From Syllogism to Common Sense Normal Modal Logic K r i p k e S e m a n t i c s Completeness Mehul Bhatt and Correspondence Theory Oliver Kutz Thomas Schneider Lecture 9 Department of Computer Science & Research Center on Spatial Cognition (SFB/TR 8) University of Bremen Examples of Modal Logics Examples of Modal Logics Classic Distinctions between Modalities Modern interpretations of modalities ‣ Mathematical Logic: ‣ Alethic modality: necessity, possibility, contingency, impossibility ‣ The logic of proofs GL: [] A means: In PA it is provable that ‘A’. ‣ distinguish further: logical - physical - metaphysical, etc. ‣ Computer Science: ‣ Temporal modality: always, some time, never ‣ Linear Temporal Logic LTL: Formal Verification ‣ Deontic modality: obligatory, permissible ‣ X A : in the next moment ‘A’ ‣ Epistemic modality: it is known that ‣ A U B: A is true until B becomes true ‣ Doxastic modality: it is believed that ‣ G = ‘always’ , F = ‘eventually’, ‣ liveness properties state that something good keeps happening: Technically, all these modalities are treated ‣ G F A or also G (B -> F A) in the same way, by using unary modal operators ‣ Linguistics / KR / etc. Modal Logic: Some History A Hilbert system for Modal Logic K ‣ The following is the standard Hilbert system for the modal logic K. ‣ Modern modal logic typically begins with the systems devised by Axioms !1 → (!2 → !1) C. I. LEWIS, intended to model strict implication and avoid the (!1 → !2) → (!1 → (!2 → !3)) → (!1 → !3) paradoxes of material implication, such as the ‘ex falso quodlibet’. !1 → !1 ∨ !2 ‣ “ If it never rains in Copenhagen, then Elvis never died.” !2 → !1 ∨ !2 (!1 → !3) → (!2 → !3) → (!1 ∨ !2 → !3) ‣ (No variables are shared in example => relevant implication) (!1 → !2) → (!1 →¬!2) →¬!1 two classical tautologies ‣ For strict implication, we define A ~~> B by [] (A --> B) ¬¬!1 → !1 ! ∧ ! → ! instead of ! " p in INT ‣ These systems are however mutually incompatible, and no base 1 2 1 logic was given of which the other logics are extensions of. !1 ∧ !2 → !2 !1 → !2 → !1 ∧ !2 ‣ The modal logic K is such a base logic, named after SAUL KRIPKE, □(! → ") → (□! → □") new axiom of and which serves as a minimal logic for the class of all its (normal) Box Distribution extensions - defined next via a Hilbert system. ! ! → ! ! Rules 1 1 2 !2 □! new rule of Necessitation Table 1. AFregesystemsforthemodallogic# modal logic axioms #4 # + □! → □□! KB # + ! → □♢! GL # + □(□! → !) → □! $4 #4+ □! → ! Some More Modal frege systems $4GrzKripke$4+ Semantics□(□(! → □!) → !) → □! ‣ Hilbert systems for other modal logics are obtained by adding axioms. ‣ A Kripke frameTable consists 2. Frege of systems a set W for, the important set of `possible modal logics worlds’, and a modal logic axioms binary relation R between worlds. A valuation ! assigns propositional variables to worlds. A pointed model M is a frame, K4 K + ✷p ✷✷p x → together with a valuation and a distinguished world x. KB K + p ✷✸p As in classical logic, if we augment frames with assignments,wearriveatthe GL K + ✷(→✷p p) ✷p Mx = p q Mx = p and Mx = q → → notion of a model. | ∧ ⇐⇒ | | S4 K4+ ✷p p Mx = p q Mx = p or Mx = q → Definition 12. A model| ∨ for⇐⇒ the modal language| is| a pair (!, # ) where S4Grz S4+✷(✷(p ✷p) p) ✷p M = p q if M = p then M = q → → → x | → ⇐⇒ x | x | – ! =($, %) is a frame and ‣ More generally, in a fixed language, the class of all normal modal logics is defined Mx = p Mx = p – # : Var ($ ) |is a¬ mapping⇐⇒ assigning| to each propositional variable & a as any set of formulae that !→ # Mx = ✷p for all xRy : My = p set # (&) of worlds| ( ($ )⇐⇒denotes the power set of| $ ). ‣ (1) contains K (2) is closed under substitution and (3) Modus Ponens Mx = #✸p exists xRy : My = p ‣ In particular, any normal extension of K contains the Axiom of Box-Distribution: With the notion of| models⇐⇒ we can now define the| semantics of modal for- ✷(p q) (✷p ✷q) mulas: → → → Definition 13. Let ', ( be modal formulas, let ) =($, %, # ) be a model and * $ be a world. Inductively we define the notion of a formula to be satisfied ∈ in ) at world *: – ),* = & if * # (&) where & Var , – ),* ∣= ' if not∈ ),* = ', ∈ – ),* ∣= '¬ ( if ),* =∣ ' and ),* = ( – ),* ∣= ' ∧ ( if ),* ∣= ' or ),* =∣ ( – ),* ∣= □'∨ if for all +∣ $ with (*,∣ +) % we have ),+ = '. ∣ ∈ ∈ ∣ Amodalformula' is satisfiable if there exists a model ) =($, %, # )and aworld* $ such that ),* = '.Dually,' is a modal tautology if for every ∈ ∣ model ) =($, %, # )andevery* $ we have ),* = '. ∈ ∣ It can be shown that the Frege system from the previous sectionisindeeda proof system for the modal logic ,,i.e.,itissoundandcompleteforallmodal tautologies. 