Countermodels from Sequent Calculi in Multi-Modal Logics

Deepak Garg Valerio Genovese Sara Negri Max Planck Institute for Software Systems University of Luxembourg, Luxembourg and University of Helsinki Kaiserslautern and Saarbrucken,¨ Germany University of Torino, Italy Finland Email: [email protected] Email: [email protected] Email: sara.negri@helsinki.fi

Abstract—A novel countermodel-producing decision procedure closure relation or SCR. As examples, we show that the that applies to several multi-modal logics, both intuitionistic and classical and intuitionistic variants of the following multi- classical, is presented. Based on backwards search in labeled modal logics are constructively decidable by our method: sequent calculi, the procedure employs a novel termination condition and countermodel construction. Using the procedure, K (the basic normal modal logic), T (reflexive frames), K4 it is argued that multi-modal variants of several classical and (transitive frames), S4 (reflexive and transitive frames) and intuitionistic logics including K, T, K4, S4 and their combinations their combinations with D (serial frames). We further show with D are decidable and have the finite model property. At least that several interaction axioms between modalities such as in the intuitionistic multi-modal case, the decidability results are I((2 α) → (2 2 α))[9, 14], unit (α → (2 α))[7] new. It is further shown that the countermodels produced by A B A A the procedure, starting from a set of hypotheses and no goals, and subsumption ((2Aα) → (2Bα)) result in decidable characterize the atomic formulas provable from the hypotheses. logics. Constructive decidability also implies the finite model property, so our results also show this property for all the I.INTRODUCTION logics listed earlier. For multi-modal intuitionistic logics, not Modal logics are widely used in several fields of Computer only our method, but also the decidability results are new. Science and their decidability is a subject of deep interest to Technical approach and challenges: We present our method the academic community. The subject has been investigated separately for classical and intuitionistic logics due to minor through various techniques, notably semantic filtrations [5,8], technical differences between the two. Our method uses the semantic tableaux [4, 11, 12], and translation into decidable labeled approach to proof theory of modal logic as developed, fragments of first-order logic [2], yet many areas related to among others, by [17, 24, 26], and more specifically [17], decidability of modal logics remain open. Two such areas are: to produce labeled sequent calculi with strong analyticity (1) Decidability of multi-modal intuitionistic logics, especially properties. We define a multi-modal intuitionistic (classical) when modalities interact with each other, and (2) Decision logic MMIχ (MMχ) as the set of all formulas that are valid in procedures based on sequent calculi that can be directly all intuitionistic (classical) Kripke frames satisfying stipulated implemented. Both these areas are challenging. Decidability conditions, represented as a set χ. Conditions in χ can be of intuitionistic modal logics is challenging because standard arbitrary, but are restricted in two ways: (1) They must be of 0 0 techniques like semantic filtrations and tableaux have not the form ∀~x.((∧i=1,...,n xiRixi) → (xRx )), where R,Ri been studied extensively in the intuitionistic setting, whereas range over the relations of a Kripke frame, and (2) The sequent calculi are difficult to use for decision procedures conditions χ must have a SCR, which is a relation over Kripke in modal logic because of a well-known problem of loop- frames satisfying some stipulated properties, as discussed later ing [11, 17, 26], which is exacerbated by the interaction in the paper. Briefly, the existence of the SCR implies that between modalities and intuitionistic implication. frame relations can be deduced from the conditions χ only in Spanning both these areas, we present a uniform decision some specific ways. We then build a non-terminating, standard procedure for several propositional multi-modal logics (both labeled sequent calculus for the logic, which we refine in two intuitionistic and classical), based on backwards search in steps to obtain a constructive decision procedure that, for a labeled sequent calculi. Our decision procedure is constructive, given sequent, either produces a proof of it, or a finite set which means that for any given formula it either produces a of finite countermodels for it. Next, we build an extension of derivation which shows that the formula is true in all (Kripke) our method that works for any logic on which our original models or produces a finite set of finite countermodels on all method works, extended with seriality. Finally, we prove an of which the formula is false. interesting property of our decision procedure: The set of The decision procedure is also general; it applies to any countermodels it produces for a given hypotheses without a intuitionistic modal logic without possibility (diamond) modal- specific goal completely characterizes the atomic formulas ities and any classical modal logic (even with possibility that can be proved from the hypotheses. Thus the set of modalities), provided the logic satisfies a specific technical countermodels produced is, in a sense, complete. We call this condition, namely the existence of what we call a suitable property comprehensiveness. The first challenge for our work is to find a general modal logic is briefly discussed in SectionIV and its de- termination condition for backwards search in labeled sequent tails are deferred to our online technical report (TR in the calculi. Our termination condition is based on containment sequel) [10]. In SectionII, we define the syntax, semantics of the sets of formulas labeling worlds, which we show to and a standard, but non-terminating labeled sequent calculus be complete so long as the logic has an SCR. This idea for the intuitionistic multi-modal logic MMIχ. Section III is a non-trivial generalization of existing work on logic- presents the main technical work. Starting from an informal specific termination conditions for many uni-modal classical introduction, we proceed to a description of the decision logics [6, 11, 17]. Finding an appropriate definition for SCRs procedure (Sections III-A– III-D), its comprehensiveness (Sec- that is both sufficient to obtain termination and general enough, tion III-E), its extension with seriality (Section III-F), and its is the main technical challenge of our work and also its main instantiation to several known intuitionistic multi-modal logics technical