1 / 30

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

neviah
Download Presentation

L ogics for D ata and K nowledge R epresentation

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  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

More Related