definition of saturated history, the clause for A pos analogous to clause (12) reads:

Ifx: A pos , then either:

(a) There is ay such that xPAy M and y: or(b) There is ay such that y=x, y x and y x.

Observe that in part (b) above, we have the condition y x instead of x y in clause (12b).This difference is enough to break the termination property (Theorem3.12); in particular, the factcalled (A) in its proof cannot be established with possibility modalities.

In classical logic, where the relation is absent, possibility modalities are easily handled byour method. In fact, they need no special treatment as they are just the DeMorgan duals ofnecessitation (AppendixB).

Symmetry condition. To the best of our knowledge, there is no SCR for the symmetry condition

(x,y.(xNAy) (yNAx)), the cornerstone of the modal logic S5. Consequently, our method cannotbe used to prove decidability for this logic, in either the uni-modal or multi-modal setting. Thisis not surprising because Baldoni has shown that, in general, multi-modal logics with symmetryconditions are undecidable [3]. So, a general method like ours is unlikely to be able to handle theseconditions. Nonetheless it is dissatisfying that our method is unable to handle even uni-modal S5,which is known to be decidable.

Other label producing conditions. We showed in Section3.6that our method is compatiblewith the seriality frame condition. A question is whether it is also compatible with other frameconditions that produce labels, like seriality does. For example, can it be extended to handle density:x,y.((xNAy) z.((xNAz) (zNAy)))? Although we have not investigated this question in

detail, there seems to be no obvious method to extend our technique to include such conditionsin general. Nonetheless, it may be possible to combine our method with work on decidability forlayered modal logics [11], that are derived from a specific subclass of label-producing conditions.We leave this combination to future work.

5 Classical Logic

Although we developed our decision procedure with intuitionistic modal logic in mind, it can bemodified to apply to classical multi-modal logics as well. The overall approach of using SCRs andthe structure of the calculi remains the same. However, because Kripke frames in classical logicdo not require the preorder , we must change the definition of CM(; M; ; ) (Definition3.4)

that relies on . This is not difficult: Instead of adding x y when x

y, we add the relationxNAz when yNAz and x y. AppendixBdescribes in detail our method as it applies to classicallogics, together will all relevant proofs. In classical logic, the possibility modality of index A canbe defined as the DeMorgan dual of the necessitation modality of the same index, so on classicallogic, our method applies to possibility modalities as well.

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6 Related Work

The applicability conditions of rules (R) and (necR) in Figure 3, based on the relation , areinspired by the work of Gasquet et al. [10] in which tableaux-based decision procedures are given forclassical uni-modal logics with the following frame conditions: Transitivity, reflexivity, symmetry,Euclideanness, seriality and confluence. Our method is based on labeled sequent calculi and itapplies to both classical and intuitionistic modal logics with any number of modalities. As aconsequence, we had to develop new proof techniques to establish our results, particularly in theintuitionistic setting.

Our method of extracting countermodels is inspired by Negris proof of completeness of labeledsequent calculi for uni-modal logic with respect to their Kripke semantics [ 17][18, Chapter 11].In that proof, it is shown how to extract (possibly infinite) countermodels from non-terminatingbranches of a failed proof search taking the union of all M occurring along the branch. Here,instead, countermodels are built in the context of a decision procedure and finitecountermodelsare built by adding additional edges based on the relation and saturating with respect to theframe conditions.

Boretti and Negri [6] develop a countermodel producing decision procedure similar to ours

for a Priorean linear time fixed point calculus (a variant of linear time temporal logic, LTL),which also includes two rules like seriality. They also use a notion of saturation and constructcountermodels from histories. The main difference between this work and [6] is that this workhandles general frame conditions and, additionally, intuitionistic connectives. Boretti and Negrialso discuss previous tableaux-style approaches to the generation of countermodels for LTL, suchas [22].

Countermodel producing sequent calculi, also known in the literature as refutation calculi,have been given for intuitionistic logic, bi-intuitionistic logic, and provability logics [12,14,19]andin a way closer to the present papers approach in[20]. One of the peculiarities of our method inrelation to previous work is that the countermodel construction is made part of the calculus itself.