contribution. (Section III-G). SectionIV briefly lists modifications needed to The second challenge in our work is to actually build the adapt the method to classical logic. Related work is discussed countermodel when we know that backwards proof search in SectionV and SectionVI concludes the paper. Proofs of has unsuccessfully terminated. To this end, we observe that a theorems and complete descriptions of all calculi are presented straightforward extension of the model inherent in the sequent in our TR [10]. at which search terminates (with a few more relations) is χ actually a countermodel to the sequent. As far as we know, II.MMI :MULTI-MODAL INTUITIONISTIC LOGIC this construction is novel. We start by defining formally the family of intuitionistic Contributions: Our work makes the following contributions: multi-modal logics we consider in this paper. The family is (1) It proves, by uniform method, the decidability of the parametrized by a set χ of conditions on Kripke frames that necessitation-only, multi-modal intuitionistic variants of the must have a specific (standard) form, as described later in this logics K, T, K4 and S4 and their combinations with the logic section. The logic obtained by instantiating our definition of D. Our decision procedure produces countermodels, and also syntax and semantics with a specific set χ is called MMIχ. establishes the finite model property for these logics. (2) We Let I = {A, B, . . .} be a finite set of indices for modalities provide the first sequent calculus based constructive decision and p denote an atomic formula, drawn from a countable set procedure for multi-modal logics. (3) At a technical level, we of such formulas. Then, the syntax of formulas of the logic provide a general method for forcing termination in labeled MMIχ is: sequent calculi for modal logics and a sufficient condition (the existence of SCRs) under which it works without loss ϕ, α, β ::= p | > | ⊥ | α ∧ β | α ∨ β | α → β | A nec α of completeness. We also present a simple method to extract Connectives >, ⊥, ∧, and ∨ have their usual meanings. countermodels when search terminates. (4) As far as we know, Implication → is interpreted intuitionistically. A nec α is the ours is the first decision procedure which produces a set of necessitation modality of index A. This is commonly written countermodels that is comprehensive in the sense described 2Aα, but we prefer the more descriptive notation. Negation above. is not primitive, but may be defined in a standard way as Limitations: There are existing undecidability results for ¬α = α → ⊥. We do not consider possibility modalities in modal logics with frame conditions such as symmetry and the intuitionistic setting, since they are incompatible with our transitivity [3]. Consequently, we cannot hope for a method method. that proves decidability for all logics MMIχ or all logics Semantics: We provide Kripke (frame) semantics to formu- MMχ. Nonetheless, there are some classes of frame conditions las of MMIχ and assume in our presentation that the reader which are not known to immediately result in undecidability, has basic familiarity with this style of semantics. Several but to which our method does not apply, either because different Kripke semantics for intuitionistic modal logic have they do not fit our definition of χ or because they do not been proposed [1, 24, 27]. In what follows, we use semantics have SCRs. First, due to an interaction with intuitionistic that are similar to those of Wolter et al. [27]. This choice implication, our method cannot handle possibility modalities makes the technical development easier. in intuitionistic logic, even though it works fine with them in classical logic. Second, we do not know whether our Definition II.1 (Kripke model). An intuitionistic method can handle “label-producing” conditions like density model, Kripke model or, simply, model M is a tuple or confluence. However, it can be easily extended to work (W, ≤, {NA}A∈I , h) where, with seriality, as discussed in Section III-F. Finally, like many - (W, ≤) is a preorder. Elements of W are called worlds other methods in this domain, an analysis of our proofs does and written x, y, z, w. Since ≤ is a preorder, it satisfies the not necessarily produce good upper bounds on the actual following conditions: complexity of modal logics. ∀x.(x ≤ x) (refl) Organization: Since our constructive decision procedure ∀x, y, z.(((x ≤ y) ∧ (y ≤ z)) → (x ≤ z)) (trans) and its correctness proof are almost identical for intuitionistic - Each NA is a binary relation on W that satisfies the condition and classical modal logics, in this paper, we present our (≤ ◦NA) ⊆ NA, i.e. results only for the intuitionistic case. The case of classical ∀x, y, z.(((x ≤ y) ∧ (yNAz)) → (xNAz)) (mon-N) - h assigns to each atom p the set of worlds h(p) ⊆ W where init Σ; , x ≤ y;Γ, x : p ⇒χ y : p, ∆ p holds. We require h to be monotone w.r.t. ≤, i.e., if x ∈ h(p) M and x ≤ y then y ∈ h(p). χ Σ, y; M, x ≤ y;Γ, y : α ⇒ y : β, x : α → β, ∆ A model without the assignment, i.e., the tuple χ →R Σ; M;Γ ⇒ x : α → β, ∆ (W, ≤, {NA}A∈I ) is also called a frame and the conditions χ on relations above, e.g., (mon-N), are called frame conditions. Σ; M, x ≤ y;Γ, x : α → β ⇒ y : α, ∆ χ Σ; M, x ≤ y;Γ, x : α → β, y : β ⇒ ∆ The frame conditions χ: In addition to the frame conditions →L Σ; , x ≤ y;Γ, x : α → β ⇒χ ∆ (refl), (trans) and (mon-N), we allow a countable number of M additional frame conditions as rules of the following form: χ Σ, y; M, xNAy;Γ ⇒ y : α, x : A nec α, ∆ 0 0 χ necR ∀~x.((∧i=1,...,n xiRixi) → (xRx )), where R1,...,Rn,R are Σ; ;Γ ⇒ x : A nec α, ∆ 0 0 M from the set {NA | A ∈ I}∪{≤} and all variables xi, xi, x, x χ are in ~x. A set of such additional frame conditions is denoted Σ; M, xNAy;Γ, x : A nec α, y : α ⇒ ∆ χ χ necL χ. MMI is the logic whose valid formulas are exactly those Σ; M, xNAy;Γ, x : A nec α ⇒ ∆ that are valid (in the sense defined below) in frames that satisfy χ all conditions in χ. Σ; M, x ≤ y, yNAz, xNAz;Γ ⇒ ∆ χ mon-N Σ; M, x ≤ y, yNAz;Γ ⇒ ∆ Definition II.2 (Satisfaction). Given a model M = 0 0 (W, ≤, {NA}A∈I , h) and a world w ∈ W , we define the (∀~x.