1. Slide-1 Modal Logic and Its applications Cheng-Chia Chen Department of Computer Science, National Cheng-Chi University

2. Slide-2 Contents Classical propositional logic (CPL) Basic modal logic logic of knowledge and belief deontic logic logic of actions and programs(PDL)

3. Slide-3 Elements of a Logic Language syntax (formal language) semantics (model theory) axiomatics (proof theory) decidability & complexity (computation theory) automated deduction (Theorem proving)

4. Slide-4 Classical Propositional Logic(CPL) The language L: – a set of proposition symbols (PV) : –p,q, r... means it-is-raining, it-is-cloudy,... logical connectives: /\ (and), ~ (negation) (well-formed) formulas (abstract syntax): P ::= p | P /\ Q | ~P Definitions: P \/ Q abbreviates ~(~P /\ ~Q) P => Q abbreviates ~(P /\ ~Q)

5. Slide-5 The semantics for CPL Goals: –1. define the contexts in which formulas can be given truth values. –2. define the truth conditions for formulas. interpretation (world, state): any assignment of truth value {1,0} to propositional symbols Truth conditions (or satisfaction relation) |= : I |= p iff I(p)=T; I |= P /\ Q iff I |= P and I |= Q I |= ~P iff not I |= P If I |= A, then say I is a model of A.

6. Slide-6 Some logical notions A formula is satisfiable iff it is true in some world. A formula is valid (a tautology) (|= A) if it is true in all worlds. A is a logical consequence of a set of formulas S (S |= A) iff A is true in all models of S. Problems : How to characterize the set {A | A is a tautology} ?

7. Slide-7 Calculus and provability A calculus C over a language L is a finite set of rules, each of the form: –(A1,A2,..., An, B) –A1,A2,...,An : Premises –B: conclusion –if n = 0 => axioms Example: (A, B, A /\B), (A, A=>B, B), (A=>B, B, A),...

8. Slide-8 Provability Given a calculus C, The set C = {A | A is C-provable(denoted |- C A)} is defined recursively as follows: –Basis:If (A) is a rule, then A in C ---axioms –Ind: If (A1,..,An,B) is a rule & – A1,...,An in C, then B in C.

9. Slide-9 An axomatization for CPL Let CPL be the calculus: (1) Axiom schema: –A => (B => A) –(A=>(B =>C)) => ((A=>B)=>(A=>C)) –(~A => ~B) => (B => A) (2) Inference rule: –from A and A => B infer B (MP) Theorem: A is valid in CPL iff A is CPL-provable

10. Slide-10 Basic Modal logic The logical study of necessity and possibility The language: –CPL augmented with two modal operators: [] (necessity) and ⃟ (possibility). –P : any proposition, then []P (<>P) means “P is necessarily (possibly) true”. –Meaning of []p: depends on the context it is used, not only determined by the truth value of p A family of logics instead of a single logic

11. Slide-11 Types of necessity logical necessity: –e.g, p \/ ~p is logically necessarily true. physical necessity: –F=ma Epistemic necessity: –e.g., It is believed(known) that... Normal necessity: – e.g., It is obligated (permitted, forbidden) that... time-related (always, eventual) Others: –After the programs terminates P must holds,...

12. Slide-12 Formal Definition The language: –Alphabet (  ): PV: a set of propositional variables. logical connectives: ~ (not), /\ (and), [] (necessity) –MF: a set of modal formulas defined inductively: A ::= p | A /\ B | ~ A | []A –Abbreviations (Macros) (A \/ B) abbreviates ~(~A /\ ~B); (A  B) abbreviates ~(A /\ ~B) ⃟ A abbreviates ~[]~A

13. Slide-13 Possible-world Semantics for modal logic Truth conditions for p /\ q, p \/ q, p  q, and ~p. –Let p = “I win the game”, – q = “It is 5 p.m.” –Assume I win the game and – the present time is 3 p.m, – then p/\q: false, p\/q: true and p  q: false. But how about the statement: []p =It must be the case that I win the game. “

14. Slide-14 Meaning of necessity and possibility: The game: –Two players A,B, each getting a card from four cards labeled 1,2,3,4 randomly. rule: –The player who get a card larger than the other’s wins.

15. Slide-15 Scenario I: A gets “2”. Then consider the following sentences: –1. “A may possibly win” = “It is possibly true that A win” = “ ⃟ A_win” –2. “A may possibly not win” –3. “A must win” –4. “B must not get “2”” Which is right ? why?

16. Slide-16 The answer: Statement 1 is right –since (2,1) may be the real world, in which A wins. Statement 2 is right –since (2,3), (2,4) are possible, in which A does not win. statement 3 is false –since there are cases (e.g., (2,3), (2,4)) in which A does not win. Statement 4 is true since in all possible cases B does not get 2.

17. Slide-17 The Rule: (2,1) A_win ~B_2 (2,3) ~A_win ~B_2 (2,4) ~A_win ~B_2 Possible worlds Impossible worlds (2,?) Real world ~[]A_win ⃟ A_win ⃟ ~A_win [] ~B_2 (3,4)

18. Slide-18 The Possible-world Semantics: Let W = the set of worlds –e.g, {(x,y) | x = 1..4, y =1..4 & x  y} Let V : W x PV -> {0,1} be a valuation function s.t., V(w,p) =1 iff p is assigned true at world w. –e.g, V((2,1), A-win) = 1 R be a binary relation (I.e., subset of WxW) s.t. wRw’ iff w’ is a possible world of w. –e.g, (2,x)R(2,1), (2,x)R(2,3), (2,x)R(2,4). The triple M= is called a (possible-world) structure.