Gasquet and Said [11]introduce a technique called dynamic filtration to establish complexity

bounds for the satisfiability problem of classical layered modal logics (LMLs), i.e., logics char-acterized by semantic properties only stating the existence of possible worlds that are in somesense further than the other. Typically, such logics include confluence-like conditions, but theydo not include transitivity-like conditions. Our work provides constructive decision procedures fora different and disjoint class of logics to which the techniques in [11] do not apply. In fact, withthe exception of seriality, none of the frame conditions considered in this paper fall in the classof LMLs. Moreover, because LMLs cannot be applied with transitivity conditions it is not clearwhether the techniques in[11] are suitable in the intuitionistic setting.

Simpson [23]presents decision procedures based on labeled sequent calculi for the intuitionisticuni-modal logics K, D, T and B together with S5. He leaves open the decidability of intuitionisticS4, K4 and KD4. Our method shows that the necessitation-only fragments of all three logics are

decidable, not only in the uni-modal case, but also in the multi-modal case and, further, that thelogics have constructive decision procedures.Schmidt and Tishkovsky [21] present a general method for synthesizing sound and complete

tableaux calculi given a semantic filtration [5] for the underlying classical modal logic. Althoughsemantic filtrations are a powerful and general technique, their definition is not clear for manyintuitionistic and multi-modal logics, so our method handles several logics that cannot be handled bySchmidt and Tishkovsky. To the best of our knowledge, the only work on filtrations for intuitionistic

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logics is limited to the uni-modal case [15] (the author of the paper notes in the conclusion thatgeneralizing to the multi-modal case is not trivial). Filtration-based methods are also technicallydifferent from our syntactic approach. Whereas filtrations manipulate Kripke models, our methodis purely syntactic and SCRs only work with sets of formulas generated during a specific backwardssearch. A consequence of this difference is that, for any of the logics considered in this paper, we

have not been able to find a suitable filtration on the obvious model whose worlds are equivalenceclasses of . In particular, it seems extremely difficult to satisfy the back condition of afiltration.

Alechina and Shakatov [2] present a general technique to prove decidability of intuitionistic(multi)-modal logics by embedding the relational definition of the semantics into Monadic SecondOrder Logic (MSO). As noted by the authors themselves and unlike our method, this approachdoes not give good decision procedures since it proceeds by reduction to satisfiability of formulasin SkS (monadic second-order theory of trees with constant branching factor k), which has non-elementary complexity. Moreover, the method applies to a logic only if its frame conditions can beexpressed as acyclic closure conditions in MSO; this makes the method inapplicable to logics withframe conditions mentioning more than one modal relation, e.g., the logic (I) from Figure 4.

Negri [16]and Vigano[24] provide sound, complete and terminating labeled sequent calculi for

uni-modal classical logics K, T, K4 and S4. Out method extends these results for a wider class ofmodal logics including axiom D, multiple modalities and intuitionistic logics.

Gore and Nguyen [13] present non-labeled tableaux calculi for seven types of classical multi-modal logics to reason about epistemic states of agents in distributed systems. The introducedtableaux require formulas of the logic to be first translated into a clausal form. We observe thatone of the axioms presented in [13] is positive introspection for beliefs and corresponds to axiom(I) in Figure4. To the best of our knowledge, [13]is the only work to provide decision proceduresfor logics including axiom (I).

7 Conclusion and Future Work

We have presented a sequent calculus-based, constructive decision procedure for several multi-modal logics, both intuitionistic and classical. Besides a novel construction of countermodels and anovel termination condition, we show, for the first time, that several standard intuitionistic multi-modal logics without diamonds, as well as several logics with interactions between modalities, aredecidable. We also show that our procedure has an interesting, novel comprehensiveness property.

