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Axiomatizations for Propositional and Modal Team Logic Axiomatizations for Propositional and Modal Team Logic Martin Lück Leibniz Universität Hannover, Germany [email protected] Abstract A framework is developed that extends Hilbert-style proof systems for propositional and modal logics to comprehend their team-based counterparts. The method is applied to classical propo- sitional logic and the modal logic K. Complete axiomatizations for their team-based extensions, propositional team logic PTL and modal team logic MTL, are presented. 1998 ACM Subject Classification F.4.1 Mathematical Logic Keywords and phrases team logic, propositional team logic, modal team logic, proof system, axiomatization Digital Object Identifier 10.4230/LIPIcs.CSL.2016.33 1 Introduction Propositional and modal logics, while their history goes back to ancient philosophers, have assumed an outstanding role in the age of modern computer science, with plentiful applications in software verification, modeling, artificial intelligence, and protocol design. An important property of a logical framework is completeness, i.e., that the act of mechanical reasoning can effectively be done by a computer. A recent extension of classical logics is the generalization to team semantics, i.e., formulas are evaluated on whole sets of assignments. So-called team based logics allow a more sophisticated expression of facts that regard multiple states of a system simultaneously as well as their internal relationship towards each other. The concept of team logic originated from the idea of quantifier dependence and independence. The question was simple andis long-known in linguistics: How can the statement For every 푥 there is 푦(푥), and for every 푢 there is 푣(푢) such that P(x,y,u,v) be formalized? The fact that 푣 should only depend on 푢 cannot be expressed with first-order quantifiers. Some suggestions were the independence-friendly logic ℐℱ by Hintikka and Sandu [7] or the dependence logic 풟 by Väänänen [15]. Hodges found that a compositional semantics of ℐℱ can be formulated with the concept of teams [8], which was adapted by Väänänen [14, 15] together with an atom of dependence, written =(푥, 푦) or dep(푥, 푦). Beside Väänänen’s dependence atom a variety of atomic formulas solely for the reasoning in teams were introduced. Galliani and others found a connection to database theory; they defined common constraints like independence ⊥, inclusion ⊆ and exclusion | in the framework of team semantics [2, 4]. Beside first-order logic, all these atoms were also adapted for modal logic ℳℒ [14] and recently propositional logic 풫ℒ [16]. As for any logic system, the question of axiomatizability arose. After all, team logics enable reasoning about sets of valuations, and predicate logic with set quantifiers (풮풪) is not axiomatizable. An important connection to team logic was found in the sense that dependence © Martin Lück; licensed under Creative Commons License CC-BY 25th EACSL Annual Conference on Computer Science Logic (CSL 2016). Editors: Jean-Marc Talbot and Laurent Regnier; Article No. 33; pp. 33:1–33:18 Leibniz International Proceedings in Informatics Schloss Dagstuhl – Leibniz-Zentrum für Informatik, Dagstuhl Publishing, Germany 33:2 Axiomatizations for Propositional and Modal Team Logic logic 풟 is as powerful as existential second-order logic 풮풪(∃) [15], and that its extension 풯 ℒ (where a semantical negation ∼ is provided) is even equivalent to full second-order logic 풮풪 [10]; therefore both are non-axiomatizable. Later Kontinen and Väänänen showed that there is a partial axiomatization in the sense that ℱ풪 consequences of 풟 formulas are derivable [11]. For many weaker team logics the question of axiomatizability is open. Exceptions are certain fragments of propositional and modal team logic. They were axiomatized by Sano and Virtema [13] and Yang [16], but these solutions rely on the absence of Boolean negation. Contribution In this paper complete axiomatizations of 풫풯 ℒ and ℳ풯 ℒ are given, the full propositional and modal team logics. A crucial step in the completeness proof is the fact that 풫풯 ℒ is not more expressive than a (team semantical) Boolean combination of classical 풫ℒ formulas, in symbols 풫풯 ℒ ≡ ℬ(풫ℒ). In the modal case analogously ℳ풯 ℒ ≡ ℬ(ℳℒ) holds. For a similar application to first-order logic see the technical report [12]. The paper is built as follows: After reminding the reader of several foundational definitions (Section 2), complete axiomatizations for the Boolean closures ℬ(풫ℒ) and ℬ(ℳℒ) are presented (Section 3). The collapses from 풫풯 ℒ and ℳ풯 ℒ to ℬ(풫ℒ) and ℬ(ℳℒ) are then proven step-wise by axiomatizing the elimination of splitting (Section 4) and modalities (Section 5). 