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L ogics for D ata and K nowledge R epresentation

L ogics for D ata and K nowledge R epresentation. Modal Logic. Originally by Alessandro Agostini and Fausto Giunchiglia Modified by Fausto Giunchiglia, Rui Zhang and Vincenzo Maltese. Outline. Introduction Syntax Semantics Satisfiability and Validity Kinds of frames

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L ogics for D ata and K nowledge R epresentation

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  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 • 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”

  4. Syntax SYNTAX :: SEMANTICS :: SATISFIABILITY AND VALIDITY :: FRAMES :: REASONING SERVICES :: THEOREMS :: TABLEAU • 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 SYNTAX :: SEMANTICS :: SATISFIABILITY AND VALIDITY :: FRAMES :: REASONING SERVICES :: THEOREMS :: TABLEAU 5

  6. Semantics: Kripke Model SYNTAX :: SEMANTICS :: SATISFIABILITY AND VALIDITY :: FRAMES :: REASONING SERVICES :: THEOREMS :: TABLEAU • 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 SYNTAX :: SEMANTICS :: SATISFIABILITY AND VALIDITY :: FRAMES :: REASONING SERVICES :: THEOREMS :: TABLEAU • 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) SYNTAX :: SEMANTICS :: SATISFIABILITY AND VALIDITY :: FRAMES :: REASONING SERVICES :: THEOREMS :: TABLEAU • 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 SYNTAX :: SEMANTICS :: SATISFIABILITY AND VALIDITY :: FRAMES :: REASONING SERVICES :: THEOREMS :: TABLEAU • 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 SYNTAX :: SEMANTICS :: SATISFIABILITY AND VALIDITY :: FRAMES :: REASONING SERVICES :: THEOREMS :: TABLEAU • 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 SYNTAX :: SEMANTICS :: SATISFIABILITY AND VALIDITY :: FRAMES :: REASONING SERVICES :: THEOREMS :: TABLEAU • 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. Kinds of frames SYNTAX :: SEMANTICS :: SATISFIABILITY AND VALIDITY :: FRAMES :: REASONING SERVICES :: THEOREMS :: TABLEAU • 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 12

  13. Kinds of frames SYNTAX :: SEMANTICS :: SATISFIABILITY AND VALIDITY :: FRAMES :: REASONING SERVICES :: THEOREMS :: TABLEAU • 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 = <W, R> serial, reflexive, symmetric or transitive according to the properties of the relation R 1 2 3 1 2 3 13

  14. Valid schemas SYNTAX :: SEMANTICS :: SATISFIABILITY AND VALIDITY :: FRAMES :: REASONING SERVICES :: THEOREMS :: TABLEAU • 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.  14

  15. Proof for D: □A  ◊A valid for serial frames SYNTAX :: SEMANTICS :: SATISFIABILITY AND VALIDITY :: FRAMES :: REASONING SERVICES :: THEOREMS :: TABLEAU In all serial frames M = <W, R>, 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 □A, ◊A 1 2 3 □A, ◊A, A 15

  16. Proof for T: □ A  A valid for reflexive frames SYNTAX :: SEMANTICS :: SATISFIABILITY AND VALIDITY :: FRAMES :: REASONING SERVICES :: THEOREMS :: TABLEAU 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 16

  17. Proof for B: A  □◊A valid for symmetric frames SYNTAX :: SEMANTICS :: SATISFIABILITY AND VALIDITY :: FRAMES :: REASONING SERVICES :: THEOREMS :: TABLEAU 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’ 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 17

  18. Reasoning services: EVAL Yes EVAL M, P No SYNTAX :: SEMANTICS :: SATISFIABILITY AND VALIDITY :: FRAMES :: REASONING SERVICES :: THEOREMS :: TABLEAU • 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 ? 18

  19. Reasoning services: SAT M SAT P No SYNTAX :: SEMANTICS :: SATISFIABILITY AND VALIDITY :: FRAMES :: REASONING SERVICES :: THEOREMS :: TABLEAU • 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 19

  20. Reasoning services: UNSAT w VAL M, P No SYNTAX :: SEMANTICS :: SATISFIABILITY AND VALIDITY :: FRAMES :: REASONING SERVICES :: THEOREMS :: TABLEAU • 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 does not exist w such that M, w ⊨ P 20

  21. Reasoning services: VAL Yes VAL P No SYNTAX :: SEMANTICS :: SATISFIABILITY AND VALIDITY :: FRAMES :: REASONING SERVICES :: THEOREMS :: TABLEAU • 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 21

  22. Equivalence Modal logics – First Order Logic SYNTAX :: SEMANTICS :: SATISFIABILITY AND VALIDITY :: FRAMES :: REASONING SERVICES :: THEOREMS :: TABLEAU • We can define a translation function  : LML LFO as follows: (P) = P(x) for all propositions P in LML (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 LML, P is modally valid iff (P) is valid in FOL. 22

  23. Equivalence Modal logics – Description logics SYNTAX :: SEMANTICS :: SATISFIABILITY AND VALIDITY :: FRAMES :: REASONING SERVICES :: THEOREMS :: TABLEAU • Take ALCEU: <Atomic> ::= A | B | ... | P | Q | ... | ⊥ | ⊤ <wff> ::= <Atomic> | ¬ <wff> | <wff> ⊓ <wff> | <wff> ⊔ <wff> | ∀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 LML, P is modally valid iff(P) is valid in DL. 23

  24. Modal logics Tableau: introduction SYNTAX :: SEMANTICS :: SATISFIABILITY AND VALIDITY :: FRAMES :: REASONING SERVICES :: THEOREMS :: TABLEAU • 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 ∈ LML 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 24

  25. Modal logics Tableau: rules SYNTAX :: SEMANTICS :: SATISFIABILITY AND VALIDITY :: FRAMES :: REASONING SERVICES :: THEOREMS :: TABLEAU • 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 25

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

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

  28. Modal logics Tableau Example (III) SYNTAX :: SEMANTICS :: SATISFIABILITY AND VALIDITY :: FRAMES :: REASONING SERVICES :: THEOREMS :: TABLEAU • □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) 28

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

  30. Modal logics Tableau Example (IV) SYNTAX :: SEMANTICS :: SATISFIABILITY AND VALIDITY :: FRAMES :: REASONING SERVICES :: THEOREMS :: TABLEAU • □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) 30

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