Although we believe that our work is a significant step in using sequent calculi (especially,labeled sequent calculi) in constructive decision procedures for modal logics, a lot still needs to bedone. First, we believe that our method can be extended to label-producing frame conditions moregeneral than seriality. In particular, it should be possible to extend the technique of Section3.6to all layered modal logics of Gasquet and Said [11], which we discussed in Section 6. Second, wewould like to either extend our method, or find a new one that can handle possibility modalities in

intuitionistic logic.Another direction of research is to exploit the embedding of intuitionistic modal logics into

classical bi-modal logics to establish decidability results for the former, a direction pursued in [26]but with semantic rather than syntactic methods.

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References

[1] Natasha Alechina, Michael Mendler, Valeria de Paiva, and Eike Ritter. Categorical and kripkesemantics for constructive S4 modal logic. In Proceedings of the Annual Conference of theEuropean Association for Computer Science Logic (CSL), pages 292307, 2001.

[2] Natasha Alechina and Dmitry Shkatov. A general method for proving decidability of intu-itionistic modal logics. Journal of Applied Logic, 4(3):219230, 2006.

[3] Matteo Baldoni. Normal multimodal logics with interaction axioms. In Labelled Deduction,pages 3353. Kluwer Academic Publishers, 2000.

[4] Matteo Baldoni, Laura Giordano, and Alberto Martelli. A tableau calculus for multimodallogics and some (un)decidability results. In Proceedings of the International Conference onAutomated Reasoning with Analytic Tableaux and Related Methods (TABLEAUX), pages 4459, 1998.

[5] Patrick Blackburn, Maarten de Rijke, and Yde Venema. Modal Logic. Cambridge University

Press, 2001.

[6] Bianca Boretti and Sara Negri. Decidability for Priorean linear time using a fixed-point labelledcalculus. In Proceedings of the 18th International Conference on Automated Reasoning withAnalytic Tableaux and Related Methods (TABLEAUX), pages 108122, 2009.

[7] Matt Fairtlough and Michael Mendler. Propositional lax logic. Information and Computation,137:133, 1997.

[8] Dov M. Gabbay. A general filtration method for modal logics. Journal of Philosophical Logic,1(1):2934, 1972.

[9] Deepak Garg. Proof Theory for Authorization Logic and Its Application to a Practical File

System. PhD thesis, Carnegie Mellon University, 2009.

[10] Olivier Gasquet, Andreas Herzig, and Mohamad Sahade. Terminating modal tableaux withsimple completeness proof. In Advances in Modal Logic, pages 167186, 2006.

[11] Olivier Gasquet and Bilal Said. Tableaux with dynamic filtration for layered modal logics.In Proceedings of the 16th International Conference on Automated Reasoning with AnalyticTableaux and Related Methods (TABLEAUX), pages 107118, 2007.

[12] Valentin Goranko. Refutation systems in modal logic. Studia Logica, 53(2):299324, 1994.

[13] Rajeev Gore and Linh Anh Nguyen. Clausal tableaux for multimodal logics of belief. Funda-menta Informaticae, 94(1):2140, 2009.

[14] Rajeev Gore and Linda Postniece. Combining derivations and refutations for cut-free com-pleteness in bi-intuitionistic logic. Journal of Logic and Computation, 20(1):233260, 2010.

[15] Yasusi Hasimoto. Finite model property for some intuitionistic modal logics. Bulletin of theSection of Logic, 30(2):8797, 2001.

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[16] Sara Negri. Proof analysis in modal logic. Journal of Philosophical Logic, 34:507544, 2005.

[17] Sara Negri. Kripke completeness revisited. In G. Primiero and S. Rahman, editors, Acts ofKnowledge History, Philosophy and Logic. College Publications, 2009.

[18] Sara Negri and Jan von Plato. Proof Analysis: A Contribution to Hilberts Last Problem.

Cambridge University Press, 2011.

[19] Luis Pinto and Roy Dyckhoff. Loop-free construction of counter-models for intuitionisticpropositional logic. In Symposia Gaussiana, pages 225232, 1995.

