Axiomatizing propositional dependence logics

Katsuhiko Sano

and Jonni Virtema

1 Japan Advanced Institute of Science and Technology, Japan 2 Leibniz Universität Hannover, Germany

3 University of Tampere, Finland

{katsuhiko.sano,jonni.virtema}@gmail.com

Abstract

We give sound and complete Hilbert-style axiomatizations for propositional dependence logic (PD), modal dependence logic (MDL), and extended modal dependence logic (EMDL) by ex- tending existing axiomatizations for propositional logic and modal logic. In addition, we give novel labeled tableau calculi for PD, MDL, and EMDL. We prove soundness, completeness and termination for each of the labeled calculi.

1998 ACM Subject Classification F.4.1 Mathematical logic

Keywords and phrases propositional dependence logic, modal dependence logic, axiomatization, tableau calculus

Digital Object Identifier 10.4230/LIPIcs.CSL.2015.292

1 Introduction

Functional dependences occur everywhere in science, e.g., in descriptions of discrete systems, in database theory, social choice theory, mathematics, and physics. Modal logic is an important formalism utilized in the research of numerous disciplines including many of the fields mentioned above. With the aim to express functional dependences in the framework of logic Väänänen [9] introduceddependence logic. Dependence logic extends first-order logic with novel atomic formulae calleddependence atoms. The intuitive meaning of the first-order dependence atom = (t₁, . . . , t_n) is that the value of the termt_n is functionally determined by the values of the termst1, . . . ,tn−1. With the aim to express functional dependences in the framework of modal logic, Väänänen [10] introducedmodal dependence logic (MDL). Modal dependence logic extends modal logic withpropositional dependence atoms. A propositional dependence atom dep(p1, . . . , pn, q) intuitively states that the truth value of the proposition qis functionally determined by the truth values of the propositionsp₁, . . . , p_n. It was soon realized thatMDLlacks the ability to express temporal dependencies; there is no mechanism inMDLto express dependencies that occur between different points of the model. This is due to the restriction that only proposition symbols are allowed in the dependence atoms of modal dependence logic. To overcome this defect Ebbing et al. [1] introducedextended modal dependence logic(EMDL) by extending the scope of dependence atoms to arbitrary modal formulae. Dependence atoms in extended modal dependence logic are of the form dep(ϕ₁, . . . ϕ_n, ψ), whereϕ₁, . . . , ϕ_n, ψ are formulae of modal logic.

∗ The work of the first author was partially supported by JSPS KAKENHI Grant-in-Aid for Young Scientists (B) Grant Number 15K21025 and by JSPS Core-to-Core Program (A. Advanced Research Networks). The work of the second author was supported by grant 266260 of the Academy of Finland, and by Jenny and Antti Wihuri Foundation.

© Katsuhiko Sano and Jonni Virtema;

licensed under Creative Commons License CC-BY

24th EACSL Annual Conference on Computer Science Logic (CSL 2015).

Editor: Stephan Kreutzer; pp. 292–307

Leibniz International Proceedings in Informatics

Schloss Dagstuhl – Leibniz-Zentrum für Informatik, Dagstuhl Publishing, Germany

In recent years the research around modal dependence logic has been active. The focus has been in the computational complexity and in the expressive powers of related formalisms.

Sevenster [8] proved that the satisfiability problem for modal dependence logic isNEXPTIME- complete, whereas Ebbing and Lohmann [2] showed that the related model checking problem isNP-complete. Ebbing et al. [1] extended these results to handle alsoEMDL. Subsequently Virtema [11] showed that the validity problems for MDLandEMDL areNEXPTIME-hard and contained inNEXPTIME^NP. Moreover he showed that the corresponding problem for the propositional fragmentPD(see Section 2.1 for a definition) ofMDLisNEXPTIME-complete.

Hella et al. [4] gave a van Benthem–style characterization of the expressive power of EMDL via the so-called team k-bisimulation. In the article it was also shown that the expressive powers of EMDL and ML(6) (modal logic extended with intuitionistic disjunction) coincide. More recently Kontinen et al. (in the manuscript [6]) gave another van Benthem–style characterization for the expressive power of the so-calledmodal team logic.

Moreover, in the manuscript [7], Sano and Virtema gave a Goldblatt–Thomason theorem forMDLandEMDL. They also showed that with respect to frame definabilityMDLand EMDL coincide with a fragment of modal logic extended with the universal modality in which the universal modality occurs only positively. These characterization truly demonstrate the naturality of the related languages.

In this paper we give sound and complete axiomatizations for variants of propositional and modal dependence logics (PD,PL(>),MDL,EMDL, andML(>)). We give Hilbert-style axiomatizations for these logics by extending existing axiomatizations for propositional logic and modal logic. In addition, we give novel labeled tableau calculi for these logics. This paper is one of the first articles on proof theory of propositional and modal dependence logics.

The only other work known by the authors of this article is the PhD thesis of Fan Yang [12] and the subsequent manuscript [13]. Among other things, in her thesis, Yang presents axiomatizations of variants of propositional dependence logic based on natural deduction.

Our Hilbert style axiomatization of PD coincides in essence with the natural deduction system given by Yang. However our axiomatization avoids the complexity of the system of Yang by concentrating on the proof-theoretic essence of the axiomatization. Provided that a Hilbert-style axiomatization for the negation normal form fragment of propositional logic is given, we specifyoneinference rule which gives us an axiomatization ofPD.

The article is structured as follows. In Section 2 we introduce the required notions and definitions. In Section 3 we give Hilbert-style axiomatizations for propositional and modal dependence logics. In Section 4 we present labeled tableau calculi for these logics.

2 Preliminaries

The syntax of propositional logic (PL) and modal logic (ML) could be defined in any standard way. However, when we consider extensions ofPLandMLby dependence atoms, it is useful to assume that all formulae are innegation normal form, i.e., negations occur only in front of atomic propositions. Thus we will define the syntax ofPLandMLin negation normal form. Whenϕis a formula ofPLorML, we denote by ϕ^⊥ the equivalent formula that is obtained from ¬ϕby pushing all negations to the atomic level. Furthermore, we defineϕ^>:=ϕ. When~ais a tuple of symbols of lengthk, we denote bya_j thejth element of

~a,j≤k. Whenϕis a formula,|ϕ|denotes the number of symbols inϕexcluding negations and brackets. WhenAis a set|A|denotes the number of elements inA. Whenf :A→Bis a function andC⊆A, we definef[C] :={f(a)|a∈C}.

