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

2. 2 Outline  Introduction  Syntax  Semantics  Satisfiability and Validity  Kinds of frames  Reasoning services  Theorem of equivalence with FOL  Theorem of equivalence with DL  Tableau calculus 2

3. Introduction  We want to model situations like this one: 1. “Fausto is always happy” 2. “Fausto is happy under certain circumstances”  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” 3

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: ::= A | B |... | P | Q |... | ⊥ | ⊤ | ::= | ¬ | ∧ | ∨ |  |  | □ | ◊ 4 SYNTAX :: SEMANTICS :: SATISFIABILITY AND VALIDITY :: FRAMES :: REASONING SERVICES :: THEOREMS :: TABLEAU

5. Different interpretations 5 Philosophy □ P : “P is necessary” ◊P : “P is possible” Epistemic □ a P : “Agent a believes P ” or “Agent a knows P” Temporal logics □ P : “P is always true” ◊P : “P is sometimes true” Dynamic logics or logics of programs □ a P : “P holds after the program a is executed” Description logics □ HASCHILD MALE  ∀ HASCHILD.MALE ◊ HASCHILD MALE  ∃ HASCHILD.MALE SYNTAX :: SEMANTICS :: SATISFIABILITY AND VALIDITY :: FRAMES :: REASONING SERVICES :: THEOREMS :: TABLEAU

6. Semantics: Kripke Model  A Kripke Model is a triple M = 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, is called frame and is a relational structure. 6 SYNTAX :: SEMANTICS :: SATISFIABILITY AND VALIDITY :: FRAMES :: REASONING SERVICES :: THEOREMS :: TABLEAU

7. Semantics: Kripke Model  Consider the following situation:  M = W = {1, 2, 3, 4} R = {,,,, } I(BeingHappy) = {2} I(BeingSad) = {1} I(BeingNormal) = {3, 4} 7 123 4 BeingHappy BeingSad BeingNormal SYNTAX :: SEMANTICS :: SATISFIABILITY AND VALIDITY :: FRAMES :: REASONING SERVICES :: THEOREMS :: TABLEAU

8. Truth relation (true in a world)  Given a Kripke Model M =, a proposition P ∈ L ML 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 atomicw ∈ I(P) 2. P =  QM, w ⊭ Q 3. P = Q  TM, w ⊨ Q and M, w ⊨ T 4. P = Q  TM, w ⊨ Q or M, w ⊨ T 5. P = Q  TM, 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 SYNTAX :: SEMANTICS :: SATISFIABILITY AND VALIDITY :: FRAMES :: REASONING SERVICES :: THEOREMS :: TABLEAU

9. Semantics: Kripke Model  Consider the following situation:  M = W = {1, 2, 3, 4} R = {,,,, } I(BeingHappy) = {2} I(BeingSad) = {1} I(BeingNeutral) = {3, 4} M, 2 ⊨ BeingHappy M, 2 ⊨  BeingSad M, 4 ⊨ □ BeingHappy M, 1 ⊨ ◊BeingHappy M, 1 ⊨  ◊BeingSad 9 123 4 BeingHappy BeingSad BeingNormal SYNTAX :: SEMANTICS :: SATISFIABILITY AND VALIDITY :: FRAMES :: REASONING SERVICES :: THEOREMS :: TABLEAU

10. Satisfiability and Validity  Satisfiability A proposition P ∈ L ML is satisfiable in a Kripke model M = if M, w ⊨ P for all worlds w ∈ W. We can then write M ⊨ P  Validity A proposition P ∈ L ML is valid if P is satisfiable for all models M (and by varying the frame ). We can write ⊨ P 10 SYNTAX :: SEMANTICS :: SATISFIABILITY AND VALIDITY :: FRAMES :: REASONING SERVICES :: THEOREMS :: TABLEAU

11. Satisfiability  Consider the following situation:  M = W = {1, 2, 3, 4} R = {,,, } I(BeingHappy) = {2} I(BeingSad) = {1} I(BeingNormal) = {3, 4} M, w ⊨ □ BeingHappy for all w ∈ W, therefore □ BeingHappy is satisfiable in M. 11 123 4 BeingHappy BeingSad BeingNormal SYNTAX :: SEMANTICS :: SATISFIABILITY AND VALIDITY :: FRAMES :: REASONING SERVICES :: THEOREMS :: TABLEAU