[20] Luis Pinto and Tarmo Uustalu. Proof search and counter-model construction for bi-intuitionistic propositional logic with labelled sequents. In Proceedings of the 18th Inter-national Conference on Automated Reasoning with Analytic Tableaux and Related Methods(TABLEAUX), pages 295309, 2009.

[21] Renate A. Schmidt and Dmitry Tishkovsky. Automated synthesis of tableau calculi. LogicalMethods in Computer Science, 7(2):132, 2011.

[22] Stephan Schwendimann. A new one-pass tableau calculus for PLTL. In Proceedings of theInternational Conference on Automated Reasoning with Analytic Tableaux and Related Methods(TABLEAUX), pages 277291, 1998.

[23] Alex K. Simpson. The Proof Theory and Semantics of Intuitionistic Modal Logic. PhD thesis,University of Edinburgh, 1994.

[24] Luca Vigano. A framework for non-classical logics. PhD thesis, Universitat des Saarlandes,1997.

[25] Luca Vigano. Labelled Non-Classical Logics. Kluwer Academic Publishers, 2000.

[26] F. Wolter and M. Zakharyaschev. Intuitionistic modal logics. In Logic and Foundations of

Mathematics, pages 227238. Kluwer Academic Publishers, 1999.

A Proofs from Section3

Lemma A.1 (Lemma3.5). If; M; ; is a saturated history, then CM(;M;;)has a mono-tonic valuationh, i.e., x h(p) andx y CM(;M;;)implyy h(p).

Proof. LetCbe the set {x y | x y}. Suppose thatx y CM(;M; ; ), i.e.,x y M Cand x h(p). From the latter, there is a z such that z x Mand z: p . From the definitionof SCR, clause (4) it follows that x( C)y, where all the relations are in M. Hence, we havea chain x = x0( C)x1 . . . ( C)xn= y where all relations are in M. We induct onito showthat T(p) Sfor(xi).

- For i = 0, x0 = x and we know that z : p and z x M. It follows from definition ofSfor that T(p) Sfor(x), as required.

- For the induction step, assume that T(p) Sfor(xi). We prove that T(p) Sfor(xi+1). Weconsider two possible cases on the relation xi( C)xi+1.

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xi xi+1 M. Because T(p) Sfor(xi), there is a z such that z xi M and

z :p . Hence, also z xi+1 M. So T(p) Sfor(xi+1).

(xi, xi+1) C. Because of the definition ofC, Sfor(xi) Sfor(xi+1), so T(p) Sfor(xi)immediately implies T(p) Sfor(xi+1).

This completes the inductive proof that T(p) Sfor(xi). In particular, T(p) Sfor(xn). Bydefinition of Sfor, there is a z such that z xn M and z : p . This immediately impliesxn h(p), i.e., y h(p), as required.

Lemma A.2 (Lemma3.6). The following hold for any saturated history;M; ; :

A. IfT() Sfor(;M; ; , x), then CM(;M;;)|=x :

B. IfF() Sfor(;M; ; , x), then CM(;M;;)|=x :

Proof. We prove both properties simultaneously by lexicographic induction, first on, and then onthe partial (tree-like) order ofM. (Note that we cannot induct on either M or the relation inCM(;M; ; ), because both of these may potentially be cyclic.) Since the context ;M; ; isfixed here, we abbreviate Sfor(;M; ; , x) to Sfor(x). Let Cbe the set {(x, y)| x y}.

Proof of A.

Case. = p. We are given thatT(p) Sfor(x) and want to prove that CM(;M;;)|= x : p.SinceT(p) Sfor(x), we know from definition of the function Sfor that there is a y withy x Mand y: p . Since y x M, we know from definition of CM(;M;;) that x h(p). Hence,by definition of|=, we have CM(;M;;)|=x : p.

Case. = . Here, CM(;M;;)|=x : is trivial by the definition of|=.

Case. = . Then the pre-condition T() Sfor(x) or, equivalently, x: is impossible by

clause (3) of the definition of saturated history. So this case is vacuous.