2 Preliminaries If in the following 퐴 is a set, then P(퐴) refers to its power set. The notation [푛] will be used for the set {1, . , 푛}, assuming 푛 ∈ N. We define a logic as a triple ℒ = (훷, A, ). The component 훷 is a countable set consisting of finite words over some alphabet 훴, the so-called formulas of ℒ. The set A contains possible valuations of formulas in 훷, and the binary relation is the truth or satisfaction relation between A and 훷. To distinguish between different satisfaction relations we sometimes write ℒ. We use the same symbol for the entailment relation, 휙 휓 meaning that ∀퐴 ∈ A : 퐴 휙 implies 퐴 휓. These relations are as usual generalized to sets, 퐴 훷 meaning ∀휙 ∈ 훷 : 퐴 휙, and 훷 휓 meaning that ∀퐴 ∈ A : 퐴 훷 implies 퐴 휓. The reader is assumed to be familiar with the foundations of classical propositional and modal logics. We define classical propositional logic via a countable set 풫풮 := {푥1, 푥2,...} of atomic propositional statements and the connectives → and ¬. Truth ⊤ and falsum ⊥ are defined as (푥1 → 푥1) and ¬⊤, respectively. On top of propositional logic, modal logic is defined with the additional unary modality with the standard Kripke semantics. 2.1 Team logics Let ℒ be a logic. We introduce two new operators to ℒ: The unary strong negation ∼ and the binary material implication (under the assumption that they are not symbols in formulas of ℒ). The logic ℬ(ℒ) is_ the Boolean closure of ℒ and is defined by the following grammar, where 훼 stands for any ℒ-formula: 휙 ::= 훼 | ∼휙 | (휙 휙). Note that in particular any atom of the logic ℒ is an atom of ℬ(ℒ), but connecting_ ℒ-formulas with ∼ or always yields formulas not in ℒ. We further use the symbol ⊥⊥ (strong falsum), ⊥⊥ := _∼(휓 휓), and the abbreviations (휙 휓) := (∼휙 휓), (휙 휓) := ∼(휙 ∼휓) and (휙 휓) :=_ (휙 휓) (휓 휙). The semantics6 of ℬ(ℒ) _extend ℒ 7by, given some_ valuation ] _ 7 _ 퐴 ∈ A of ℒ, as follows: 퐴 ∼휙 ⇔ 퐴 2 휙 and 퐴 휙 휓 ⇔ 퐴 2 휙 or 퐴 휓. _ M. Lück 33:3 The next operator introduced in team logic is the binary operator (, called linear implication, similar as in Väänänen’s first-order team logic 풯 ℒ [15]. Assume that the logic 3 ℒ = (훷, A, ) has a splitting relation 휎ℒ ⊆ A . If (퐴, 퐵, 퐶) ∈ 휎ℒ then we say that (퐵, 퐶) is a splitting or division of 퐴. The semantics is that 퐴 휙 ( 휓 if for all (퐵, 퐶) with (퐴, 퐵, 퐶) ∈ 휎ℒ it holds 퐵 2 휙 or 퐶 휓. Abbreviate 휙 휓 := ∼(휙 ( ∼휓). If a logic ℒ has a splitting relation, then the syntax of 풮(ℒ) is the extension of ℬ(ℒ) by the grammar rule 휙 ::= (휙 ( 휙). Propositional team logic 풫풯 ℒ is the logic of 풮(풫ℒ)-formulas. A valuation of 풫풯 ℒ is a team 푇 which is a (possibly empty) set of propositional assignments 푠: 풫풮 → {0, 1}. If 휙 ∈ 풫ℒ then 푇 휙 if 푠 휙 in 풫ℒ semantics for all 푠 ∈ 푇 . A division of a team 푇 is simply a pair (푆, 푈) such that 푆 ∪ 푈 = 푇 . Modal team logic ℳ풯 ℒ is the closure of ℳℒ under ∼, , ( (as above) and the unary _ modalities and ∆. Abbreviate ♦ := ∼∆∼. In contrast to ℳℒ, valuations are not pointed Kripke structures (풦, 푤) but have the form (풦, 푇 ), where 푇 ⊆ 푊 is called a team. For 휙 ∈ ℳℒ it holds (풦, 푇 ) 휙 if (풦, 푤) 휙 in ℳℒ semantics for all 푤 ∈ 푇 . A division of (풦, 푇 ) is a pair ((풦, 푆), (풦, 푈)) such that 푆 ∪ 푈 = 푇 . If (푊, 푅, 푉 ) is a Kripke structure, then we define the image 푅[푇 ] of a team 푇 ⊆ 푊 as { 푤 ∈ 푊 | ∃푣 ∈ 푇 : 푣푅푤 } and the pre-image 푅−1[푇 ] as { 푤 ∈ 푊 | ∃푣 ∈ 푇 : 푤푅푣 }.A ′ ′ −1 ′ successor team 푇 of 푇 is a team such that 푇 ⊆ 푅[푇 ] and 푇 ⊆ 푅 [푇 ]. The semantics of ′ and ∆ is (퐾, 푇 ) 휙 if (퐾, 푅[푇 ]) 휙, and (퐾, 푇 ) ∆휙 if for all successor teams 푇 of 푇 it ′ holds (퐾, 푇 ) 휙. In the following we drop parentheses according to the usual precedence rules; further we assume →, and ( as right-associative and ∧, , ∨, , as left-associative. Also we reserve_ the letters 훼, 훽, 훾, . for classical7풫ℒ,6ℳℒ formulas; and we use 휙, 휓, 휗, . for general 풫풯 ℒ and ℳ풯 ℒ formulas. 2.2 Proof systems Proof systems or calculi are connected to the so-called Entscheidungsproblem, the problem of algorithmically deciding if a given formula 휙 of a logic ℒ is valid. Formally we define a proof system as a triple 훺 = (훯, 훹, 퐼) where 훯 is a set of formulas, 훹 ⊆ 훯 is a set of axioms, and 퐼 ⊆ P(훯) × 훯 is a set of inference rules. 훯, 훹 and 퐼 are all countable and decidable.
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