12. 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 12 123 12 12 3 SYNTAX :: SEMANTICS :: SATISFIABILITY AND VALIDITY :: FRAMES :: REASONING SERVICES :: THEOREMS :: TABLEAU

13. 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’’  We call a frame F = serial, reflexive, symmetric or transitive according to the properties of the relation R 13 123 12 3 SYNTAX :: SEMANTICS :: SATISFIABILITY AND VALIDITY :: FRAMES :: REASONING SERVICES :: THEOREMS :: TABLEAU

14. 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  ◊ Avalid for serial frames T: □ A  Avalid for reflexive frames B:A  □ ◊ Avalid for symmetric frames 4: □ A  □□ Avalid for transitive frames 5: ◊ A  □ ◊ Avalid 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.  14 SYNTAX :: SEMANTICS :: SATISFIABILITY AND VALIDITY :: FRAMES :: REASONING SERVICES :: THEOREMS :: TABLEAU

15. Proof for D: □A  ◊A valid for serial frames In all serial frames M =, we have that if (1) then (2) (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 15 12 3 SYNTAX :: SEMANTICS :: SATISFIABILITY AND VALIDITY :: FRAMES :: REASONING SERVICES :: THEOREMS :: TABLEAU □A, ◊A □A, ◊A, A

16. 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)) 16 □A, A 12 SYNTAX :: SEMANTICS :: SATISFIABILITY AND VALIDITY :: FRAMES :: REASONING SERVICES :: THEOREMS :: TABLEAU

17. 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 accessible world w’ ∈ W, i.e. 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’ and for 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. 17 12 3 A, □◊A ◊A SYNTAX :: SEMANTICS :: SATISFIABILITY AND VALIDITY :: FRAMES :: REASONING SERVICES :: THEOREMS :: TABLEAU

18. Reasoning services: EVAL  Model Checking (EVAL) Given a (finite) model M = and a proposition P ∈ L ML we want to check whether M, w ⊨ P for all w ∈ W M, w ⊨ P for all w ? 18 EVAL M, P Yes No SYNTAX :: SEMANTICS :: SATISFIABILITY AND VALIDITY :: FRAMES :: REASONING SERVICES :: THEOREMS :: TABLEAU

19. Reasoning services: SAT  Satisfiability (SAT) Given a proposition P ∈ L ML we want to check whether there exists a (finite) model M = such that M, w ⊨ P for all w ∈ W Find M such that M, w ⊨ P for all w 19 SAT P M No SYNTAX :: SEMANTICS :: SATISFIABILITY AND VALIDITY :: FRAMES :: REASONING SERVICES :: THEOREMS :: TABLEAU

20. Reasoning services: UNSAT  Unsatisfiability (unSAT) Given a (finite) model M = and a proposition P ∈ L ML we want to check that does not exist any world w such that M, w ⊨ P Verify that does not exist w such that M, w ⊨ P 20 VAL M,  P w No SYNTAX :: SEMANTICS :: SATISFIABILITY AND VALIDITY :: FRAMES :: REASONING SERVICES :: THEOREMS :: TABLEAU

21. Reasoning services: VAL  Validity (VAL) Given a a proposition P ∈ L ML we want to check that M, w ⊨ P for all (finite) models M = and w ∈ W Verify that M, w ⊨ P for all M and w 21 VAL P Yes No SYNTAX :: SEMANTICS :: SATISFIABILITY AND VALIDITY :: FRAMES :: REASONING SERVICES :: THEOREMS :: TABLEAU

22. Equivalence Modal logics – First Order Logic  We can define a translation function  : L ML  L FO as follows:  (P) = P(x) for all propositions P in L ML  (  P) =   (P) for all propositions P  (P * Q) =  (P) *  (Q) for all propositions P, Q and * ∈ { , ,  }  ( □ P) = ∀ x  (P) for all propositions P 5  (◊P) = ∃ x  (P) for all propositions P THEOREM: For all propositions P in L ML, P is modally valid iff  (P) is valid in FOL. 22 SYNTAX :: SEMANTICS :: SATISFIABILITY AND VALIDITY :: FRAMES :: REASONING SERVICES :: THEOREMS :: TABLEAU