Case. = . We are given thatT( ) Sfor(x) or, equivalently, that x: . Byclause (5) of the definition of saturated history, x : and x : . Hence, T() Sfor(x)and T() Sfor(x). By the i.h., CM(;M;;) |= x : and CM(;M;;) |= x : . Hence,CM(;M;;)|=x : , as required.

Case. = . We are given that T( ) Sfor(x) or, equivalently, that x : .By clause (7) of the definition of saturated history, either x : or x : . Hence,either T() Sfor(x) or T() Sfor(x). By the i.h., either CM(;M;;) |= x : orCM(;M;;)|=x : . In either case, CM(;M;;)|=x : , as required.

Case. = . We are given that T( ) Sfor(x). We need to show that for any ysuch that x y in the model and CM(;M;;)|=y : , we have CM(;M;;)|=y : . Pickany y such that x y in the model and CM(;M;;) |= y : . From the definition of SCR,clause (4), it follows that x( C)y, where the relations are in M. Hence, there is a chainx= x0( C)x1 . . . ( C)xn= y, where the relations are in M. We induct on i to prove thatT( ) Sfor(xi) for each i.

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For i= 0, x0 = x and we are given that T( ) Sfor(x), so we are done.

For the inductive case, assume that T( )Sfor(xi) for some i. We show that T( ) Sfor(xi+1). We consider two possible cases on the relation xi( C)xi+1:

xi xi+1 M: From the i.h., we know that T( ) Sfor(xi). Hence, there

is a z such that z xi M and z : . Clearly, z xi+1 M, soT( ) Sfor(xi+1).

(xi, xi+1) C: Because of the definition of C, Sfor(xi) Sfor(xi+1), so T( ) Sfor(xi) immediately implies T( ) Sfor(xi+1).

This completes the inductive proof. It follows, in particular, that T( ) Sfor(xn). Conse-quently, there is some z such that z xn = y M and z

: . Hence, by clause (9)of the definition of saturated history, we must have either y : or y : . The for-mer implies, by the i.h., that CM(; M;;) |= y : , which contradicts our assumption thatCM(;M;;)|= y : . So, we must have y : . This implies T() Sfor(y) and hence, bythe i.h., that CM(;M;;)|=y : .

Case. = A nec . We are given that T(A nec ) Sfor(x). We need to show thatCM(;M;;)|=x : A nec, i.e., for any ysuch that xNAyin the model, we have CM(;M;;)|=y : . Pick any y such that xNAy in the model. Because of the definition of SCR, clause (3), wehavex((R(A)C) NA )y, where the relationsR(A) andNAare in M. So there are x0, . . . , xn, y

such that x= x0(R(A) C)x1 . . . (R(A) C)xnNAy y . We now prove, by induction on i, that

T(A nec) Sfor(xi) for each i.

For i= 0, x0 = x and we are given that T(A nec) Sfor(x).

For the inductive case, assume that T(A nec) Sfor(xi) for some i. We show that T(A nec) Sfor(xi+1) by case analyzing the relation xi(R(A) C)xi+1.

xi(R(A))xi+1 M: By the i.h., T(A nec ) Sfor(xi) so there is some z such thatz(R(A))xi M and z : A nec . Clearly, we have z(R(A))

xi+1 M, so T(A nec) Sfor(xi+1).

(xi, xi+1) C: Because of the definition of C, Sfor(xi+1) Sfor(xi). Thus, T(A nec) Sfor(xi) immediately implies T(A nec) Sfor(xi+1).

Since we just proved that T(A nec ) Sfor(xi), it follows, in particular, that T(A nec ) Sfor(xn). Consequently, there is some z

such that z(R(A))xn M and z : A nec .

Then, we also have (within M) that: z(R(A))xnNAy. So, by clause (2) of the definition of

SCR, zNAy M. Hence, by clause (11) of the definition of saturated history, we must have

y : . Therefore, T() Sfor(y) and by the i.h., CM(;M;;) |= y : . Since

y y CM(;M; ; ), by Lemma2.3, CM(;M;;)|=y : .

Proof of B.