((∧i (xiRixi)) → (xRx ))) ∈ χ 0 0 χ satisfaction relation M |= w : α, read “the world w satisfies xiRixi ∈ M Σ; M, xRx ;Γ ⇒ ∆ χ χ formula α in model M” by induction on α (we omit standard Σ; M;Γ ⇒ ∆ definitions for ∨, ∧ and >): - M |= w : p iff w ∈ h(p) Fig. 1. Seq-MMIχ: A labeled sequent calculus for MMIχ, selected rules - M |= w : α → β iff for every w0 such that w ≤ w0 and M |= w0 : α, we have M |= w0 : β. 0 0 - M |= w : A nec α iff for every w such that wNAw , we have presented in what is known as the “labeled” style of calculi for M |= w0 : α. modal logics, which means that the calculus proves formulas We say that M 6|= w : α if it is not the case that M |= w : α. labeled with symbolic worlds. A labeled formula contains a In particular, for every M and every w, M 6|= w : ⊥. symbol x, y, z, w, u denoting a world and a formula α, written A formula α is true in a model M, written M |= α, if for together as x : α. A sequent in our calculus has the form every world w ∈ M, M |= w : α. A formula α is valid in Σ; M;Γ ⇒χ ∆, where: Σ is a finite set of world symbols MMIχ, written |= α, if M |= α for every model M satisfying appearing in the rest of the sequent (world symbols are also all conditions in χ. called labels); M is a finite multi-set of relations of the forms Valid axioms: We list below some valid axioms and admis- x ≤ y and xNAy; Γ and ∆ are finite multi-sets of labeled sible rules that are common to all logics MMIχ. If a rule/axiom formulas. Intuitively, if Σ; M;Γ ⇒χ ∆ is valid, then every is standard in literature, its common name is mentioned to the model with a world set containing at least Σ, satisfying all extreme right. (α ≡ β means (α → β) ∧ (β → α).) relations in M and all labeled formulas in Γ must satisfy (all tautologies of intuitionistic propositional logic) (IPC) at least one labeled formula in ∆. This is formalized in the If |= α, then |= A nec α (nec) following definition. |= (A nec (α → β)) → ((A nec α) → (A nec β)) (K) |= ((A nec α) ∧ (A nec β)) ≡ (A nec (α ∧ β)) Definition II.4 (Sequent satisfaction and validity). A model The frame conditions χ can be used to force additional M and a mapping ρ from elements of Σ to worlds of χ axioms in a standard way, which has been explored in great M satisfy a sequent Σ; M;Γ ⇒ ∆, written M, ρ |= χ detail in literature on correspondence theory [5]. For example, (Σ; M;Γ ⇒ ∆), if at least one of the following holds: (1) There is an xRy ∈ with R ∈ {≤} ∪ {N | A ∈ I} the condition ∀x, y.((xNAy) → (x ≤ y)) corresponds to M A the axiom α → (A nec α), commonly called (unit) and of such that ρ(x) R ρ(y) 6∈ M; (2) There is an x : α ∈ Γ central importance in lax logic [7]. Similarly, the condition such that M 6|= ρ(x): α; (3) There is an x : α ∈ ∆ such that M |= ρ(x): α. ∀x, y, z.(((xNAy) ∧ (yNAz)) → (xNAz)) corresponds to χ the axiom (A nec α) → (A nec A nec α). A model M satisfies a sequent Σ; M;Γ ⇒ ∆, written χ The following is a fundamental property of Kripke models M |= (Σ; M;Γ ⇒ ∆), if for every mapping ρ, we have χ χ in intuitionistic modal logics, proved by induction on α. M, ρ |= (Σ; M;Γ ⇒ ∆). Finally, a sequent Σ; M;Γ ⇒ ∆ is valid, written |= (Σ; M;Γ ⇒χ ∆), if for every model M Lemma II.3 (Monotonicity). If M |= w : α and w ≤ w0 ∈ χ we have M |= (Σ; M;Γ ⇒ ∆). M, then M |= w0 : α. Rules of the sequent calculus: Selected rules of the sequent χ χ A. Seq-MMI : A Labeled Sequent Calculus for MMI calculus for MMIχ are shown in Figure1. Following standard We present a sound, complete, cut-free sequent calculus approach in labeled calculi, the rules for each connective for MMIχ. Following the work of Negri [17], our calculus is mimic the (Kripke) semantic definition of the connective. The rules (→R) and (necR) introduce fresh worlds into Σ, in backwards proof search. It only remains to show that consistent with the semantic definition (Definition II.2). We this condition forces termination. This follows from the fact say that ` (Σ; M;Γ ⇒χ ∆) if Σ; M;Γ ⇒χ ∆ has a proof in that Sfor(Σ; M; Γ; ∆, x) increases monotonically for each x the calculus. The sequent calculus is both sound and complete in a backwards proof search and the fact that the number with respect to the semantics. of possible values of Sfor is finite, which, in turn, is a consequence of the weak subformula property (Theorem II.6). Theorem II.5 (Soundness). If ` (Σ; M;Γ ⇒χ ∆), then |= χ We further show that if no rule applies backwards to (Σ; M;Γ ⇒ ∆). χ a sequent Σ; M;Γ ⇒ ∆ (after imposing our termination The converse of Theorem II.5, completeness, also holds but checks), then we can obtain a countermodel to the sequent we do not need this result in the rest of our development so Σ; M;Γ ⇒χ ∆ by adding an edge x ≤ y whenever we do not present it here. The following property is critical Sfor(Σ; M; Γ; ∆, x) ⊆ Sfor(Σ; M; Γ; ∆, y). This forms the to the design and correctness of our constructively complete basis of our countermodel extraction. (As further explained in decision procedure. SectionV, this method of extracting countermodels is inspired by Negri’s proof of completeness of labeled sequent calculi for Theorem II.6 (Weak subformula property). If a formula uni-modal logic with respect to Kripke semantics [18].) ϕ appears in any proof tree (possibly infinite) obtained by The definition of the function Sfor depends on the condi- applying the rules of Figure1 backwards starting from a χ tions χ that define the logic. In our formal development, we concluding sequent Σ; M;Γ ⇒ ∆, then ϕ is a subformula of define a suitable Sfor for every logic for which there exists some formula in either Γ or ∆. what we call a suitable closure relation (SCR). Technically, a SCR is a family of relations on frames, which satisfies some III.CONSTRUCTIVELY COMPLETE DECISION PROCEDURE stipulated properties. Our entire method applies to any logic As argued by Negri [17], and also illustrated in our TR, MMIχ for whose χ a SCR exists. This begs the question constructing a decision procedure based on the sequent calcu- of how general the existence of a SCR is. As we show in lus of Figure1 is non-trivial because backwards search in the Section III-G, several (multi-)modal logics with reflexivity, calculus can loop forever due to unbounded creation of new transitivity, and modality interaction conditions, including the worlds in the rules (→R) and (necR). Although prior work multi-modal logics K, K4, S4, T, and I [9, 14] have SCRs. We such as that of Negri describes loop-detection methods for further show in Section III-F that our method can be extended many specific uni-modal classical logics individually, there is to any logic with a SCR plus the seriality condition on its no known general solution. In this section we present a general accessibility relations. technique that not only detects such loops in a wide variety Our technical presentation consists of the following steps: of logics MMIχ, but also produces Kripke countermodels (1) We define the term “SCR for χ”. Its existence is the only witnessing the non-validity of the end-sequent when such condition that must hold for our method to apply to the logic loops are detected. Although the end-result of our technique, MMIχ. (2) We define the function Sfor using SCRs. We also i.e., the decision procedure itself, is quite straightforward, define a predicate on sequents, whose elements (sequents) building up to it requires some non-standard machinery, which are called saturated histories. Roughly, a saturated history we motivate here by presenting an informal outline of our is a sequent on which backwards application of all rules is method. useless, i.e., applying any rule other than (→R) and (necR) To control the use of rules (→R) and (necR) in backwards backwards on the sequent does not add any new labeled proof search, we generalize a technique from existing work formulas, and the application of these two rules is blocked by on tableaux calculi for classical uni-modal logics [11]. The the containment condition on Sfor. We prove, by construction, technique prevents loops by checking for containment of the key property of our entire method: If a sequent is a formulas that label a world in those labeling another. We start saturated history, then it has a finite countermodel. (3) We χ χ by observing that in any sequent Σ; M;Γ ⇒ ∆ obtained define an intermediate sequent calculus Seq-MMICM with χ during backwards proof search starting from a single goal judgments Σ; M;Γ ⇒CM ∆ & S. Here, S is a (possibly formula, all worlds in M lie on a rooted, directed tree, whose empty) finite set of finite countermodels. The correctness χ edges are relations in M that were introduced by the rules property of this calculus is: If Σ; M;Γ ⇒CM ∆ & {} has χ χ (→R) and (necR) in earlier steps of the search. We call the a proof, then ` (Σ; M;Γ ⇒ ∆) and if Σ; M;Γ ⇒CM ∆ & S reflexive-transitive closure of this tree . Next, we define has a proof for S 6= {}, then every M ∈ S satisfies: a function Sfor(Σ; M; Γ; ∆, x) that lists, approximately, all M 6|= (Σ; M;Γ ⇒χ ∆). Backwards search in this calculus formulas labeled by x in Γ and ∆ (the exact definition of Sfor does not necessarily terminate. However, we find the calculus depends on χ, and is one of our key technical contributions). a useful intermediate step to prove many properties. (4) We χ This function satisfies a pivotal property whose proof requires observe that a specific strategy for proof search in Seq-MMICM induction on : If there is a world y (y 6= x) such that terminates. This strategy is presented as a sequent calculus χ y  x and Sfor(Σ; M; Γ; ∆, x) ⊆ Sfor(Σ; M; Γ; ∆, y), then Seq-MMIT , which is a countermodel producing decision pro- it is useless to apply any of the rules (→R) and (necR) on any cedure. principal formula labeled by x in the sequent Σ; M;Γ ⇒χ ∆ In what follows, we present the technical details of each of these four steps. In particular, Section III-D describes our F (ϕ) as “ϕ should be false”. Given a history Σ; M; Γ; ∆ and decision procedure. x ∈ Σ, the signed formulas of x, written Sfor(Σ; M; Γ; ∆, x), are defined as follows: A. Suitable Closure Relations (SCRs) Sfor(Σ; M; Γ; ∆, x) = Our constructive decision procedure applies to any logic {T (ϕ) | x : ϕ ∈ Γ} ∪ {F (ϕ) | x : ϕ ∈ ∆}∪ χ ∗ MMI whose χ has a SCR. Call a frame M closed if it is {T (A nec ϕ) | ∃y. y(R(A)) x ∈ M and y : A nec ϕ ∈ Γ}∪ closed under the conditions (refl), (trans), (mon-N) and χ. {T (ϕ → ψ) | ∃y. y ≤ x ∈ M and y : ϕ → ψ ∈ Γ}∪ {T (p) | ∃y. y ≤ x ∈ M and y : p ∈ Γ} Given a frame M, let M denote its closure obtained by closing the frame under the conditions (refl), (trans), (mon-N) and χ, When Σ, M, Γ, ∆ are clear from context, we abbreviate obtained as the least fixed point of the application of these Sfor(Σ; M; Γ; ∆, x) to Sfor(x). We say that x 4 y iff rules. Informally, a SCR is a family of relations (R(A))A∈I Sfor(x) ⊆ Sfor(y). that, given any closed frame M and any extension M0 of it with We call a frame M tree-like if it can be derived from a finite additional edges of the form ≤, characterizes all relations NA tree of the relations ≤ and NA and (possibly partial) closure in M0 in terms of the relations in M and the difference M0−M. by frame rules. This tree is called the underlying tree of M and we say that x  y (in ) iff there is a directed path from Definition III.1 (Suitable closure relation (SCR)). A family of M x to y in the tree underlying . binary relations (R(A)) is called a suitable closure relation M A∈I The key concept of our method is a saturated history, or SCR for χ if the following hold: (0) Each R(A) is definable a generalization of Hintikka sets to labeled, intuitionistic in first-order logic in terms of the relations ≤∪{N | A ∈ I}. A sequents. Intuitively, a history Σ; ; Γ; ∆ is saturated if we can (1) For a finite frame and x, y ∈ , it can be decided M M M directly define a countermodel for the sequent Σ; ;Γ ⇒χ ∆. whether x(R(A))y or not. (2) ((R(A))∗ ◦ N ) ⊆ N can be M A A (The definition of this countermodel is given immediately after derived from the frame conditions (refl), (trans), (mon-N) and the definition of a saturated history.) χ. (3) For any closed frame M and any C ⊆ {x ≤ y | x, y ∈ ∗ M}, if xNAy ∈ M ∪ C, then x((R(A)∪C) ◦NA ◦≤)y, where Definition III.3 (Saturated history). A history Σ; M; Γ; ∆ is all relations R(A) and NA on the right are in M. (4) For called saturated if the following hold: (1) M is tree-like and any closed frame M and any C ⊆ {x ≤ y | x, y ∈ M}, if saturated with respect to the conditions (refl), (trans), (mon-N) x ≤ y ∈ M ∪ C, then x(≤ ∪ C)∗y, where all ≤ relations on and χ. In particular, because M is tree-like, it has a relation the right are in M. Condition (4) depends only on χ, not on  defined on it. (2) If x : p ∈ Γ, then there is no y such that the choice of R(A), but we include it here for uniformity. x ≤ y ∈ M and y : p ∈ ∆. (3) There is no x such that x : > ∈ ∆. (4) There is no x such that x : ⊥ ∈ Γ. (5) If x : α ∧ β ∈ Γ, Example III.2 (SCR for transitivity). Let trans(A) be the then x : α ∈ Γ and x : β ∈ Γ. (6) If x : α ∧ β ∈ ∆, then either frame condition ∀x, y, z.(((xN y) ∧ (yN z)) → (xN z)) A A A x : α ∈ ∆ or x : β ∈ ∆. (7) If x : α ∨ β ∈ Γ, then either and let χ = {trans(A) | A ∈ I}. Then, the relation R(A) = x : α ∈ Γ or x : β ∈ Γ. (8) If x : α ∨ β ∈ ∆, then x : α ∈ ∆ N ∪ ≤ is a SCR for χ. To prove this, we verify each of the A and x : β ∈ ∆. (9) If x : α → β ∈ Γ and x ≤ y ∈ , then conditions (0)–(4) in the definition of SCR. Conditions (0) and M either y : α ∈ ∆ or y : β ∈ Γ. (10) If x : α → β ∈ ∆, then (1) are trivially true. (2) is equivalent to ((N ∪ ≤)∗ ◦ N ) ⊆ A A either: (10a) There is a y such that x ≤ y ∈ , y : α ∈ Γ N , which follows from the frame conditions (mon-N) and χ. M A and y : β ∈ ∆, or (10b) There is a y such that y 6= x, y  x To prove (3), suppose that xNAy ∈ M ∪ C. Then, because the and x 4 y. (11) If x : A nec α ∈ Γ and xNAy ∈ M, then only way to derive a relation NA is to use either (mon-N) or ∗ y : α ∈ Γ. (12) If x : A nec α ∈ ∆, then either: (12a) There is trans(A), it follows that in M ∪ C, we have x((NA ∪ ≤) ◦ ∗ a y such that xNAy ∈ M and y : α ∈ ∆, or (12b) There is a NA)y. So, we also have x((NA ∪≤∪C) ◦NA)y, where all ≤ ∗ y such that y 6= x, y  x and x 4 y. 