2.1 Propositional logic with team semantics

LetPROP={zi|i∈N}denote the set of exactly allpropositional variables, i.e.,proposition symbols. We mainly use metavariables p, q, p1, p2, q1, q2, etc., in order to refer to the variable symbols inPROP. LetDbe a finite, possibly empty, subset ofPROP. A functions:D→ {0,1}is called anassignment. A setX of assignmentss:D→ {0,1}is called apropositional team. The setD is thedomain ofX. Note that the empty team∅does not have a unique domain; any subset ofPROPis a domain of the empty team. By{0,1}^D, we denote the set of all assignmentss:D→ {0,1}.

Let Φ be a set of proposition symbols. The set of formulae for propostional logicPL(Φ) is generated by the grammar: ϕ::=p| ¬p|(ϕ∧ϕ)|(ϕ∨ϕ), wherep∈Φ.

By|=PL, we denote the ordinary satisfaction relation of propositional logic defined via assignments. Next we define the team semantics of propositional logic.

IDefinition 1. Let Φ be a set of atomic propositions and let X be a propositional team.

The satisfaction relationX |=ϕforPL(Φ) is defined as follows. Note that, we always assume that the proposition symbols that occur inϕare also in the domain ofX.

X|=p ⇔ ∀s∈X :s(p) = 1.

X|=¬p ⇔ ∀s∈X :s(p) = 0.

X |= (ϕ∧ψ) ⇔ X |=ϕandX |=ψ.

X |= (ϕ∨ψ) ⇔ Y |=ϕandZ|=ψ, for someY, Z such thatY ∪Z=X.

IProposition 2([8]). Let ϕbe a formula of propositional logic andX a propositional team.

Then X |=ϕ ⇔ ∀s∈ X : s|=PL ϕ. In particular the equivalence {s} |=ϕ ⇔ s|=PL ϕ holds for every assignments.

The syntax of propositional logic with intuitionistic disjunction PL(>)(Φ) is obtained by extending the syntax ofPL(Φ) by the grammar ruleϕ::= (ϕ>ϕ).The syntax ofpropositional dependence logicPD(Φ) is obtained by extending the syntax ofPL(Φ) by the grammar rules ϕ::= dep(p1, . . . , p_n, q), wherep₁, . . . , p_n, q ∈Φ andn∈N. The intuitive meaning of the propositional dependence atomdep(p1, . . . , pn, q) is that the truth value of the proposition symbolqis completely determined by the truth values of the proposition symbolsp₁, . . . , p_n. We define the semantics for the intuitionistic disjunction and the propositional dependence atoms as follows:

X|= (ϕ>ψ) ⇔ X |=ϕorX |=ψ

X |= dep(p1, . . . , pn, q) ⇔ ∀s, t∈X:s(p1) =t(p1), . . . , s(pn) =t(pn) implies thats(q) =t(q).

The next proposition is very useful. The proof is very easy and analogous to the corresponding proof for first-order dependence logic [9].

IProposition 3(Downwards closure). Let ϕbe a formula of PL(>)orPD and letY ⊆X be propositional teams. Then X|=ϕimpliesY |=ϕ.

Note that, by downwards closure,X|= (ϕ∨ψ) iffY |=ϕandX\Y |=ψfor someY ⊆X.

2.2 Modal logics

In order to keep the notation light, we restrict our attention to mono-modal logic, i.e., to modal logic with just the modal operators♦and. However this is not really a restriction,

since the definitions, results, and proofs of this article generalize, in a straightforward manner, to facilitate any number of modalities.

Let Φ be a set of atomic propositions. The set of formulae for modal logic ML(Φ) is generated by the grammar: ϕ::=p| ¬p|(ϕ∧ϕ)|(ϕ∨ϕ)|♦ϕ|ϕ, where p∈Φ.

Note that, since negations are allowed only in front of proposition symbols,and♦are not interdefinable. The syntax ofmodal logic with intuitionistic disjunction ML(6)(Φ) is obtained by extending the syntax ofML(Φ) by the grammar ruleϕ::= (ϕ6ϕ).

Theteam semantics for modal logicis defined viaKripke modelsandteams. In the context of modal logic, teams are subsets of the domain of the model.

IDefinition 4. Let Φ be a set of proposition symbols. AKripke model K over Φ is a tuple K = (W, R, V), whereW is a nonempty set ofworlds, R⊆W×W is a binary relation, and V: Φ→ P(W) is avaluation. A subsetT ofW is called ateam of K. Furthermore define

R[T] :={w∈W | ∃v∈T s.t. vRw}, R⁻¹[T] :={w∈W | ∃v∈T s.t. wRv}.

For teamsT, S ⊆W, we write T[R]S if S ⊆R[T] andT ⊆R⁻¹[S]. Thus, T[R]S holds if and only if for every w∈T there exists some v ∈S such that wRv, and for everyv ∈S there exists somew∈T such thatwRv.

We are now ready to define the team semantics for modal logic and modal logic with intuitionistic disjunction.

IDefinition 5. Let Φ be a set of atomic propositions, K a Kripke model andT a team of K. The satisfaction relation K, T |=ϕforML(Φ) is defined as follows.

K, T |=p ⇔ w∈V(p) for everyw∈T. K, T |=¬p ⇔ w6∈V(p) for everyw∈T. K, T |= (ϕ∧ψ) ⇔ K, T |=ϕand K, T |=ψ.

K, T |= (ϕ∨ψ) ⇔ K, T1|=ϕand K, T2|=ψ for some T1 andT2

such thatT1∪T2=T .

K, T |=♦ϕ ⇔ K, T⁰|=ϕfor someT⁰ such thatT[R]T⁰. K, T |=ϕ ⇔ K, T⁰|=ϕ, where T⁰ =R[T].

ForML(6) we have the following additional clause:

K, T |= (ϕ6ψ) ⇔ K, T |=ϕor K, T |=ψ.

By|=ML, we denote the ordinary satisfaction relation of modal logic defined via pointed Kripke models.