Case. = p. We are given that F(p) Sfor(x) or, equivalently, that x : p . Suppose,for the sake of contradiction, that CM(;M;;) |= x : p. Then, x h(p) and hence, from the

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construction of the countermodel, there is a z such thatz x M andz : p . This immediatelycontradicts clause (2) of the definition of saturated history because we have z x M, z : p and x: p . Hence we must have CM(;M;;)|=x : p.

Case. = . Then the pre-condition F() Sfor(x) or, equivalently, x: is impossible by

clause (3) of the definition of saturated history. So this case is vacuous.

Case. = . Here, CM(;M;;)|=x : is trivial by the definition of|=.

Case. = . Suppose F( ) Sfor(x). Then, x: . Hence, by clause (6) of thedefinition of saturated history, either x: or x: . Therefore, eitherF() Sfor(x) orF() Sfor(x). By i.h., either CM(;M;;)|=x : or CM(;M;;)|=x : . In either case,CM(;M;;)|=x : .

Case. = . Suppose F( ) Sfor(x). Then, x : . Hence, by clause (8)of the definition of saturated history, x : and x : . Therefore, F() Sfor(x) andF() Sfor(x). By i.h., CM(;M;;)|=x : and CM(;M;;)|=x : . By definition of|=,

we have CM(;M;;)|=x : .

Case. = . Suppose F( ) Sfor(x). This implies, by definition of Sfor, thatx: . By clause (10) of the definition of saturated history, we have that either:

1. There is a y such that x y M, y: and y: or

2. There is a y such that y=x, y x and x y.

If (a) holds, then by the i.h., CM(;M;;)|= y : and CM(;M;;)|= y : . Further,x y , so CM(;M;;)|=x : .

If (b) holds, then sincex y, F( ) Sfor(y). By the i.h. on the worldy , which is strictly

smaller than x in the relation (since y =x), it follows that CM(;M;;)|=y : . Notethat in CM(;M;;), x y. So, by Lemma2.3, CM(;M;;)|=x : , as required.

Case. = A nec . Suppose F(A nec ) Sfor(x). This implies, by definition of Sfor thatx: A nec . By clause (12) of the definition of saturated history, we have that either:

(a) There is a y such that xNAy M and y: or

(b) There is a y such that y=x, y x and x y.

If (a) holds, then by the i.h., CM(;M;;)|=y : . Since xNAy, it immediately follows thatCM(;M;;)|=x : A nec.

If (b) holds, then since x y, F(A nec ) Sfor(y). By the i.h. on the world y, which isstrictly smaller in the order (since x=y), it follows that CM(;M;;)|=y : A nec. Sincein CM(;M; ; ) we have x y , Lemma2.3immediately implies CM(;M;;)|=x : A nec,as required.

Lemma A.3 (Comprehensiveness 1). Suppose (;M; CM S). Suppose x and p aresuch thatM S. M |=x : p. Then, (; M; CMx: p, {}).

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Proof. By induction on the given derivation of ;M; CM S and case analysis of its lastrule (the rules are listed in Figure 2).

Case.;M; ; is a saturated history

;M; CM {CM(;M;;)}CM

Here S = {CM(;M;;)}. The given condition M S. M |= x : p implies (by def-inition of CM) that there is a z such that z x and z : p . Therefore, by rule (init),(; M; CM x : p, {}), as required.

Case.;M, y y; , y :qCM y : q, {}

init

By rule (init), we have (;M, y y; , y : qCM x : p, y : q, {}), which is what weneed to prove.

Case.;M; , y: CM {}

L

By rule (L),(;M; , y:

CM x : p, {}), as required.

Case.;M; CM y: , {}

R

By rule (R), (;M; CM x: p, y: , {}), as required.

Case.;M; CM y: , y: , S1 ;M;

CM y: , y: , S2

;M; CM y: , S1, S2R

Here,S=S1, S2. We are given that M (S1, S2). M |=x : p.