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 Definition III.4 (Countermodel of a saturated history). For a required. The proof of (4) is similar to that of (3). saturated history Σ; M; Γ; ∆, the countermodel of the history, CM(Σ; M; Γ; ∆) is defined as follows: (1) The worlds of B. Saturated Histories CM(Σ; M; Γ; ∆) are exactly those in Σ. (2) The relations of χ Our method applies only to those logics MMI whose χ has CM(Σ; M; Γ; ∆) are M ∪ C, where C = {x ≤ y | x 4 y}. a SCR, so in the sequel we fix a set of frame conditions χ and (3) h(p) = {x | ∃y. (y ≤ x ∈ M) ∧ (y : p ∈ Γ)}. assume there is a SCR (R(A)) for this χ. Although we A∈I It is not obvious that CM(Σ; ; Γ; ∆) is a model. It does present the technical material generically with respect to an M satisfy all frame conditions. However, we must also show abstract χ and SCR, we advise the reader to choose a specific monotonicity: If x ≤ y ∈ CM(Σ; ; Γ; ∆) and x ∈ h(p), χ and its SCR (for instance, those from Example III.2), and M then y ∈ h(p). The following lemma states that this is the instantiate our definitions and theorems on them. case. A history is a tuple Σ; M; Γ; ∆ (equivalently, a sequent Σ; M;Γ ⇒χ ∆) such that all labels in M, Γ and ∆ occur in Σ. Lemma III.5. If Σ; M; Γ; ∆ is a saturated history, then Let T (ϕ) and F (ϕ) be two uninterpreted unary relations on CM(Σ; M; Γ; ∆) has a monotonic valuation h, i.e., x ∈ h(p) formulas. Informally, we read T (ϕ) as “ϕ should be true” and and x ≤ y ∈ CM(Σ; M; Γ; ∆) imply y ∈ h(p). The next Lemma states the central property of our Σ; M; Γ; ∆ is a saturated history method. In particular, the Lemma immediately implies that χ CM Σ; M;Γ ⇒CM ∆ &{CM(Σ; M; Γ; ∆)} if Σ; M; Γ; ∆ is a saturated history, then CM(Σ; M; Γ; ∆) is a χ countermodel to the sequent Σ; M;Γ ⇒ ∆. χ init Σ; M, x ≤ y;Γ, x : p ⇒CM y : p, ∆ & {}

Lemma III.6. The following hold for any saturated his- χ Σ, y; M, x ≤ y;Γ, y : α ⇒CM y : β, x : α → β, ∆ & S tory Σ; M; Γ; ∆: (A) If T (ϕ) ∈ Sfor(Σ; M; Γ; ∆, x), then χ →R Σ; M;Γ ⇒CM x : α → β, ∆ & S CM(Σ; M; Γ; ∆) |= x : ϕ. (B) If F (ϕ) ∈ Sfor(Σ; M; Γ; ∆, x), then CM(Σ; ; Γ; ∆) 6|= x : ϕ. χ M Σ; M, x ≤ y;Γ, x : α → β ⇒CM y : α, ∆ & S1 χ Σ; M, x ≤ y;Γ, x : α → β, y : β ⇒CM ∆ & S2 Proof: We prove both properties simultaneously by lexi- →L Σ; , x ≤ y;Γ, x : α → β ⇒χ ∆ & S ,S cographic induction, first on ϕ, and then on the partial (tree- M CM 1 2 like) order  of M. χ Σ, y; M, xNAy;Γ ⇒CM y : α, x : A nec α, ∆ & S necR Σ; ;Γ ⇒χ x : A nec α, ∆ & S Corollary III.7 (Existence of countermodel). If Σ; M; Γ; ∆ M CM χ is a saturated history, then CM(Σ; M; Γ; ∆) 6|= (Σ; M;Γ ⇒ χ Σ; M, xNAy;Γ, x : A nec α, y : α ⇒CM ∆ & S ∆). χ necL Σ; M, xNAy;Γ, x : A nec α ⇒CM ∆ & S

χ χ χ Σ; , x ≤ y, yNAz, xNAz;Γ ⇒ ∆ & S C. Seq-MMICM: Countermodels for MMI M CM χ mon-N Σ; M, x ≤ y, yNAz;Γ ⇒CM ∆ & S Having defined a saturated history, i.e., a sequent for which 0 0 a countermodel exists (Corollary III.7), we now define a se- (∀~x.((∧i (xiRixi)) → (xRx ))) ∈ χ χ χ x R x0 ∈ Σ; , xRx0;Γ ⇒χ ∆ & S quent calculus Seq-MMICM, written ⇒CM, which uses this fact i i i M M CM χ χ to emit countermodels from unprovable sequents. Although Σ; M;Γ ⇒CM ∆ & S this calculus is not a decision procedure, we find it a useful χ χ step in proving several results, in particular, the results of Fig. 2. Seq-MMICM: Countermodel producing sequent calculus for MMI , Section III-E. selected rules χ χ Sequents of Seq-MMICM have the form Σ; M;Γ ⇒CM ∆ & S, where S is a finite set of finite models. We write χ χ χ D. Seq-MMIT : Termination and Countermodel Extraction ` (Σ; M;Γ ⇒CM ∆ & S) if Σ; M;Γ ⇒CM ∆ & S has a χ proof. The meaning of Σ; M;Γ ⇒CM ∆ & S depends on S. Next, we describe a particular backwards proof search strat- χ χ χ If ` (Σ; M;Γ ⇒CM ∆ & {}), then ` (Σ; M;Γ ⇒ ∆) and egy in Seq-MMICM that always terminates without losing com- χ if ` (Σ; M;Γ ⇒CM ∆ & S) with S 6= {}, then every model pleteness, thus obtaining a countermodel producing decision M ∈ S is a countermodel to Σ; ;Γ ⇒χ ∆ in the sense of procedure for MMIχ. This strategy is described as a calculus M χ χ (the converse of) Definition II.4. Seq-MMIT , with sequents of the form Σ; M;Γ ⇒T ∆ & S. χ The rules of the calculus can be interpreted backwards as a Selected rules of the sequent calculus Seq-MMICM are shown in Figure2. First, every rule in the ordinary sequent decision procedure with inputs Σ, M, Γ, and ∆ and output χ calculus (Figure1) is modified to have in the conclusion the S. For a given Σ, M, Γ, and ∆, (Σ; M;Γ ⇒ ∆) is provable union of the sets of (counter)models in the premises. This is iff S = {}, else every model in S is a countermodel to the sound because the rules of the sequent calculus are invertible sequent. χ (i.e., the conclusion of each rule holds iff the premises Selected rules of the calculus Seq-MMIT are shown in hold). Second, there is a new rule (CM) that produces the Figure3. Each rule in the calculus corresponds to a rule of the χ countermodel CM(Σ; M; Γ; ∆) when Σ; M; Γ; ∆ is a saturated same name in Seq-MMICM (Figure2). The only significant history. difference between the two calculi is that the premise of χ We emphasize again that this calculus is not necessarily a the rule (CM) in Seq-MMICM requires that Σ; M; Γ; ∆ be χ χ decision procedure because it includes all rules of ⇒ and a saturated history, but the rule (CM) applies in Seq-MMIT hence admits all of the latter’s infinite backwards derivations only when no other rule applies. To ensure that “no other as well. rule applies” implies that Σ; M; Γ; ∆ is a saturated history, we spread the negations of the conditions (1) and (5)–(12) from χ Theorem III.8 (Soundness 1). If ` (Σ; M;Γ ⇒CM ∆ & {}), the definition of saturated history to rules other than (CM) χ then ` (Σ; M;Γ ⇒ ∆). as pre-conditions, called applicability conditions (written in χ boxes in Figure3). Conditions (2), (3) and (4) obviously hold Theorem III.9 (Soundness 2). If ` (Σ; M;Γ ⇒CM ∆ & S), when the rules (init), (>R) and (⊥L) do not apply, respectively. then for every model M ∈ S, M 6|= (Σ; M;Γ ⇒χ ∆). Hence, when no rule other than (CM) applies, all 12 conditions Proof: By induction on the given derivation of of the definition of saturated history must hold, so Σ; M; Γ; ∆ χ Σ; M;Γ ⇒CM ∆ & S. The case of rule (CM) follows by must be a saturated history. It only remains to show that the Corollary III.7. calculus with these applicability conditions does not admit 2) For any Σ; M; Γ; ∆ with M tree-like, there is an S such No other rule applies χ χ CM that ` (Σ; M;Γ ⇒T ∆ & S) and such an S can be Σ; ;Γ ⇒ ∆ &{CM(Σ; ; Γ; ∆)} M T M finitely computed.