IProposition 6 ([8]). Let ϕbe an ML-formula, K be a Kripke model, and T be a team of K. ThenK, T |=ϕ ⇔ ∀w∈T : K, w|=MLϕ In particular, for every pointw ofK, the equivalenceK,{w} |=ϕ ⇔ K, w|=_MLϕholds.

The syntax formodal dependence logic MDL(Φ) is obtained by extending the syntax of ML(Φ) by the rules ϕ ::= dep(p1, . . . , p_n, q), where p₁, . . . , p_n, q ∈ Φ and n ∈ N, for propositional dependence atoms. The syntax forextended modal dependence logic EMDL(Φ) is obtained by extending the syntax ofML(Φ) by the rulesϕ::= dep(ϕ1, . . . , ϕn, ψ), where ϕ1, . . . , ϕn, ψ∈ ML(Φ) andn∈N, formodal dependence atoms. The intuitive meaning of

the modal dependence atom dep(ϕ1, . . . , ϕn, ψ) is that the truth value of the formulaψ is completely determined by the truth values of the formulaeϕ1, . . . , ϕn. Formally:

K, T |= dep(ϕ1, . . . , ϕn, ψ) ⇔ ∀w, v∈T : ^

1≤i≤n

(K,{w} |=ϕi⇔K,{v} |=ϕi) implies (K,{w} |=ψ⇔K,{v} |=ψ).

The following result for MDL and ML(6) is due to [10] and [3], respectively. For EMDLit follows via a translation fromEMDLinto ML(6), see [1].

IProposition 7(Downwards closure). Let ϕbe a formula ofML(6)orEMDL, letK be a Kripke model and letS⊆T be teams ofK. ThenK, T |=ϕimpliesK, S|=ϕ.

2.3 Equivalence and validity in team semantics

We say that formulaeϕandψof PL(>)(Φ) orPD(Φ) are equivalent and writeϕ≡ψ, if the equivalenceX |=ϕ ⇔ X |=ψholds for every propositional teamX. Likewise, we say that formulaeϕand ψofML(>)(Φ) orEMDL(Φ) areequivalent and writeϕ≡ψ, if the equivalenceK, T |=ϕ ⇔ K, T |=ψ holds for every Kripke modelKand team T ofK.

A formulaϕofPL(>)(Φ) orPD(Φ) is said to bevalid, ifX |=ϕholds for every teamX such that the proposition symbols that occur inϕare in the domain ofX. Analogously, a formulaψofEMDL(Φ) orML(6)(Φ) is said to bevalid, if K, T |=ϕholds for every Kripke model K (such that the proposition symbols inϕare mapped by the valuation of K) and every teamT of K. Whenϕis a valid formula ofL, we write|=_L ϕ.

The following proposition shown in [11, 12] will later prove to be very useful.

IProposition 8(>-disjunction property). LetL ∈ {PL(>),ML(>)}. For everyϕ, ψ inL,

|=_L(ϕ>ψ)iff|=_Lϕor |=_Lψ.

3 Extending axiomatizations of PL and ML

In this section we show how to extend sound and complete axiomatizations forPL and ML into sound and complete axiomatizations forPL(6) and ML(6), respectively. We use the fact that bothPL(6) andML(6) have the >-disjunction property. In addition, we obtain axiomatizations forPD,MDL, andEMDL. The axiomatizations are based on compositional translations fromPD intoPL(6), and fromMDLandEMDLintoML(6).

3.1 Axiomatizations for PL( > ) and ML( > )

In the definition below, we treat different occurrences of the same formulae as distinct entities.

IDefinition 9. Letϕbe a formula ofPL(>) orML(>). Let SubOcc(ϕ) denote theset of exactly all occurrences of subformulas of ϕ. Define

SubOcc

>(ϕ) :={(ψ>θ)|(ψ>θ)∈SubOcc(ϕ)}.

We call a functionf : SubOcc_>(ϕ)→SubOcc(ϕ) a>-selection function forϕiff (ψ>θ)

∈ {ψ, θ}, for every (ψ>θ)∈SubOcc

>(ϕ). Iff is a>-selection function forϕ, thenϕ^f denotes the formula that is obtained fromϕby replacing simultaneously each (ψ>θ)∈SubOcc_>(ϕ) byf(ψ>θ).

Note that ifϕ∈ PL(>)(Φ), ψ∈ ML(>)(Ψ),f is a >-selection function for ϕ, and g is a

>-selection function forψ, thenϕ^f ∈ PL(Φ) and ψ^g∈ ML(Ψ).

IProposition 10([11]). Letϕ be a formula ofPL(>)or ML(>), and let F be the set of exactly all>-selection functions forϕ. Then,ϕ≡

6

LetH_PLandH_ML denote sound and complete axiomatizations of the negation normal form fragments ofPLandML, respectively. For a logicL, anL-context is a formula of the logicLextended with the grammar ruleϕ::=∗. Byϕ(ψ /∗) we denote the formula that is obtained formϕby uniformly substituting each occurrence of ∗inϕbyψ. We are now ready to define the axiomatizations forPL(>) andML(>). We usePL(>)- andML(>)-contexts in the following rules:

ϕ(ψ_i/∗)

(I>i)

ϕ (ψ₁>ψ₂)/ ∗ i∈ {1,2}.

Let H_PL(

>)(H_ML(

>), resp.) be the calculus H_PL (H_ML, resp.) extended with the rules (I>1) and (I>2). In the calculiH_PL(

>) andH_ML(

>), the axioms and inference rules of HPLandHML may only be applied to formulae ofPLand ML(i.e, to formulae without

>), respectively.

ITheorem 11. H_PL(

>)andH_ML(

>) are sound and complete.

Proof. We will proof the soundness and completeness forH_PL(

>). The case for H_ML(

>)

is completely analogous. Note first that from Proposition 2 it follows directly thatHPLis complete forPLalso in the context of team semantics.

For soundness, it suffices to show that the rule (I>1) preserves validity. The case for (I>2) is symmetric. Let ϕ be a PL(>)-context and let ψ1 andψ2 be PL(>)-formulae.

Assume that γ₁ := ϕ(ψ₁/∗) is valid. We will show that then γ₂ := ϕ (ψ₁>ψ₂)/ ∗ is valid. LetF andGbe the sets of exactly all>-selection functions forγ1 andγ2, respectively.