1. M S1.M |=x : p (From assumption M (S1, S2). M |=x : p)

2. (; M; CM x : p, y: , y: , {}) (i.h. on 1st premise and (1))

3. M S2.M |=x : p (From assumption M (S1, S2). M |=x : p)

4. (; M; CM x : p, y: , y: , {}) (i.h. on 2nd premise and (2))

5. (; M; CM x : p, y: , {}) (Rule (R) on 2,4)

Case. All other cases are similar to the case of (R) above: We apply the i.h. to the premises andreapply the rule.

Lemma A.4 (Comprehensiveness 2). Suppose (;M; CM S). Suppose x and p aresuch that(;M; CMx: p, {}). Then, M S. M |=x : p.

Proof. Suppose M S. From Theorem3.9,we know that (1) w, w .(wRw M) (wRw M), (2) (w : ) . M |=w : and (3) (w : ) . M |=w : . By Theorem3.8applied tothe assumption (; M; CM x : p, {}), we know that (; M;

x : p, ). ApplyingTheorem2.5, we get that M, |= (;M; x: p, ) for every and, in particular, for (x) =x.Using (1)(3) and the definition of|= on sequents, we immediately get M |=x : p, as required.

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Theorem A.5 (Comprehensivenss, Theorem 3.16). Suppose (;M; CM S). Then(; M; CMx: p, {}) iffM S. M |=x : p.

Proof. LemmasA.3and A.4each state one direction of this theorem.

B Constructive Decidability for Classical Multi-Modal LogicThis section re-develops our method for classical logic instead of intuitionistic logic. To eliminateconfusion between names the classical logic with frame conditions is written MM, not MMI.

Syntax. The syntax of the formulas of the logic MM is shown below. I={A , B , . . .} is a finiteset of indices for modalities and p denotes an atomic formula, drawn from a countable set of suchformulas.

Formulas ,, ::= p | | | | A nec

Other standard connectives not listed above can be defined: = , =(() ()),

= () and A pos = (A nec()). A pos is the possibility modality of index A.

B.1 Semantics

We provide Kripke (frame) semantics to formulas of MM and assume in our presentation that thereader has basic familiarity with this style of semantics.

Definition B.1 (Kripke model). A classical model, Kripke model or, simply, model,M is a tuple(W, {NA}AI, h) where,

W is a set, whose elements x, y , z , w are called worlds.

Each NA is a binary relation on W

h assigns to each atom p the set of worlds h(p) W where pholds.

A model without the assignment, i.e., the tuple (W, {NA}AI) is also called a frame.

The frame conditions . We allow a countable number of additional frame conditions denotedby rules of the following form: x.((i=1,...,n xiRix

i) (xRx

)), where R1, . . . , Rn, R are from theset {NA | A I} and all variables xi, x

i, x , x

are in x. A set of such additional frame conditionsis denoted . MM is the logic whose valid formulas are exactly those that are valid (in the sensedefined below) in frames that satisfy all conditions in .

Definition B.2 (Satisfaction). Given a model M = (W, {NA}AI, h) and a world w W, we

define the satisfaction relationM |=w : , read the world w satisfies formula in model M byinduction on as follows:

M |=w : p iffw h(p)

M |=w : (unconditionally)

M |=w : iffM |=w : and M |=w :

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Axiom Rules

;M; , x: p x: p, init

Logical Rules

;M; x: , R

;M; x: , x: , ;M; x: , x: ,

;M; x: , R

;M; , x: , x: , x:

;M; , x: L

;M; , x: x: ,

;M; x: , R

;M; , x: x: ,

;M; , x: L

, y;M, xNAy; y: , x: A nec

;M; x: A nec , necR

;M, xNAy; , x: A nec , y:

;M, xNAy; , x: A nec

necL

Frame Rules

(x.((i(xiRix

i)) (xRx))) xiRix

i M ;M,xRx;

;M;

Figure 5: Seq-MM: A labeled sequent calculus for MM

M |=w : iffM |=w :

M |=w : A nec iff for every w such that wNAw, we have M |=

garg.2012.countermodelsfromsequentcalculiinmultimodallogics

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