Introduction to Modal Logics (Draft) Stefan Wolfl¨ July 22, 2015 2 Contents 1 From Propositional to Modal Logic5 1.1 Propositional logic................................5 1.2 A simple modal logic...............................7 2 Modal Language, Frames, and Models 13 2.1 Relational structures............................... 13 2.2 Modal languages................................. 16 2.3 Relational models and satisfaction........................ 19 2.4 Constructing models............................... 24 2.5 Translating modal logic into first-order logic................... 27 2.6 Consequence relation and compactness...................... 29 2.7 Expressiveness.................................. 31 3 Normal Modal Logics, Frame Classes, and Definability 39 3.1 Normal modal logics............................... 39 3.2 Kripke frames and definability.......................... 41 3.3 Proving theorems................................. 45 3.4 Soundness and completeness........................... 48 3.5 Canonical frames, definability, and compactness................. 53 3.6 The modal logic S4 ................................ 56 3.7 The modal logic S5 ................................ 58 3.8 The modal logic KL ............................... 60 4 Decidability and Complexity 65 4.1 Finite Model Property............................... 65 4.2 Filtration..................................... 68 4.3 Complexity.................................... 70 5 Decision Procedures 81 5.1 A tableaux procedure for K(m) .......................... 81 5.2 Tableaux procedures for other modal logics................... 87 5.3 Optimization techniques............................. 89 3 4 1 From Propositional to Modal Logic 1.1 Propositional logic Let P be a set of propositional variables. The language LPL(P) has the following list of symbols as alphabet: variables from P, the logical symbols ?, >, :, !, ^, _, $, and brackets. The set of LPL(P)-formulae, then, is defined as the smallest set of words over this alphabet that contains ?, >, each p 2 P, and is closed under the formation rule: If j and y are formulae, then so are :j, (j ^y), (j _y), and (j !y). We may express this in an abbreviated manner as: j ::= ? j > j p j :j j (j ^ j0) j (j _ j0) j (j ! j0) j (j $ j0) — where p ranges over P. Propositional variables are also referred to as atomic formulae. LPL(P) will also denote the set of LPL(P)-formulae, and for abbreviation we will use the nota- tions LPL, L(P), or simply L if this is understood from the context. The semantics of propositional logic is defined in terms of truth assignments. A truth assign- ment is simply a function V : P ! f0;1g, i.e., V assigns to each atomic formula a truth value 0 (“false”) or 1 (“true”). Given a truth assignment V, the satisfaction relation is then defined as follows: V 6j= ? V j= > V j= p () V(p) = 1 V j= :j () V 6j= j V j= j ^ y () V j= j and V j= y V j= j _ y () V j= j or V j= y V j= j ! y () V 6j= j or V j= y V j= j $ y () V j= j iff V j= y Definition 1.1. An LPL-formula j is satisfiable if there exists a truth assignment V such that V j= j. It is valid if for each truth assignment V, it holds V j= j and contingent if it is neither valid nor unsatisfiable. Two formulae j and y are (logically) equivalent (or: PL-equivalent) if they are satisfied in the same sets of truth assignments. Given a set of LPL-formulae, S, a formulae j, and a truth assignment V, we write: V j= S if V satisfies each j 2 S, and S j= j if for each truth assignment V with V j= S, it holds that V j= j. j is then said to be a PL- consequence of S. We remark that most of these concepts also make sense if we restrict them to classes of truth assignments: Let C be a class of truth assignments (on the same set of propositional variables P). Then j is called C -satisfiable if j is satisfied in some V chosen from C , C -valid if j is satisfied in each V in C , etc. From these definitions, it is clear that for all formulae j and y, (j ^y) is equivalent to :(j ! :y) and (j _ y) equivalent to (:j ! y), etc. That is, one can drop, for example, ^, _, $, 5 >, and ? from the alphabet and instead use them as abbreviations in the meta-language: (j ^ y) := :(j !:y) (j _ y) := (:j ! y) (j $ y) := ((j ! y) ^ (y ! j)) > := (p0 ! p0) ? := :> where p0 is a fixed chosen propositional variable