By Proposition 10, γ₁ ≡

6

f

1 and γ₂ ≡

6

g

2. Since γ₁ is valid, it follows by Proposition 8, thatγ₁^f0 is valid for somef⁰ ∈F. Since clearly, for everyf ∈F, there exists someg∈Gsuch thatγ^f₁ =γ^g₂, it follows that there exists someg⁰∈Gsuch thatγ₂^g0 is valid.

Thusγ2 is valid.

In order to prove completeness, assume that a PL(6)-formula ϕ is valid. Let F be the set of exactly all>-selection functions forϕ. By Propositions 10 and 8, there exists a functionf ∈F such that thePL-formulaϕ^f is valid. SinceH_PLis complete andH_PL(

6)

extendsH_PL,ϕ^f is provable also inH_PL(

6). Clearly by using the rules (I>1) and (I>2) repetitively toϕ^f, we eventually obtain ϕ. Thus we conclude thatH_PL(

>) is complete. J

3.2 Axiomatizations for PD, MDL, and EMDL

The following equivalence was observed by Väänänen in [10]:

dep(p1, . . . , pn, q)≡ _

a1,...,an∈{⊥,>}

^ p^a₁¹, . . . , p^a_nⁿ,(q6q^⊥) . (1)

Ebbing et al. ([1]) extended this observation of Väänänen into the following equivalence concerningEMDL:

dep(ϕ₁, . . . , ϕ_n, ψ)≡ _

a1,...,an∈{⊥,>}

^ ϕ^a₁¹, . . . , ϕ^a_nⁿ,(ψ6ψ^⊥) . (2)

These equivalences demonstrate the existence of compositional translations fromPD into PL(>), and from MDLandEMDLinto ML(>), respectively.

We will use the insight that rises from combining the above equivalences with Propositions 8 and 10 in order to construct axiomatizations forPD, MDL, andEMDL, respectively.

Recall that when~ais a finite tuple of symbols, we useaj to denote thejth member of~a. For each natural numbern∈Nand function f :{⊥,>}ⁿ→ {>,⊥}, we have the following rules:

ϕ W

~a∈{⊥,>}ⁿ

V p^a₁¹, . . . , p^a_nⁿ, q^f(~a) /∗

PLdep(f) ϕ dep(p1, . . . , pn, q)/∗

ϕ W

~ a∈{⊥,>}ⁿ

V ϕ^a₁¹, . . . , ϕ^a_nⁿ, ψ^f(~a) /∗

MLdep(f)

† ϕ dep(ϕ1, . . . , ϕn, ψ)/∗

where †means that ϕ₁, . . . , ϕ_n, ψ are required to be modal formulae.¹ Define PLdep :=

{ PLdep(f)

|f : {⊥,>}ⁿ → {>,⊥}, wheren ∈N} andMLdep:= { MLdep(f)

| f : {⊥,>}ⁿ → {>,⊥}, wheren ∈N}. Let H_PD andH_MDL be the extensions of the calculi HPLandHML by the rules ofPLdep, respectively. LetHEMDL be the extension ofHML

by the rules ofMLdep.

The proof of the following theorem is analogous to that of Theorem 11.

ITheorem 12. Let L ∈ {PD,MDL,EMDL},HL is sound and complete.

4 Labeled tableaus for propositional dependence logics

The calculi presented in Section 3 have a few clear shortcomings. Foremost, the calculi miss the team semantic nature of these logics. Thus the calculi are in some parts quite complicated. Especially this is the case for the rulesPLdepandMLdep. This seems to be the case also for any concrete implementations of the axiomatizationsH_PLandH_ML of the negation normal form fragments ofPLandML, respectively.

In this section we give axiomatizations forPD,MDL, andEMDLthat do not have the shortcomings of the calculi of Section 3. The proof rules of the labeled tableau calculi that are given in this section have a natural and simple correspondence with the truth definitions of connectives and modalities in team semantics.

4.1 Checking validity via small teams

The following result (observed, e.g., in [11]) follows directly from the fact thatPL(>) and PDare downwards closed, i.e., from Proposition 3.

IProposition 13. Letϕbe a formula of PL(>) orPD and letD be the set of exactly all proposition symbols that occur inϕ. Then ϕis valid iff {0,1}^D|=ϕ.

Adapting a notion that was introduced by Jarmo Kontinen in [5] for first-order dependence logic, we say that anML(>)- orEMDL-formulaϕisn-coherent if the condition

K, T |=ϕ ⇔ K, T⁰|=ϕfor allT⁰⊆T such that|T⁰| ≤n holds for all Kripke models K and teamsT of K.

1 In the special case whereϕis∗in the rule PLdep(f)

, the obtained rule coincides with the rule of Dependence Atom Introduction in [12, p.75].

The following result forML(>) was shown in [4]. The result for EMDLfollows from the result forML(>) essentially via the following equivalence.

dep(ϕ₁, . . . , ϕ_n, ψ)≡ _

a₁,...,a_n∈{⊥,>}

^ ϕ^a₁¹, . . . , ϕ^a_nⁿ,(ψ6ψ^⊥) .

For ϕ ∈ ML(>), we define Rank

>(ϕ) to be the number of intuitionistic disjunctions in ϕ. Forψ∈ EMDL, we define Rank_>(ψ) to be the number of intuitionistic disjunctions in theML(>) formula obtained by using the above equivalence. Note that Rank_>(ϕ)≤ |ϕ|, whereas Rank

>(ψ)≤2^|ψ|.

ITheorem 14. Every formulaϕof ML(>)or EMDLis 2^Rank>(ϕ)-coherent.

The following result follows directly from Theorem 14.

ICorollary 15. Let ϕbe a formula ofML(>)orEMDL. The following holds:

ϕ is valid iff K, T |=ϕfor every Kripke model Kand every team T ofK such that |T| ≤2^Rank>(ϕ).

4.2 Tableau Calculi for PL, PL( > ), and PD

We will now present labeled tableau calculi for PL,PL(>), andPD. In Section 4.3 we will extend these calculi to deal withML,MDL, andEMDL.

