L ogics for D ata and K nowledge R epresentation

1. Logics for Data and KnowledgeRepresentation Modal Logic Originally by Alessandro Agostini and Fausto Giunchiglia Modified by Fausto Giunchiglia, Rui Zhang and Vincenzo Maltese

2. Outline • Introduction • Syntax • Semantics • Satisfiability and Validity • Kinds of frames • Correspondence with FOL 2

3. Introduction • We want to model situations like this one: 1. “Fausto is always happy” circumstances” 2. “Fausto is happy under certain • In PL/ClassL we could have: HappyFausto • In modal logic we have: 1. □ HappyFausto 2. ◊ HappyFausto As we will see, this is captured through the notion of “possible words” and of “accessibility relation”

4. Syntax • We extend PL with two logical modal operators: □ (box) and ◊ (diamond) □P : “Box P” or “necessarily P” or “P is necessary true” ◊P : “Diamond P” or “possibly P” or “P is possible” Note that we define □P = ◊P, i.e. □ is a primitive symbol • The grammar is extended as follows: <Atomic Formula> ::= A | B | ... | P | Q | ... | ⊥ | ⊤ | <wff> ::= <Atomic Formula> | ¬<wff> | <wff>∧ <wff> | <wff>∨ <wff> | <wff>  <wff> | <wff>  <wff> | □ <wff> | ◊ <wff> 4

5. Different interpretations 5

6. Semantics: Kripke Model • A Kripke Model is a triple M = <W, R, I> where: • W is a non empty set of worlds • R ⊆ W x W is a binary relation called the accessibility relation • I is an interpretation function I: L  pow(W) such that to each proposition P we associate a set of possible worlds I(P) in which P holds • Each w ∈ W is said to be a world, point, state, event, situation, class … according to the problem we model • For "world" we mean a PL model. Focusing on this definition, we can see a Kripke Model as a set of different PL models related by an "evolutionary" relation R; in such a way we are able to represent formally - for example - the evolution of a model in time. • In a Kripke model, <W, R> is called frame and is a relational structure. 6

7. Semantics: Kripke Model • Consider the following situation: • M = <W, R, I> W = {1, 2, 3, 4} R = {<1, 2>, <1, 3>, <1, 4>, <3, 2>, <4, 2>} I(BeingHappy) = {2} I(BeingSad) = {1} I(BeingNormal) = {3, 4} BeingHappy 1 2 3 BeingSad BeingNormal 4 BeingNormal 7

8. Truth relation (true in a world) • Given a Kripke Model M = <W, R, I>, a proposition P ∈ LML and a possible world w ∈ W, we say that “w satisfies P in M” or that “P is satisfied by w in M” or “P is true in M via w”, in symbols: M, w ⊨ P in the following cases: 1. P atomic w ∈ I(P) 2. P = Q M, w ⊭ Q 3. P = Q  T M, w ⊨ Q and M, w ⊨ T 4. P = Q  T M, w ⊨ Q or M, w ⊨ T 5. P = Q  T M, w ⊭ Q or M, w ⊨ T 6. P = □Q for every w’∈W such that wRw’ then M, w’ ⊨ Q 7. P = ◊Q for some w’∈W such that wRw’ then M, w’ ⊨ Q NOTE: wRw’ can be read as “w’ is accessible from w via R” 8

9. Semantics: Kripke Model • Consider the following situation: • M = <W, R, I> W = {1, 2, 3, 4} R = {<1, 2>, <1, 3>, <1, 4>, <3, 2>, <4, 2>} I(BeingHappy) = {2} I(BeingSad) = {1} I(BeingNeutral) = {3, 4} M, 2 ⊨ BeingHappy M, 2 ⊨ BeingSad M, 4 ⊨ □BeingHappy M, 1 ⊨ ◊BeingHappyM, 1 ⊨ ◊BeingSad BeingHappy 1 2 3 BeingSad BeingNormal 4 BeingNormal 9

10. Satisfiability and Validity • Satisfiability A proposition P ∈ LML is satisfiable in a Kripke model M = <W, R, I> if M, w ⊨ P for all worlds w ∈ W. We can then write M ⊨ P • Validity A proposition P ∈ LML is valid if P is satisfiable for all models M (and by varying the frame <W, R>). We can write ⊨ P 10

11. Satisfiability • Consider the following situation: • M = <W, R, I> W = {1, 2, 3, 4} R = {<1, 2>, <2, 2>, <3, 2>, <4, 2>} I(BeingHappy) = {2} I(BeingSad) = {1} I(BeingNormal) = {3, 4} M, w ⊨ □BeingHappy for all w ∈ W, therefore □BeingHappy is satisfiable in M. BeingHappy 1 2 3 BeingSad BeingNormal 4 BeingNormal 11