in P. A complete system of truth-functional connectives is any set of truth-functional connectives in which each truth-functional connective of arbitrary arity is definable. Examples include the systems f:;!g, f:;^g, f:;_g, and f?;!g. In what follows we will often switch between different such complete systems as appropriate in the respective context. Most of the times, however, we will stick to the system f:;!g. 0 0 0 For truth assignments V and V , we say that V and V coincide on Q ⊆ P (and write V =Q V ) if for each q 2 Q, V(q) = V 0(q). Lemma 1.1 (Coincidence Lemma). Let V and V 0 be truth assignments that coincide on Q ⊆ P. Then for each formula j in which only propositional variables from Q occur, it holds: 0 V j= j () V j= j: C Definition 1.2. A substitution is a map s : P ! LPL. Each substitution can be extended to a s function · : LPL ! LPL via the following recursive definition: ps := s(p) (:j)s := :js (j ! y)s := js ! ys ::: If a substitution s simply “replaces” each occurrence of variable p by a formula j, we simply s write y[p=j] instead of y . Analogously, we will use the notation y[p1=j1;:::; pk=jk] to indicate that the variables p1;:::; pn are simultaneously replaced by j1;:::;jn, respectively. Lemma 1.2 (Substitution Lemma). Let s be a substitution and V be a truth assignment. Define V s by V s (p) = 1 if and only if V j= s(p) for p 2 P. Then for each formula j, s s V j= j () V j= j : C Lemma 1.3 (Interpolation Theorem). For any LPL-formulae j and y with j j= y, there exists an LPL-formula c such that (a) in c only such propositional variables occur that occur in both j and y, and (b) it holds j j= c and c j= y. C In the situation of the lemma the formula c is called an interpolant of j and y. Actually, the proof requires that one of the symbols ? and/or > are in the language LPL. Otherwise, the claim only holds true for j j= y, where at least one propositional variable occurs in both j and y. 6 Definition 1.3. The degree of an LPL-formula j, degj, is the number of logical connectives occurring in j. The length of j is the number of alphabet symbols occurring j. Note that both notions depend on the chosen system of truth-functional connectives. Remark 1.1. It is an NP-complete problem (referred to as SAT) to decide for a given LPL- formula j whether it is satisfiable. The problem is also NP-complete if satisfiability is to be checked for propositional formulae in conjunctive normal form (CNF). A formula is in CNF if it is a conjunction of disjunctions of literals (a literal is a negated or unnegated atomic formula), i.e., the formula has the form n mi ^ _ li ji ; (CNF) i=1 ji where li ji is of the form p or :p for some propositional variable p. Accordingly, the problem to decide whether a given formula is valid is co-NP-complete. 1.2 A simple modal logic In what follows we will introduce a very simple “modal logic”. Again let P be a set of atomic propositions (or propositional variables). The language L(P) has the following list of sym- bols as alphabet: propositions of P, the logical symbols :, !, and , and brackets. The set of L(P)-formulae, then, is defined as follows: 0 j ::= p j :j j (j ! j ) j j — where p ranges over P. L(P) will also denote the set of L(P)-formulae, and for abbrevi- ation we will use the notations L, L(P), or simply L if this is understood. On the basis of the connectives of L(P), we can define: ♦j := ::j (j ! y) := (j ! y) Definition 1.4. A valuation model is a non-trivial family V = fVsgs2S of truth assignments of L (“non-trivial” means that S is a non-empty set). The elements of S are referred to as states (or possible worlds). V can be thought of as a function (called P-valuation on S) V : S ! 2P, i.e., it assigns to each state s in S a (possibly empty) set of propositional variables (namely those that are true in Vs). Given a valuation model V, we define a satisfaction relation as follows: V j=s p () p 2 V(s) V j=s :j () V 6j=s j V j=s j ! y () V 6j=s j or V j=s y 0 V j=s j () V j=s0 j; for each s 2 S By this definition and the definition of ♦, the truth condition for ♦j is then given by: V j=s ♦j () V j=s ::j () V 6j=s :j 0 () V 6j=s0 :j; for some s 2 S 0 () V j=s0 j; for some s 2 S 7 s1 :p, q, r, :s p, p, q, q, r, :r, :s :s s0 s2 Figure 1.1: An example of a valuation model Definition 1.5.