Any finite, possibly empty, subset α ⊆ N is called a label. We mainly use symbols α, β, α₁, α₂, β₁, β₂, etc, in order to refer to labels and symbolsi, j, i₁, i₂, j₁, j₂, etc, in order to refer to natural numbers. Our tableau calculi are labeled, meaning that the formulae occurring in the tableau rules arelabeled formulae, i.e., of the formα:ϕ, whereαa label andϕis a formula of some logicL. Labels correspond to teams and the elements of labels, i.e., natural numbers, correspond to points in a model. The intendedtop downreading of the labeled formulaα:ϕis thatαdenotes some team thatfalsifiesϕ. A tableau in these calculi is just a well-founded, finitely branching tree in which each node is labeled by a labeled formula, and the edges represent applications of the tableau rules. The tableau rules needed for axiomatizingPL,PL(>), andPDare given in Figure 1.

In the construction of tableaus, we impose a rule that a labeled formula is never added to a tableau branch in which it already occurs. Asaturated branch is a tableau branch in which no rules can be applied or the application of the rules have no effect on the branch. A saturated tableau is a tableau in which every branch is saturated. A branch of a tableau is calledclosed if it contains at least one of the following:

1. Both {i}:pand{i}:¬p, for some proposition symbolpand natural numberi∈N. 2. ∅:ϕ, for some formulaϕ.

3. {i}: dep(p₁, . . . , pn, q), for some proposition symbolsp₁, . . . , pn, q andi, n∈N.

If a branch of a tableau is not closed it is called open. A tableau is called closed if every branch of the tableau is closed. A tableau is calledopen if at least one branch in the tableau is open.

Let T_PL denote the calculi consisting of the rules (P rop), (¬P rop), (∧), and (∨) of Figure 1. Let T_PL(

>) denote the extension ofT_PL by the rule (>) of Figure 1, and T_PD denote the extension ofTPLby the rules (Split) and (PLdep) of Figure 1.

Let ϕbe a formula ofL(Φ)∈ {PL(Φ),PL(>)(Φ),PD(Φ)} andk:= min(|Φ|,Rank

>(ϕ)).

We say that a tableauT is atableau forϕif the root ofT is{1, . . . ,2^k}:ϕandT is obtained

{i1, . . . , ik}:p

(P rop) {i1} :p |. . . | {ik} :p

{i1, . . . , ik}: ¬p

(¬P rop) {i1} :¬p| . . . | {ik}: ¬p

α : (ϕ∧ψ) α :ϕ |α : ψ (∧) α : (ϕ∨ψ)

(∨) whereβ⊆α β :ϕ| α\β : ψ

α : (ϕ_>ψ) (_>) α :ϕ α : ψ α : dep(p1, . . . , pn, q)

(Split)†

α1 : dep(p1, . . . , pn, q) |. . . | α_k : dep(p1, . . . , pn, q)

†: α1, . . . , αk are exactly all subsets ofαof cardinality 2.

{i1, i2} : dep(p1, . . . , pn, q)

(PLdep)‡

{i1}: p^g₁¹⁽¹⁾ | . . . | {i1} :p^g₁^k(1) {i2}: p^g₁¹⁽¹⁾ | . . . | {i2} :p^g₁^k(1)

.. .

.. .

.. . {i1}:p^g_n¹⁽ⁿ⁾ | . . . | {i1} :p^g_n^k(n) {i2}:p^g_n¹⁽ⁿ⁾ | . . . | {i2} :p^g_n^k(n) {i1, i2}: q |. . . | {i1, i2}: q {i1, i2}: ¬q |. . . | {i1, i2}: ¬q

‡: g1, . . . gkare exactly all functions with domain{1, . . . , n}and co-domain{>,⊥}.

Figure 1Tableau Rules forTPL,TPL(>), andTPD.

by applying the rules ofT_L. We say thatϕisprovableinT_Land write`TLϕif there exists a closed tableau forϕ.

IExample 16. We show that the PD-formula dep(p, p) is provable T_PD. Figure 2 is an illustration of a closedTPD-tableau for dep(p, p).

Since the number of proposition symbols that occur in dep(p, p) is one, the root of the tableau is{1,2}: dep(p, p). We first apply the rule (PLdep) to{1,2}: dep(p, p) and branch into two branches as depicted in Figure 2. In the left (right) branch we apply the rule (¬P rop) to{1,2}:¬p((P rop) to{1,2}:p). Consequently, each branch of the tableau becomes closed due to the labeled formulae of the type{i}:pand{i}:¬p,i∈ {1,2}. Therefore, dep(p, p) is a theorem ofT_PD.

ITheorem 17(Termination ofT_PL,T_PL(

>), andT_PD).LetLbe a logic in{PL,PL(>),PD}

andϕ anL-formula. Every tableau forϕin TL is finite.

Proof. LetT be a tableau for ϕ. By definition, the root ofT is α:ϕ, for some finite α.

Clearly every application of the tableau rules either decreases the size of the label or the length of the formula. Note also that the rule (∨) can be applied to any β : ψ∈ T only

finitely many times. ThusT must be finite. J

ILemma 18. If there exists a saturated open branch forϕthen ϕis not valid.

Proof. LetBbe a saturated open branch forϕand let Φ be the set of proposition symbols that occur inϕ. Letα:ϕdenote the root of the branchB. It is easy to check that ifβ :ψis a labeled formula inBthenβ ⊆α. For eachi∈αwe define an assignmentsi : Φ→ {0,1}

such that s_i(p) :=

(1 if the labeled formula{i}:¬poccurs in the branchB, 0 otherwise.

{1,2}: dep(p, p)

{1}:p {2}:p {1,2}:p {1,2}:¬p

{1}:¬p

×

{2}:¬p

×

{1}:¬p {2}:¬p {1,2}:p {1,2}:¬p {1}:p

×

{2}:p

×

Figure 2A tableau showing that thePD-formula dep(p, p) is provable inTPD.

It is easy to show by induction that if a labeled formulaβ:ψoccurs in the branchBthen Xβ 6|=ψ, where Xβ={si|i∈β}. Thusϕis not valid. J I Theorem 19 (Completeness of T_PL, T_PL(

>), and T_PD). Let L be any of the logics in {PL,PL(>),PD}. The calculus TL is complete.