χ init Proof: By a straightforward counting argument. Σ; M, x ≤ y;Γ, x : p ⇒T y : p, ∆ & {} Note that Theorem III.12(2) does not stipulate that the S ∀y ∈ Σ.(x ≤ y ∈ ) ⇒ (y : α 6∈ Γ y : β 6∈ ∆) be unique. Indeed, depending on the order in which the rules M or χ of the calculus ⇒T are applied to a given sequent, S may be ∀y ∈ Σ.(y  x) ⇒ (x = y or x 46 y) χ different. However, the fact that at least one such S exists and Σ, y; M, x ≤ y;Γ, y : α ⇒T y : β, x : α → β, ∆ & S χ χ →R can be computed is enough to get decidability for MMI . Σ; M;Γ ⇒T x : α → β, ∆ & S Lemma III.13 (Simulation). If M is tree-like and ` χ χ y : α 6∈ ∆ and y : β 6∈ Γ (Σ; M;Γ ⇒T ∆ & S), then ` (Σ; M;Γ ⇒CM ∆ & S). χ Σ; M, x ≤ y;Γ, x : α → β ⇒T y : α, ∆ & S1 χ Proof: By induction on the given derivation of Σ; M, x ≤ y;Γ, x : α → β, y : β ⇒T ∆ & S2 χ →L Σ; ;Γ ⇒ ∆ & S. The case of rule (CM) follows from Σ; , x ≤ y;Γ, x : α → β ⇒χ ∆ & S ,S M T M T 1 2 Lemma III.10. The rest of the cases are immediate from the i.h. The only fact to take care of is that the tree-like property holds ∀y ∈ Σ.(xN y ∈ ) ⇒ y : α 6∈ ∆ A M for each i.h. application. This follows from Lemma III.11. ∀y ∈ Σ.(y  x) ⇒ (x = y or x 46 y) χ Theorem III.14 (Decidability). For a tree-like M, suppose Σ, y; M, xNAy;Γ ⇒T y : α, x : A nec α, ∆ & S χ necR that S is such that ` (Σ; ;Γ ⇒ ∆ & S) (such an S must Σ; ;Γ ⇒χ x : A nec α, ∆ & S M T M T exist and can be computed using Theorem III.12). Then: (1) If S = {}, then |= (Σ; ;Γ ⇒χ ∆). (2) If S 6= {}, then y : α 6∈ Γ Σ; , xN y;Γ, x : A nec α, y : α ⇒χ ∆ & S M M A T every model M in S is a countermodel to the sequent, i.e., χ necL Σ; , xN y;Γ, x : A nec α ⇒ ∆ & S χ M A T M 6|= (Σ; M;Γ ⇒ ∆). χ Proof: By Lemma III.13, we have that ` (Σ; ;Γ ⇒χ xNAz 6∈ M Σ; M, x ≤ y, yNAz, xNAz;Γ ⇒T ∆ & S M CM χ mon-N ∆ & S). Now, (1) follows from Theorems III.8 and II.5 and Σ; M, x ≤ y, yNAz;Γ ⇒T ∆ & S (2) follows from Theorem III.9. 0 0 (∀~x.((∧i (xiRixi)) → (xRx ))) ∈ χ Corollary III.15 (Decidability and finite model property). If 0 0 0 χ χ xiRixi ∈ M xRx 6∈ M Σ; M, xRx ;Γ ⇒T ∆ & S a SCR exists for χ, then MMI is decidable, has the finite χ χ Σ; M;Γ ⇒T ∆ & S model property and has a constructive decision procedure. E. Comprehensiveness of Constructed Countermodels Fig. 3. Seq-MMIχ: Terminating, countermodel producing sequent calculus T χ χ χ for MMI , selected rules. Applicability conditions are written in boxes . Countermodels generated by Seq-MMICM (and Seq-MMIT ) Wherever mentioned, the relation is the equivalence relation of the contexts χ 4 have an interesting property: If ` (Σ; M;Γ ⇒CM ∆ & S), Σ; M; Γ; ∆ in the conclusion of the rule. Similarly,  is the order of the then ` (Σ; ;Γ ⇒χ x : p, ∆) if and only if ∀M ∈ S. M |= underlying tree of M. M x : p. Thus, if we can produce a set of countermodels S by running without an actual goal (like x : p), then the set of infinite backwards derivations, as in Theorem III.12. atoms that are actually true are exactly those that are in the intersection of the valuation of all models in the set S. Further, Lemma III.10 (Correctness of CM). Let Σ, M, Γ and ∆ χ because the result applies to derivations in Seq-MMICM, it also be such that M is tree-like and no rule except (CM) applies χ χ applies to derivations in Seq-MMIT due to Lemma III.13 and backwards to Σ; M;Γ ⇒T ∆ & .... Then, Σ; M; Γ; ∆ is a the latter can be used to actually produce the set S. We call saturated history. this result comprehensiveness. Lemma III.11 (Tree-like M). Let M be tree-like. Then, the Suppose 0 0 0 0 χ 0 Theorem III.16 (Comprehensiveness). M in any sequent Σ ; M ;Γ ⇒T ∆ & ... appearing in χ χ χ ` (Σ; M;Γ ⇒CM ∆ & S). Then ` (Σ; M;Γ ⇒ x : p, ∆) iff a backwards search starting from Σ; M;Γ ⇒T ∆ & ... is ∀M ∈ S. M |= x : p. tree-like. F. Adding Seriality Note that the underlying tree of M in any sequent of a In this section we show that if χ has a SCR, then our backward proof search starting from a single formula consists χ of exactly those edges that are introduced in one of the rules method applies not only to the logic MMI (Corollary III.15), (→R) and (necR). but also to the logic which, in addition, forces seriality with respect to some of its relations NA. Seriality for index A is Theorem III.12 (Termination). The following hold: the condition ∀x.∃y.(xNAy). This corresponds to the axiom χ 1) Any backwards search in Seq-MMIT starting from a ¬(A nec ⊥), also called D in literature [5]. Note that seriality χ sequent Σ; M;Γ ⇒T ∆ with M tree-like terminates. does not fit our definition of χ because frame conditions in χ cannot contain existentials, so it cannot be handled in the representative, and (2) Our method also applies to any of method described so far. Consequently, we must modify our these logics combined with seriality from Section III-F due method slightly to include seriality as a frame condition. The to Theorem III.17. only new challenge is to control creation of worlds due to the Theorem III.18 (Decidability of Common Logics). The in- seriality condition during backwards search; for this we use an tuitionistic logics shown in Figure4 have the SCRs also approach similar to that for controlling the use of rules (→R) shown in that figure. Consequently, all these logics (and their and (necR). Proofs do not change significantly. combination with the seriality condition from Section III-F) Suppose we wish to make relations N for A ∈ J ⊆ I A are constructively decidable by our method. serial. We first add the following rule to our sequent calculus χ Seq-MMI (Figure1) for every A ∈ J : IV. CLASSICAL LOGIC χ Σ, x, y; M, xNAy;Γ ⇒ ∆ Our decision procedure can be modified to apply to classical χ D Σ, x; M;Γ ⇒ ∆ multi-modal logics as well. The overall approach of using Next, we change clause (1) of the definition of saturated SCRs and the structure of the calculi remains the same. history not to require closure under this new frame condition, However, because Kripke frames in classical logic do not which would cause infinite models, but instead new conditions require the preorder ≤, we must change the definition of based on 4: CM(Σ; M; Γ; ∆) (Definition III.4) that relies on ≤. This is not difficult: Instead of adding x ≤ y when x y, we add the (1) M is tree-like and saturated with respect to the 4 conditions (refl), (trans), (mon-N) and χ. In addition, relation xNAz when yNAz and x 4 y. Our TR describes in at least one of the following must hold for each x ∈ detail our method as it applies to classical logics, together will Σ and each index A ∈ J : (1a) There is a y ∈ Σ all relevant proofs. In classical logic, the possibility modality of index A can be defined as the DeMorgan dual of the such that xNAy ∈ M, or (1b) There is a y ∈ Σ such necessitation modality of the same index, so on classical logic, that y 6= x, y  x and x 4 y. our method applies to possibility modalities as well. With this new clause (1), we can show by induction on  that CM(Σ; ; Γ; ∆) is closed under seriality for A ∈ J , M V. RELATED WORK hence it is a model of our (modified) logic. Next, we add the following rule for every A ∈ J to the terminating calculus The applicability conditions of rules (→R) and (necR) χ in Figure3, based on the relation , are inspired by the Seq-MMIT and a corresponding rule without the applicability 4 χ work of Gasquet et al. [11] in which tableaux-based decision conditions to Seq-MMICM. procedures are given for classical uni-modal logics with the ∀y ∈ Σ. (xNAy 6∈ M) following frame conditions: Transitivity, reflexivity, symmetry, ∀y ∈ Σ.