12. Validity • Prove that P: □A  ◊A is valid • In all models M = <W, R, I>, (1) □A means that for every w∈W such that wRw’ then M, w’ ⊨ A (2) ◊A means that for some w∈W such that wRw’ then M, w’ ⊨ A It is clear that if (1) then (2) in the example (as we will see this is valid in serial frames) A 1 2 3 A 12

13. Kinds of frames • Given the frame F = <W, R>, the relation R is said to be: • Serial iff for every w ∈ W, there exists w’ ∈ W s.t. wRw’ • Reflexive iff for every w ∈ W, wRw • Symmetric iff for every w, w’ ∈ W, if wRw’ then w’Rw • Transitive iff for every w, w’, w’’ ∈ W, if wRw’ and w’Rw’’ then wRw’’ • Euclidian iff for every w, w’, w’’ ∈ W, if wRw’ and wRw’’ then w’Rw’’ • We call a frame <W, R> serial, reflexive, symmetric or transitive according to the properties of the relation R 13

14. Kinds of frames • Serial: for every w ∈ W, there exists w’ ∈ W s.t. wRw’ • Reflexive: for every w ∈ W, wRw • Symmetric: for every w, w’ ∈ W, if wRw’ then w’Rw 1 2 3 1 2 1 2 3 14

15. Kinds of frames • Transitive: for every w, w’, w’’ ∈ W, if wRw’ and w’Rw’’ then wRw’’ • Euclidian: for every w, w’, w’’ ∈ W, if wRw’ and wRw’’ then w’Rw’’ 1 2 3 1 2 3 15

16. Valid schemas • A schema is a formula where I can change the variables • THEOREM. The following schemas are valid in the class of indicated frames: D: □A  ◊A valid for serial frames T: □A  A valid for reflexive frames B: A  □◊A valid for symmetric frames 4: □A  □□A valid for transitive frames 5: ◊A  □◊A valid for Euclidian frames NOTE: if we apply T, B and 4 we have an equivalence relation • THEOREM. The following schema is valid: K: □(A  B)  (□A  □B) Distributivity of □ w.r.t.  16

17. Proof for T: □ A  A valid for reflexive frames Assuming M, w ⊨ □A, we want to prove that M, w ⊨ A. From the assumption M, w ⊨ □A, we have that for every w’∈W such that wRw’ we have that M, w’ ⊨ A (1). Since R is reflexive we also have w’Rw, we then imply that M, w ⊨ A (by substituting w to w’ in (1)) □A, A 1 2 17

18. Proof for B: A  □◊A valid for symmetric frames Assume M, w ⊨ A. To prove that M, w ⊨ □◊A we need to show that for every w’ ∈ W such that wRw’ then M, w ⊨ ◊A. M, w ⊨ ◊A is that for some w’’∈W such that w’Rw’’ then M, w’’ ⊨ A. Therefore we need to prove that for every w’∈W such that wRw’ andfor some w’’∈W such that w’Rw’’ then M, w’’ ⊨ A Since R is symmetric, from wRw’ it follows that w’Rw. For w’’∈W such that w’’ = w, we have that w’Rw’’ and M, w’’ ⊨ A. Hence M, w ⊨ A. A, □◊A ◊A 1 2 3 18

19. Reasoning services: EVAL Yes EVAL M, P No • Model Checking (EVAL) Given a (finite) model M = <W, R, I> and a proposition P ∈ LML we want to check whether M, w ⊨ P for all w ∈ W M, w ⊨ P for all w ? 19

20. Reasoning services: SAT M SAT P No • Satisfiability (SAT) Given a proposition P ∈ LML we want to check whether there exists a (finite) model M = <W, R, I> such that M, w ⊨ P for all w ∈ W Find M such that M, w ⊨ P for all w 20

21. Reasoning services: UNSAT w VAL M, P No • Unsatisfiability (unSAT) Given a (finite) model M = <W, R, I> and a proposition P ∈ LML we want to check that does not exist any world w such that M, w ⊨ P Verify that ∃ w such that M, w ⊨ P 21

22. Reasoning services: VAL Yes VAL P No • Validity (VAL) Given a a proposition P ∈ LML we want to check that M, w ⊨ P for all (finite) models M = <W, R, I> and w ∈ W Verify that M, w ⊨ P for all M and w 22

23. Correspondence between □ and ∀ (◊ and ∃) • We can define a translation function T: LML LFO as follows: 1. T(P) = P(x) for all propositions P in LML 2. T(P) = T(P) for all propositions P 3. T(P * Q) = T(P) * T(Q) for all propositions P, Q and *∈{,,} 4. T(□P) = ∀x T(P) for all propositions P 5. T(◊P) = ∃x T(P) for all propositions P THEOREM: For all propositions P in LML, P is modally valid iff T(P) is valid in FOL. 23