Proof. FixL ∈ {PL,PL(>),PD}. Assume6`T_L ϕ. Thus every tableau forϕis open. From Theorem 17 it follows that there exists a saturated open tableau forϕ. Thus there exists a saturated open branch forϕ. Thus, by Lemma 18,6|=_Lϕ. J I Definition 20. Let B be a tableau branch and Index(B) the set of exactly all natural numbers that occur inB. We say thatBisfaithfulto a propositional teamX by a mapping f : Index(B)→X if, for allα:ϕ∈ B,f[α]6|=ϕ.

ILemma 21. LetL be a logic in {PL,PL(>),PD}. Ifϕ∈ Lis not valid then there is an open saturated branch in every saturated tableau ofϕin TL.

Proof. Assume6|=_L ϕ. Let Φ be the set of exactly all proposition symbols that occur inϕ.

By Proposition 13,{0,1}^Φ6|=ϕ. Putα:={1, . . . ,2^|Φ|}and fix a bijection f :α→ {0,1}^Φ. LetT be an arbitrary saturated tableau forϕ. By Theorem 17,T is finite and, by definition, the root ofT isα:ϕ. Note that Index(B) = α, for every branchBwith the rootα:ϕ. We will show that there is an open saturated branch inT.

First, we establish thatB₀ :={α:ϕ}is faithful to{0,1}^Φbyf. But, this is easy since f[α] = {0,1}^Φ. Second, assume that we have constructed a branch Bn such that Bn is faithful to{0,1}^Φ byf. We will show that at least one extension ofBn by rules of T_L is faithful to {0,1}^Φby f. Here we are concerned with the rule of (∨) alone. Assume that, fromβ1: (ψ1∨ψ2)∈ Bn and the rule of (∨), we obtain two extensions{β2:ψ1} ∪ Bn and {β1\β₂:ψ₂} ∪ Bn forβ₂⊆β₁. Our goal is to show that one of the extensions is faithful to {0,1}^Φ by f. By assumption, we obtain f[β1] 6|= (ψ1∨ψ2). By the semantic clause for ∨, f[β₂] 6|=ψ₁ or f[β₁]\f[β₂] 6|= ψ₂. Since f[β₁]\f[β₂] ⊆f[β₁\β₂], it follows from downwards closure thatf[β2]6|=ψ1 orf[β1\β2]6|=ψ2. This implies that at least one of the two extensions is faithful to{0,1}^Φbyf. We choose one of the faithful extensions as Bn+1.

SinceT is finite and saturated,Bj is a saturated branch inT for somej ∈N. Moreover,

sinceBj is faithful to{0,1}^Φbyf,Bj is open. J

I Theorem 22 (Soundness of T_PL, T_PL(

>), and T_PD). Let L be any of the logics in {PL,PL(>),PD}. The calculus TL is sound.

i1Rj1

.. . inRjn

{i1, . . . , in}:♦ϕ (♦) {j1, . . . , jn}:ϕ

α : ϕ

()†

f1(1)Ri1 |. . . | f_k(1)Ri1

.. .

.. .

.. . f1(t)Rit |. . . | f_k(t)Rit

{i1, . . . it}: ϕ| . . . | {i1, . . . it}: ϕ {i1, i2} : dep(ϕ1, . . . , ϕn, ψ)

(MLdep)‡

{i1} :ϕ^h₁¹⁽¹⁾ |. . . | {i1}:ϕ^h₁^k(1) {i2} :ϕ^h₁¹⁽¹⁾ |. . . | {i2}:ϕ^h₁^k(1)

.. .

.. .

.. . {i1}: ϕ^hn¹⁽ⁿ⁾ |. . . | {i1}:ϕ^hn^k(n)

{i2}: ϕ^hn¹⁽ⁿ⁾ |. . . | {i2}: ϕ^hn¹⁽ⁿ⁾

{i1, i2} :ψ |. . . | {i1, i2} :ψ {i1, i2}: ψ^⊥ |. . . | {i1, i2} :ψ^⊥

†: t= 2^Rank>(ϕ)andf1, . . . , fkdenote exactly all functions with domain{1, . . . , t}and co-domainα, and i1, . . . , it are fresh and distinct.

‡:h1, . . . h_kdenotes all the functions with domain{1, . . . , n}and co-domain{>,⊥}.

Figure 3Additional Tableau Rules forTML,TML(>),TMDL andTEMDL.

Proof. FixL ∈ {PL,PL(>),PD}. Assume that 6|=_L ϕ. By Lemma 21, there is an open saturated branch in every saturated tableau ofϕinTL. Therefore, and since, by Theorem 17, every tableau ofϕinT_L is finite, there does not exists any closed tableau forϕinT_L.

Thus6`TL ϕ. J

4.3 Tableau Calculi for ML, ML( > ), MDL, and EMDL

In addition to labeled formulae, the tableau rules for modal logics contain accessibility formulaeof the formiRj, wherei, j∈N. The intended interpretation ofiRjis that the point denoted byjis accessible by the relation Rfrom the point denoted byi. The tableau rules for the calculi are given in Figures 1 and 3.

In the construction of tableaus, in addition to the rules given in Section 4.2, we impose that the tableau rule () is never applied twice to the same labeled formula in any branch.

The definitions of open, closed and saturated tableau and branch are as in Section 4.2 with the following additional rule: A branch is calledclosed also if it contains a labeled formula {i}: dep(ϕ1, . . . , ϕn, ψ), for somei, n∈Nandϕ1, . . . , ϕn, ψ∈ ML.

LetTML,T_ML(>), andTMDL denote the extensions ofTPL,T_PL(>), andTPDby the rules (♦) and () of Figure 3, respectively. LetT_EMDL denote the extension ofT_ML by the rules (Split) of Figure 1 and (MLdep) of Figure 3.

Letϕbe a formula ofL ∈ {ML,ML(>),MDL,EMDL}. We say that a tableau T is atableau for ϕif the root ofT is{1, . . . ,2^Rank>(ϕ)}:ϕandT is obtained by applying the rules ofT_L. We say thatϕisprovableinT_L and write`TL ϕif there exists a closed tableau forϕ.

IExample 23. This example illustrates one difference betweenTPLandTMDLeven for the same formula dep(p, p). Figure 4 is an illustration of a closedT_MDL-tableau for dep(p, p).

