74.419 Artificial Intelligence Modal Logic

1. 74.419 Artificial IntelligenceModal Logic see reference last slide

2. Syntax of Modal Logic (□ and ◊) Formulae in (propositional) Modal Logic ML: • The Language of ML contains the Language of Propositional Calculus, i.e. if P is a formula in Propositional Calculus, then P is a formula in ML. • If  and  are formulae in ML, then , , , , □, ◊ * • are also formulae in ML. * Note: The operator ◊ is often later introduced and defined through □ .

3. Semantics of Modal Logic (□ and ◊) The semantics of a modal logic ML is defined through: • a set of worlds W = {w1, w2, ..., wn}, • an accessibility relation RWW, and • an interpretation function: {0,1}

4. Semantics of Modal Logic ( and ) The interpretation in ML of a formula P, Q, ... of the propositional language of ML corresponds to its truth value in the "current world": • w (P)=1 iffI(P) is true in w. • w (PQ)=1 iff I(PQ) is true in w. • ...

5. Semantics of Modal Logic (□ and ◊) We extend the semantics with an interpretation of the operators □ and ◊, specified relative to a "current world" w. • For all wW: • w (□)=1 iff w': (w,w')R  w' ()=1 ; 0 otherwise. • w (◊)=1 iff w': (w,w')R  w' ()=1 ; 0 otherwise. Note: Often, the operator ◊ is defined in terms of □: • ◊  □

6. Semantics of Modal Logic (□ and ◊) We can also prove the equivalence of □ and ◊ for our definitions above: w (□)=1 iff (w (□)=1) (or w (□)=0) iff w': (w,w')R  w' ()=1 iff w': (w,w')R  w' ()=0 iff w': (w,w')R  w' ()=1 iff w (◊)=1 This means: □  ◊ Exercise: Proof ◊  □ !

7. Semantics of Modal Logic (□ and ◊) Other logical operators are interpreted as usual, e.g. w (□)=1 iff w (□)=0

8. Semantics of ML - Complex Formulas The interpretation of a complex formula of ML is based on the interpretation of the atomic propositional symbols, and then composed using the interpretation function  defined above, e.g. • w (□)=1 iff (w': (w,w')R  w' ()=1) iff w': (w,w')R  w' ()=0 Let's say   (PQ). • w': (w,w')R  w' (PQ)=0 • w': (w,w')R  (w' (P)=0 w' (Q)=0) "P or Q" is not necessarily true in world w, if there is a world w', accessible from w, in which P is false or Q is false.

9. Semantics of Modal Logic - Grounding The interpretation in ML of a formula P, Q, ... of the propositional language of ML corresponds to its truth value in the "current world": • w (P)=1 iffI(P) is true in w. • w (PQ)=1 iff I(PQ) is true in w. • ...

10. Semantics of Modal Logic • A formula  is satisfied in a world w of a Model M=<W,R,>, if it is true in this world wW under the given interpretation , i.e. w ()=1. M,w |=  • A formula  is truein a Model M=<W,R,>, if it is satisfied in all worlds wW of M. M |=  • A formula  is valid, if it is true in all Models. |=  • A formula  is satisfiable, if it is satisfied in at least one world wW of one Model M=<W,R,>. (or: If its negation is not valid.)

11. Semantics of Modal Logic • A formula  is satisfied in a world w of a Model M=<W,R,>, if it is true in this world under the given interpretation , i.e. w ()=1. M, w |=  • A formula  is truein a Model M=<W,R,>, if it is satisfied in all worlds wW of M. M |=  • A formula  is valid, if it is true in all Models.  |=  • A formula  is satisfiable, if it is satisfied in at least one world wW of one Model M=<W,R,>. (or: If its negation is not valid.) • A formula  is a consequence of a set of formulas  in M=<W,r,>, if in all worlds wW, in which  is satisfied,  is also satisfied. |= 

12. Semantics of Modal Logic: Terminology Sometimes the term "frame" is used to refer to worlds and their connection through the accessibility relation: • A frame<W, R> is a pair consisting of a non-empty set W (of worlds) and a binary relation R on W. • A model <F, > consists of a frame F, and a valuation  that assigns truth values to each atomic sentence at each world in W.

13. Textbooks on (Modal) Logic Richard A. Frost, Introduction to Knowledge-Base Systems, Collins, 1986 (out of print) Comments: one of my favourite books; contains (almost) everything you need w.r.t. foundations of classical and non-classical logic; very compact, comprehensive and relatively easy to understand. Allan Ramsay, Formal Methods in Artificial Intelligence, Cambridge University Press, 1988 Comments: easy to read and to understand; deals also with other formal methods in AI than logic; unfortunately out of print; a copy is on course reserve in the Science Library.

14. Textbooks on (Modal) Logic Graham Priest, An Introduction to Non-Classical Logic, Cambridge University Press, 2001 Comments: the most poplar book (at least among philosophy students) on non-classical, in particular, (propositional) modal logic. Kenneth Konyndyk, Introductory Modal Logic, University of Notre-Dame Press, 1986 (with later re-prints) Comments: relatively easy and nice to read; contains propositional as well as first-order (quantified) modal logic, and nothing else.

15. Textbooks on (Modal) Logic J.C. Beall & Bas C. van Fraassen, Possibilities andParadox, University of Notre-Dame Press, 1986 (with later re-prints) Comments: contains a lot of those weird things, you knew existed but you've never encountered in reality (during your university education). G.E. Hughes & M.J. Creswell, A NewIntroduction to Modal Logic, Routledge, 1996 Comments:Location: Elizabeth Dafoe Library, 2nd Floor, Call Number / Volume: BC 199 M6 H85 1996