23. Equivalence Modal logics – Description logics  Take ALCEU: ::= A | B |... | P | Q |... | ⊥ | ⊤ ::= | ¬ | ⊓ | ⊔ | ∀ R.C | ∃ R.C  We can define an equivalent multi-modal logic with a mapping function  as follows:  (A) = A for A atomic  (¬C) = ¬  (C)  (C ⊓ D) =  (C)   (D)  (C ⊔ D) =  (C)   (D)  ( ∃ R.C) = ◊ R  (C)  ( ∀ R.C) = □ R  (C) THEOREM: For all propositions P in L ML, P is modally valid iff  (P) is valid in DL. 23 SYNTAX :: SEMANTICS :: SATISFIABILITY AND VALIDITY :: FRAMES :: REASONING SERVICES :: THEOREMS :: TABLEAU

24. Modal logics Tableau: introduction 24  Recall modal logics semantics: P = □ Q for every w’ ∈ W such that wRw’ then M, w’ ⊨ Q P = ◊Q for some w’ ∈ W such that wRw’ then M, w’ ⊨ Q Each time we use □ or ◊ we state something about accessible worlds!  Recall satisfiability: A proposition P ∈ L ML is satisfiable if there exist a Kripke model in which it is true.  Therefore the key idea in the modal logics tableau is: If M, w ⊨ □ Q then Q must be present in all w’ accessible from w If M, w ⊨ ◊Q then Q must be present in some w’ trees accessible from w For all other formulas follow the rules of PL tableau SYNTAX :: SEMANTICS :: SATISFIABILITY AND VALIDITY :: FRAMES :: REASONING SERVICES :: THEOREMS :: TABLEAU

25. Modal logics Tableau: rules 25  We indicate with , i the fact that  must be true in world i ∈ W  Given the formula  in input we apply the rules below by verifying that not all branches are closed: (  ) (P  Q), i| P, i and Q, i (  ) (P  Q), i | P, i or Q, i (two branches) (◊) ◊P, i | iRj P, j given any (i,j) ∈ R to denote that P is true | in j given that it is accessible from i ( □ ) iRj □ P, i | P, j (duality)  □ P, i | ◊  P, i (duality)  ◊P, i | □  P, i  We start by convention with , 0 SYNTAX :: SEMANTICS :: SATISFIABILITY AND VALIDITY :: FRAMES :: REASONING SERVICES :: THEOREMS :: TABLEAU

26. Modal logics Tableau Example (I) 26  (  A   B)  B satisfiable? (  ) (  A   B)  B, 0 / \ (  ) (  A   B), 0 B, 0 | (open)  A, 0 |  B, 0 (open) SYNTAX :: SEMANTICS :: SATISFIABILITY AND VALIDITY :: FRAMES :: REASONING SERVICES :: THEOREMS :: TABLEAU

27. Modal logics Tableau Example (II) 27   ◊A satisfiable? (duality)  ◊A, 0 | ( □ ) □  P, 0 0R1 |  P, 1 (open) SYNTAX :: SEMANTICS :: SATISFIABILITY AND VALIDITY :: FRAMES :: REASONING SERVICES :: THEOREMS :: TABLEAU

28. Modal logics Tableau Example (III) 28  □ A   □ A valid? We negate and check whether ALL branches are closed. The negation is:  ( □ A   □ A)   □ A  □ A (  )  □ A  □ A, 0 | (duality)  □ A, 0 | ( □ ) □ A, 0 0R1 | (◊) ◊  A, 0 | A, 1 | 0R1  A, 1 (closed) SYNTAX :: SEMANTICS :: SATISFIABILITY AND VALIDITY :: FRAMES :: REASONING SERVICES :: THEOREMS :: TABLEAU

29. Modal logics Tableau: additional rules 29  We have extra rules to convey frame properties: (reflexivity) *| iRi (symmetry) iRj| jRi (transitivity) iRj jRk | iRk (seriality) *| iRj Euclidian properties can be given as a combination of the first three. SYNTAX :: SEMANTICS :: SATISFIABILITY AND VALIDITY :: FRAMES :: REASONING SERVICES :: THEOREMS :: TABLEAU

30. Modal logics Tableau Example (IV) 30  □ A  A valid in reflexive frames? The negation is:  ( □ A  A)   (  □ A  A)  □ A   A (  ) □ A   A, 0 |  A, 0 | ( □ ) (reflexive) □ A, 0 0R1 | A, 1 | 0R0 A, 0 (closed) SYNTAX :: SEMANTICS :: SATISFIABILITY AND VALIDITY :: FRAMES :: REASONING SERVICES :: THEOREMS :: TABLEAU