When dep(p, p) is considered as aPD-formula, the calculation starts with the label{1,2}

{1,2,3,4}: dep(p, p) {i, j}: dep(p, p)

{i}:p {j}:p {i, j}:p {i, j}:¬p

{i}:¬p

×

{j}:¬p

×

{i}:¬p {j}:¬p {i, j}:p {i, j}:¬p {i}:p

×

{j}:p

×

Figure 4A tableau showing that theMDL-formula dep(p, p) is provable inTMDL. {1,2}:dep(p)

1R3 2R4 {3,4}: dep(p)

{3,4}:p {3,4}:¬p

{3}:p .. .

{4}:p {3}:¬p

{4}:¬p ..

. ... ...

Figure 5A tableau showing that theMDL-formuladep(p) is not valid.

(see Example 16 and Figure 2). However, when dep(p, p) is considered as anMDL-formula, our definition leads us to start the calculation with the label{1,2,3,4}.

The equivalentML(>) formula that theMDL-formula dep(p, p) translates into is _

a∈{>,⊥}

^{p^a, p>¬p}.

Therefore Rank_>(dep(p, p)) = 2, and thus the root of anyTMDL-tableau for dep(p, p) is {1,2,3,4}: dep(p, p). We first apply the rule (Split) to{1,2,3,4}: dep(p, p) and obtain 6 branches. By symmetry, we may concentrate on one of the branches. We denote it by {i, j}

(i6=j). We then apply the rule (PLdep) to{i, j}: dep(p, p) and branch into two branches as depicted in Figure 4. In the left (right) branch we apply the rule (¬P rop) to {i, j}:¬p ((P rop) to{i, j}:p). Consequently, each branch of the tableau becomes closed due to the labeled formulae of the type{l}:pand{l}:¬p,l∈ {i, j}. Therefore, dep(p, p) is a theorem ofTMDL.

IExample 24. We show that theMDLformuladep(p) is not valid. Note that the equival- entML(>)-formula thatdep(p) translates into is(p>¬p). Therefore Rank

>(dep(p)) = 1, and thus the root of anyTMDL-tableau fordep(p) is{1,2}:dep(p). We are going to find an open saturated branch fordep(p).

First, we apply the rule () for{1,2}:dep(p). One of the branches that is obtained is depicted in Figure 5. We then apply the rule (PLdep) to{3,4}: dep(p). Then, by applying the rules (P rop) and (¬P rop) to{3,4}:pand {3,4}:¬p, respectively, we obtain an open saturated branch as depicted in Figure 5. From the open saturated branch, we can construct the following Kripke model K = (W, R, V) that falsifies theMDL-formuladep(p). Define W :={w1, w2, w3, w4}, R:={(w1, w3),(w2, w4)},V(p) :={w3}. One can easily verify that K,{w1, w2} 6|=dep(p).

IDefinition 25. Let L ∈ {ML,ML(>),MDL,EMDL}. LetB be a branch of a tableau inTL and let α: ϕbe the root of B. Recall that Index(B) denotes the set of exactly all natural numbers that occur inB. For i, j∈Index(B), we write i≺_B j ifiRj occurs inB.

By≺^∗_B and^∗_B, we mean the transitive closure and the reflexive and transitive closure of

≺B, respectively. Moreover, fori∈Index(B) andn∈N, define LevelB(i) :=|{j∈Index(B)|i₀≺^∗_Bj^∗_Bi, for some i₀∈α}|, Layer_B(n) :={j ∈Index(B)|Level_B(j) =n}.

It is easy to see that, for every branchB, the graph (Index(B),≺_B) is a well-founded forest.

ITheorem 26 (Termination ofT_ML, T_ML(

>), T_MDL, and T_EMDL). Letϕ be a formula of ML,ML(>),MDL, orEMDL. Every tableau for ϕis finite.

Proof. LetT be a tableau forϕand letα:ϕdenote the root ofT. By definitionαis finite.

Clearly, by the definitions of the tableau rules, ifβ:ψoccurs inT then |β| ≤ |α|. From this and from the definitions of the tableau rules, it is easy to see thatT is a finitely branching tree. Thus from König’s lemma it follows thatT is infinite if and only ifT has an infinite branch.

LetBbe an arbitrary branch ofT. We will show thatB is finite.

Claim 1. Ifα:ϕoccurs inBthen, for everyi, j∈α, Level_B(i) = Level_B(j).

Claim 2. For each k∈Nthe set Layer_B(k) is finite.

Claim 3. There is ak∈Nsuch that Layer_B(k) =∅.

Note first that if Layer_B(k) =∅ then Layer_B(n) =∅, for everyn≥k. Thus from Claims 2 and 3 it follows that only finitely many labels may occur inB. Note also that, for every labeled formulaβ:ψthat occurs inB,ψis either a subformula ofϕor a subformula of some θ^⊥, whereθis anMLsubformula of ϕ. Thus only finitely many labeled formulae may occur inB. ThusBis finite.

Proof of Claim 1 is easy. We will sketch the proofs of Claims 2 and 3.

Proof sketch of Claim 2. Claim 2 follows from Claim 1 by induction: Clearly Layer_B(0) is finite. Layer_B(k+ 1) is generated via applications of the tableau rule () to labeled formulaeβ :ψof the branchB, whereβ⊆Layer_B(k) andψis either a subformula ofϕ or a subformula of someθ^⊥, whereθ is anMLsubformula ofϕ. Since Layer_B(k) is finite, Layer_B(k+ 1) is as well.

Proof sketch of Claim 3. For finite labelsβ, define

mB(β) := max{|ϕ| |β1:ϕoccurs inBandβ1∩β6=∅}.

For finite labelsβ, defineMB(β:ψ) := (mB(β),|ψ|,|β|). The ordering between the tuples is defined as follows:

(i, j, k)<(k, l, m) iffi < kor (i=kand j < l) or (i=kandj =l andk < m).