(y  x) ⇒ (y = x or x 46 y) Euclideanness, seriality and confluence. Our method is based χ on labeled sequent calculi and it applies to both classical and Σ, x, y; M, xNAy;Γ ⇒T ∆ & S χ D intuitionistic modal logics with any number of modalities. As Σ, x; M;Γ ⇒T ∆ & S a consequence, we had to develop new proof techniques to With these changes, our entire development works with only establish our results, particularly in the intuitionistic setting. minor changes to the proofs (interestingly, the proof of Our method of extracting countermodels is inspired by Lemma III.6 does not change). Negri’s proof of completeness of labeled sequent calculi for Theorem III.17 (Constructive decidability with seriality). Let uni-modal logic with respect to their Kripke semantics [18][19, D contain seriality conditions for some set of indices and let Chapter 11]. In that proof, it is shown how to extract (possibly the frame conditions χ have a SCR. Then the logic MMIχ,D infinite) countermodels from non-terminating branches of a is constructively decidable by our method. failed proof search taking the union of all M occurring along the branch. Here, instead, countermodels are built in the G. Application to Common Intuitionistic Logics context of a decision procedure and finite countermodels are In this Section, we list some common sets of frame con- built by adding additional edges based on the relation 4 and ditions with their SCRs, thus showing that the intuitionistic saturating with respect to the frame conditions. logics corresponding to each of them is constructively decid- Boretti and Negri [6] develop a countermodel producing able by our method. Unfortunately, SCRs are not modular: We decision procedure similar to ours for a Priorean linear time cannot combine the SCRs for frame conditions χ1 and χ2 to fixed point calculus (a variant of linear time temporal logic, get a SCR for the modal logic χ1 ∪ χ2. As a result, we must LTL), which also includes two rules like seriality. They explicitly construct a SCR for every modal logic of interest. also use a notion of saturation and construct countermodels Figure4 lists some common intuitionistic logics, their frame from histories. The main difference between this work and conditions χ, the corresponding axioms they admit and the [6] is that this work handles general frame conditions and, corresponding SCRs. In cases where the name of the logic additionally, intuitionistic connectives. Boretti and Negri also is not common, we have cited the source of the logic. We discuss previous tableaux-style approaches to the generation note two things: (1) This list is not exhaustive, but merely of countermodels for LTL, such as [23]. Logic Frame conditions χ Additional Axioms SCR K {} – R(A) = (≤) T ∀A, x. xNAx (A nec α) → α R(A) = (≤) K4 ∀A, x, y, z. ((xNAy) ∧ (yNAz)) → (xNAz) (A nec α) → (A nec A nec α) R(A) = (NA ∪ ≤) S4 Conditions of K4 and T Axioms of K4 and T R(A) = (NA ∪ ≤) I[9, 14] ∀A, B, x, y, z. ((xNB y) ∧ (yNAz)) → (xNAz) (A nec α) → (B nec A nec α) R(A) = ((∪B∈I NB ) ∪ ≤) unit [7] ∀A, x, y. (xNAy) → (x ≤ y) α → (A nec α) R(A) = (≤) – ∀x, y. (xNAy) → (xNB y) (B nec α) → (A nec α) R(A) = (≤)

Fig. 4. SCRs for some multi-modal intuitionistic logics. All these logics are constructively decidable by our method.

Countermodel producing sequent calculi, also known in paper, we have not been able to find a suitable filtration on the literature as “refutation calculi”, have been given for the obvious model whose worlds are equivalence classes of intuitionistic logic, bi-intuitionistic logic, and provability log- 4 ∩ <. In particular, it seems extremely difficult to satisfy ics [13, 15, 20] and in a way closer to the present paper’s the “back condition” of a filtration. approach in [21]. One of the peculiarities of our method in Alechina and Shakatov [2] present a general technique relation to previous work is that the countermodel construction to prove decidability of intuitionistic (multi)-modal logics is made part of the calculus itself. by embedding the relational definition of the semantics into Gasquet and Said [12] introduce a technique called dynamic Monadic Second Order Logic (MSO). As noted by the authors filtration to establish complexity bounds for the satisfiability themselves and unlike our method, this approach does not problem of classical layered modal logics (LMLs), i.e., “logics give good decision procedures since it proceeds by reduction characterized by semantic properties only stating the existence to satisfiability of formulas in SkS (monadic second-order of possible worlds that are in some sense further than the theory of trees with constant branching factor k), which has other”. Typically, such logics include confluence-like condi- non-elementary complexity. Moreover, the method applies to tions, but they do not include transitivity-like conditions. Our a logic only if its frame conditions can be expressed as work provides constructive decision procedures for a different acyclic closure conditions in MSO; this makes the method and disjoint class of logics to which the techniques in [12] inapplicable to logics with frame conditions mentioning more do not apply. In fact, with the exception of seriality, none of than one modal relation, e.g., the logic (I) from Figure4. the frame conditions considered in this paper fall in the class Negri [17] and Vigano´ [25] provide sound, complete and of LMLs. Moreover, because LMLs cannot be applied with terminating labeled sequent calculi for uni-modal classical transitivity conditions it is not clear whether the techniques logics K, T, K4 and S4. Out method extends these results in [12] are suitable in the intuitionistic setting. for a wider class of modal logics including axiom D, multiple Simpson [24] presents decision procedures based on labeled modalities and intuitionistic logics. sequent calculi for the intuitionistic uni-modal logics K, D, T Gore´ and Nguyen [14] present non-labeled tableaux calculi and B together with S5. He leaves open the decidability of for seven types of classical multi-modal logics to reason intuitionistic S4, K4 and KD4. Our method shows that the about epistemic states of agents in distributed systems. The necessitation-only fragments of all three logics are decidable, introduced tableaux require formulas of the logic to be first not