14 L10.2 Modal Tableaux (P) all propositional tautologies Modal Sat / Taut / Validity A Tableaux(K) (φ ψ) system( φ ψ) for Modal Logic K → → → φφ ψ L10.2 Modal Tableaux (MP) → ‣ Hilbert systemsψ are generally considered difficult to use in a ‣ Modal Sat: A modal formula is satisfiable if there practical way. φ exists a pointed model that satisfies it. (G) (P) all propositional tautologies ‣ There areφ many proof systems for Modal Logics. One of the most popular ones are Semantic Tableaux: (K) (φ ψ) ( φ ψ) ‣ Modal Taut: A formula→ is→ a modal → tautology if it is ‣ refutation based proof system satisfied in all pointed models. Figure 1: Modal logic K φφ ψ ‣ highly developed optimisation techniques (MP) → ψ T ‣ isallows system toK extractplus models (T) φdirectlyφ from proofs ‣ Modal Validity: A formula is valid in a class of → φ S4 ‣ ispopular system Tinplus particular (4) forφ Descriptionφ Logic based formalisms frames (G)if it a modal tautology relative to that class of → frames. φ ‣ often used for establishing upper bounds for the complexity of a SAT problem for a logic. Figure 2: Some other modal logics Check validity of Figure✷(p 1: Modalq) ( logic✷p K✷q) Box Distribution → → → is a finite sequence of natural numbers. In addition, every formula on the T is system K plus (T) φ φ tableau has a sign Z F, T that indicates the truth-value we currently → ∈ { } S4 is system T plus (4) φ φ expect for the formula in our reasoning. That is, a formula in the modal → tableaux is of the form σZA Figure 2: Some other modal logics where the prefix σ is a finite sequence of natural numbers, the sign Z is in F, T , and F is a formula of modal logic. At this point, we understand a { } isA a finiteTableaux sequence of natural system numbers. Infor addition, Modal every formula Logic on the K prefix σ as a symbolicA Tableaux name for a world insystem a Kripke structure. for Modal Logic K tableau has a sign Z F, T that indicates the truth-value we currently ∈ { } expect for the formula in our reasoning. That is, a formula in the modal Definition 1 (K prefix accessibility) For modal logic K, prefix σ is accessible ‣ In prefixed tableaux, every formula starts with a prefix and a sign tableaux is of the form from prefix σ if σ‣ isA of basic the form semanticσn for tableaux some natural for K number is givenn. as follows: Z " σZA ‣ We introduce prefixes (denotingModal Tableaux possible worlds) that keep track of accessibility. L10.3 ! ‣ For every formula of a class α with a top level operator and sign (T or ‣ Prefixes (denoting possible worlds) keep track of accessibility. ‣ Formulae in the tableaux are labelled with T or F. where the prefix σ is a finite sequence of natural numbers, the sign Z is in F for true and false) as indicated, we define two successor formulas α1 and ‣ A prefix ! is a finite sequence of natural numbers F, T , and F is a formula of modal logic. At this point, we understand a α2: ‣ We differentiate the following four kinds of formulas: { } prefix‣ Formulaeσ as a symbolic in a tableaux name forare alabelled world inwith a Kripke T or F. structure. α α1 α2 β β1 β2 ν ν0 π π0 TA B TA TB TA B TA TB T A TA T A TA ∧ ∨ ♦ Definition 1 (K prefix accessibility) For modal logic K, prefix σ is accessible FA B FA FB FA B FA FB ∨ ∧ F ♦A FA F A FA from prefix σ if σ is of the form σn for some natural number n. FA B TA FB TA B FA TB → → F A TA TA T AEveryFA combinationFA of top-level operator and sign occurs in one of the ¬ ¬ For every formula of a class α with a top level operator and sign (T or above cases. Tableau proof rules by those classes are shown in Figure 3.A For the followingConjunctive cases of formulas we defineDisjunctive one successor formulaUniversal Existential F for‣ trueExample and false) 1 4 7 as9 is indicated, accessible wefrom define 1 4 7 twowhich successor is accessble formulas from 1α 41 etc.and tableau is closed if every branch contains some pair of formulas of the form α2: ‣ These tables essentiallyσTAand encodeσFA.A theproof semanticsfor modal of the logic. logic formula consists of a closed tableau LECTURE NOTES FEBRURARY 18, 2010 α α1 α2 β β1 β2 starting with the root 1FA. TA B TA TB TA B TA TB ∧ ∨ FA B FA FB FA B FA FB σα σβ σν 1 σπ 2 ∨ ∧ (α) (β) (ν∗) (π) FA B TA FB TA B FA TB σα σβ σβ σ ν σ π → → 1 1 2 0 0 F A TA TA T A FA FA σα ¬ ¬ 2 1 For the following cases of formulas we define one successor formula σ accessible from σ and σ occurs on the branch already 2 σ is a simple unrestricted extension of σ, i.e., σ is accessible from σ and no other prefix on the branch starts with σ LECTURE NOTES FEBRURARY 18, 2010 Figure 3: Tableau proof rules for QML The tableau rules can also be used to analyze F A A as follows: → ♦ 1 F A A (1) → ♦ 1 T A (2) from 1 1 F ♦A (3) from 1 stop No more proof rules can be used because the modal formulas are ν rules, which are only applicable for accessible prefixes that already occur on the branch.