Note that for every labeled formulaβ :ψ that occurs in Bit holds thatm_B(β)< m_B(α),

|ψ| ≤ |ϕ| and|β| ≤ |α|. Thus the ordering of the tuples is well-founded. Furthermore it is easy to check that an application of each tableau rule decreases the measure M_B. For finite collections of labeled formulae Γ, defineMB(Γ) := max{MB(β:ψ)|β:ψ∈Γ}. It is straightforward to show that, for everyk∈N, eitherM_B Layer_B(k+ 1)

<M_B Layer_B(k)

or Layer_B(k+ 1) =∅. From this the claim follows. J

IDefinition 27. LetB be a tableau branch. We say thatBis faithfulto a Kripke model K = (W, R, V) if there exists a mappingf : Index(B)→W such that, K, f[α]6|=ϕfor all α:ϕ∈ B, and f(i)Rf(j) holds, for everyiRj∈ B.

ILemma 28. LetL ∈ {ML,ML(>),MDL,EMDL}. If ϕ∈ Lis not valid then there is an open saturated branch in every saturated tableau of ϕinTL.

Proof. In this proof, we focus on ML(>). Assume that ϕ ∈ ML(>) is not valid. By Corollary 15, there is a Kripke model K = (W, R, V) and a teamT of K such that|T| ≤ 2^Rank>(ϕ) and K, T 6|=ϕ. Put α₀ := {1, . . . ,2^Rank>(ϕ)}. Let T be an arbitrary saturated tableau forϕ. By Theorem 26,T is finite and, by definition, the root ofT isα0:ϕ. We will show that there is an open branchBin T.

We first establish that B0:={α0:ϕ} is faithful to K. Letf :α0→W be any mapping (note: W is non-empty) such thatf[α₀] =T. Clearly K, f[α₀]6|=ϕ, and thusB₀is faithful to K. Assume then that we have constructed a branchBn such thatBn is faithful to K. Thus there is a mappingg: Index(B_n)→W such that, for allβ:ψ∈ B_n, K, g[β]6|=ψ, and, for all iRj∈ Bn,g(i)Rg(j) holds. We will show that any rule-application toBn generates at least one faithful extensionBn+1to K. Here we are concerned with the rules of (♦) and () alone.

(♦) Assume that {i1, . . . , ik} :♦ψ, i1Rj1, . . . , ikRjk ∈ Bn. Let α:={i1, . . . , ik} and β :=

{j₁, . . . , j_k}. We obtain from our assumption that K, g[α]6|=♦ψandg[α][R]g[β]. From the semantics of♦it follows that K, g[β]6|=ψ. ThusBn+1 :=Bn∪ {β:ψ} is faithful to K. Clearly Bn+1 is an extension ofBby the rule (♦).

() Assume thatα:ψ∈ Bn. We obtain from our assumption that K, g[α]6|=ψ. By the semantics of, it follows that K, R[g[α]]6|=ψ. Now, by Theorem 14, there exists a team S ⊆R[g[α]] such that 0<|S| ≤2^Rank>(ψ) and K, S6|=ψ. Fix suchS⊆R[g[α]] and let u₁, . . . , u_mbe the elements ofS. SinceS⊆R[g[α]] there exists a functionh:{1, . . . , m} → α such that g h(l)

Rul, for each l ≤ m. Let h⁰ : {1, . . . ,2^Rank>(ψ)} → α denote the expansion ofhdefined such thath⁰(l) :=h(m) form < l≤2^Rank>(ψ). We then extend our function gto a mappingg⁰ to cover new fresh indexesβ :={j1, . . . , j₂Rank

>(ψ)}. We define that g⁰(jl) :=ul, forl ≤m, and g⁰(jl) :=um for m < l≤2^Rank>(ψ). By construction, we obtain that K, g⁰[β] 6|=ψ andg⁰(h⁰(l))Rg⁰(j_l) for all 1 ≤l ≤2^Rank>(ψ). Therefore, together with our assumption,Bn+1:=Bn∪{h⁰(1)Rj1, . . . , h⁰(2^Rank>(ψ))Rj₂Rank

>(ψ), β:ψ}

is faithful to K byg⁰. ClearlyB_n+1 is an extension ofBby the rule ().

SinceT is finite and saturated,Bj is a saturated branch inT for somej ∈N. Moreover,

sinceB_j is faithful to K,B_j is open. J

I Theorem 29 (Soundness of T_ML, T_ML(

>), T_MDL, andT_EMDL). Let L be a logic in {ML,ML(>),MDL,EMDL}. The calculus TL is sound.

Proof. FixL ∈ {ML,ML(>),MDL,EMDL}. Assume that 6|=Lϕ. By Lemma 28, there is an open saturated branch in every saturated tableau ofϕinT_L. Therefore, and since, by Theorem 26, every tableau ofϕinTL is finite, there does not exists any closed tableau forϕ

inT_L. Thus6`TL ϕ. J

ILemma 30. Let L ∈ {ML,ML(>),MDL,EMDL}. If there exists an open saturated branch for ϕinTL then ϕis not valid.

Proof. LetBbe an open saturated branch in a tableauT ofTLstarting with{1, . . . ,2^Rank>(ϕ)}: ϕ. Define the induced Kripke model K_B = (W, R, V) from B as follows: W := Index(B);

iRj iff iRj ∈ B; V(p) := {i| {i}:¬p∈ B} for any p occurring in B, otherwise, V(p) :=

∅. It is straightforward to prove by induction on χ that α : χ ∈ B implies K_B, α 6|= χ.

Since{1, . . . ,2^Rank>(ϕ)}:ϕ∈ B, it follows that KB,{1, . . . ,2^Rank>(ϕ)} 6|=ϕ. Thus ϕis not

valid. J

ITheorem 31(Completeness of T_ML,T_ML(

>),T_MDL, andT_EMDL). Let Lbe a logic in {ML,ML(>),MDL,EMDL}. The calculus TL is complete.

Proof. FixL ∈ {ML,ML(>),MDL,EMDL}. Assume that6`TL ϕ. Thus every tableau forϕis open. From Theorem 26 it follows that there exists a saturated open tableau forϕ.

Thus there exists a saturated open branch forϕ. Thus, by Lemma 30,6|=Lϕ. J

5 Conclusion

We gave sound and complete Hilbert-style axiomatizations forPL, PL(>), PD, ML(>), MDL, andEMDL. In addition, we presented novel labeled tableau calculi for these logics.

We proved soundness, completeness and termination for each of the calculi presented.

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