only in the uni-modal case, but also in the multi-modal translated into a clausal form. We observe that one of the case and, further, that the logics have constructive decision axioms presented in [14] is positive introspection for beliefs procedures. and corresponds to axiom (I) in Figure4. To the best of Schmidt and Tishkovsky [22] present a general method our knowledge, [14] is the only work to provide decision for synthesizing sound and complete tableaux calculi given procedures for logics including axiom (I). a semantic filtration [5] for the underlying classical modal VI.CONCLUSIONAND FUTURE WORK logic. Although semantic filtrations are a powerful and general technique, their definition is not clear for many intuitionistic We have presented a labeled sequent calculus-based, con- and multi-modal logics, so our method handles several logics structive decision procedure for several multi-modal logics, that cannot be handled by Schmidt and Tishkovsky. To the best both intuitionistic and classical. Besides a novel construction of our knowledge, the only work on filtrations for intuitionistic of countermodels and a novel termination condition, we show, logics is limited to the uni-modal case [16] (the author of the by uniform method, that several standard multi-modal logics paper notes in the conclusion that generalizing to the multi- without possibility modalities, as well as several logics with modal case is not trivial). Filtration-based methods are also interactions between modalities, are decidable. We also show technically different from our syntactic approach. Whereas that our procedure has a novel comprehensiveness property. filtrations manipulate Kripke models, our method is purely We believe that our method can be extended to label- syntactic and SCRs only work with sets of formulas generated producing frame conditions more general than seriality. In during a specific backwards search. A consequence of this particular, it should be possible to extend the technique of difference is that, for any of the logics considered in this Section III-F to all layered modal logics of Gasquet and Said [12], which we discussed in SectionV. Second, we would [15] R. Gore´ and L. Postniece, “Combining derivations and like to either extend our method, or find a new one that can refutations for cut-free completeness in bi-intuitionistic handle possibility modalities in intuitionistic logic. logic,” Journal of Logic and Computation, vol. 20, no. 1, Another direction of research is to exploit the embedding pp. 233–260, 2010. of intuitionistic modal logics into classical bi-modal logics to [16] Y. Hasimoto, “Finite model property for some intuitionis- establish decidability results for the former, a direction pursued tic modal logics,” Bulletin of the Section of Logic, vol. 30, in [27] but with semantic rather than syntactic methods. no. 2, pp. 87–97, 2001. [17] S. Negri, “Proof analysis in modal logic,” Journal of REFERENCES Philosophical Logic, vol. 34, pp. 507–544, 2005. [1] N. Alechina, M. Mendler, V. de Paiva, and E. Ritter, [18] ——, “Kripke completeness revisited,” in Acts of Knowl- “Categorical and kripke semantics for constructive S4 edge – History, Philosophy and Logic, G. Primiero and modal logic,” in Proceedings of the Annual Conference S. Rahman, Eds. College Publications, 2009. of the European Association for Computer Science Logic [19] S. Negri and J. von Plato, Proof Analysis: A Contribution (CSL), 2001, pp. 292–307. to Hilbert’s Last Problem. Cambridge University Press, [2] N. Alechina and D. Shkatov, “A general method for prov- 2011. ing decidability of intuitionistic modal logics,” Journal [20] L. Pinto and R. Dyckhoff, “Loop-free construction of of Applied Logic, vol. 4, no. 3, pp. 219–230, 2006. counter-models for intuitionistic propositional logic,” in [3] M. Baldoni, “Normal multimodal logics with interaction Symposia Gaussiana, 1995, pp. 225–232. axioms,” in Labelled Deduction. Kluwer Academic [21] L. Pinto and T. Uustalu, “Proof search and counter-model Publishers, 2000, pp. 33–53. construction for bi-intuitionistic propositional logic with [4] M. Baldoni, L. Giordano, and A. Martelli, “A tableau labelled sequents,” in Proceedings of the 18th Interna- calculus for multimodal logics and some (un)decidability tional Conference on Automated Reasoning with Analytic results,” in Proceedings of the International Conference Tableaux and Related Methods (TABLEAUX), 2009, pp. on Automated Reasoning with Analytic Tableaux and 295–309. Related Methods (TABLEAUX), 1998, pp. 44–59. [22] R. A. Schmidt and D. Tishkovsky, “Automated synthesis [5] P. Blackburn, M. de Rijke, and Y. Venema, Modal Logic. of tableau calculi,” Logical Methods in Computer Sci- Cambridge University Press, 2001. ence, vol. 7, no. 2, pp. 1–32, 2011. [6] B. Boretti and S. Negri, “Decidability for Priorean linear [23] S. Schwendimann, “A new one-pass tableau calculus for time using a fixed-point labelled calculus,” in Proceed- PLTL,” in Proceedings of the International Conference ings of the 18th International Conference on Automated on Automated Reasoning with Analytic Tableaux and Reasoning with Analytic Tableaux and Related Methods Related Methods (TABLEAUX), 1998, pp. 277–291. (TABLEAUX), 2009, pp. 108–122. [24] A. K. Simpson, “The proof theory and semantics of [7] M. Fairtlough and M. Mendler, “Propositional lax logic,” intuitionistic modal logic,” Ph.D. dissertation, University Information and Computation, vol. 137, pp. 1–33, 1997. of Edinburgh, 1994. [8] D. M. Gabbay, “A general filtration method for modal [25] L. Vigano,` “A framework for non-classical logics,” Ph.D. logics,” Journal of Philosophical Logic, vol. 1, no. 1, dissertation, Universitat¨ des Saarlandes, 1997. pp. 29–34, 1972. [26] ——, Labelled Non-Classical Logics. Kluwer Academic [9] D. Garg, “Proof theory for authorization logic and its Publishers, 2000. application to a practical file system,” Ph.D. dissertation, [27] F. Wolter and M. Zakharyaschev, “Intuitionistic modal Carnegie Mellon University, 2009. logics,” in Logic and Foundations of Mathematics. [10] D. Garg, V. Genovese, and S. Negri, “Countermodels Kluwer Academic Publishers, 1999, pp. 227–238. from sequent calculi in multi-modal logics,” 2012, Tech- nical report online at http://www.mpi-sws.org/∼dg. [11] O. Gasquet, A. Herzig, and M. Sahade, “Terminating modal tableaux with simple completeness proof,” in Advances in Modal Logic, 2006, pp. 167–186. [12] O. Gasquet and B. Said, “Tableaux with dynamic filtra- tion for layered modal logics,” in Proceedings of the 16th International Conference on Automated Reasoning with Analytic Tableaux and Related Methods (TABLEAUX), 2007, pp. 107–118. [13] V. Goranko, “Refutation systems in modal logic,” Studia Logica, vol. 53, no. 2, pp. 299–324, 1994. [14] R. Gore´ and L. A. Nguyen, “Clausal tableaux for multimodal logics of belief,” Fundamenta Informaticae, vol. 94, no. 1, pp. 21–40, 2009.