19. Slide-19 Truth-conditions for modal formulas M = : a possible world structure; w: a world ∈ W, The statement : “A is true at world w in structure M” is defined as follows: –M,w |= p iff V(w,p) = 1 –M,w |= A /\ B iff M,w |= A and M,w |= B –M,w |= ~A iff not M,w |= A. –M,w |= ⃟ A iff – A is true at some possible world of w. –M,w |= [] A iff A is true at all possible worlds of w.

20. Slide-20 Some definitions A: modal formula, M: structure, C: a class of structures A is valid iff it is true in all worlds of all structures. A is C-valid iff it is true at all worlds of all structures of C. Problem: Given a class of structures C, –{A | A is C-valid } = ?

21. Slide-21 Interesting classes of structures Class name Property of R T reflexive: wRw. D serial: for all w, there is w’ s.t. w R w’. 4 transitive: wRw’ & w’Rw’’ ⇒ wRw’’. 5 Eulidean: wRw’ & wRw’’ ⇒ w’ R w’’. B symmetric: wRw’ ⇒ w’Rw. r: any string from {T,D,4,5,B} without repetition. Kr = the class of the structures whose R satisfying all properties mentioned in r. –(I.e., Every theorem of the logic Kr is valid in all Kr- struture, and vice versa.)

22. Slide-22 Axiomatization of modal logics Axioms definitions PC all truth-functional tautologies K [](P  Q)  ([]P  []Q) T []P  P D []P  ~[]~p 4 []P  [][]P 5 ~[]P  []~[]P B ~P  []~[]P. Inference rule: MP: from P, P  Q infer Q Nec: from P infer []P

23. Slide-23 Axiomatizations of modal logic r: any subset {T,D,4,5,B}. Kr = the axiom system (calculus) including axioms K, PC and all of r and inference rules MP and Nec. Kr-provable formulas are defined recursively as follows: –1. Every axioms of Kr is Kr-provable. –2. If P, P  Q are Kr-provable then so is Q (MP) –3. If P is Kr-provable, then so is []P (Nec). Theorem[Chellas80]: –A is Kr-valid iff A is Kr-provable.

24. Slide-24 Some useful modal logics Logical system Property of R usage S5 (KT45) equivalencelogic of knowledge KD serial deontic logic KD45 almost equ. logic of belief S4 (KT4) ref. tran.Intuitionistic logic S4.3 linear(total) temporal logic     w    w real world must be possible real world may and may not be possible Worlds inside are fully connected {w’ | w R w’}

25. Slide-25 Logic of Knowledge and Belief Modal logic of knowledge : KT45(S5) Modal logic of belief: KD45( weak S5) Epsitemic interpretation of knowledge&belief axioms –KA means A is known; BA means A is believed. – T: []A  A (knowledge axioms) – D: []A  ~[]~A (belief axiom) – 4: []A  [][] A (positive introspection) – 5:~[]A  []~[]A (negative introspection) –K:[]A /\ [](A  B)  []B (distribution axiom) –Nec: From p infer []p -- agent knows the logic

26. Slide-26 Extensions to multimodal logics: –S5 (KD45) can model only one single agent’s knowledge (believes) –Multi-agent cases: n agents: 1,2,3,...,n; 2n knowledge(and belief) operators K 1,B 1,...,K n,B n : K i A ( B i A ) means agent i knows(resp. believes) A. –Resulting logic: S5 n WS5 n N copies of S5, and N copies of KD45, each for one agent.e.g., Tj: K j A  A where j =1,..,n. – semantics: Structure M= Each Ki is an equivalence relation on W and Bi is a serial,trans. and euclidean relation.

27. Slide-27 Related Issues[Halpern85] Logical Omniscience Problem: Agents with S5 (KD45) ability are perfect logical reasoners, but human never be. Common knowledge, Distributed knowledge –[E]P = [1]P /\ [2]P.../\[n]P –[C]P = [E]P /\[E][E]P /\ [E][E][E]P /\... = [E]P /\[E][C]P –[D]P = P can be known by an agent who knows all what others known (the wisest man). –Needed and useful in many fields (Economics,distributing sys,AI...)

28. Slide-28 Deontic interpretation of modal logic Deontic logic (D or KD) –PA means A is permitted; OA means A is obligated; FA means A is forbidden. –A is (strongly) forbidden = Doing A or bringing about A will result in punishment (dangerous, disastrous) worlds. –A is obligated = not doing A or not bring about A will result in punishment. = ~A is forbidden. –A is (weekly) permitted = A is not forbidden = doing A may not result in punishment. –Another possible pairs: – weekly forbidden/strongly permitted

29. Slide-29 Semantic analysis of forbidden, obligation and permission commit-crime or dead (undesired world) ~drive-car murder~pay-tax ~dead drive-car ~dead pay-tax ~ murder drive-car murder pay-tax dead ~drive-car pay-tax ~murder ~dead ~drive-car~pay-tax dead ~murder Permitted worlds current world sets of worlds which may become the real world F murder : since all murder-worlds are red. O pay-tax: since all ~pay-tax world are red. P drive-car: some drive-car-world is white.

30. Slide-30 Formalization of Deontic logic W: The set of all possible worlds D: A set of undesired, punishment world V: WXPV -> {0,1} with the constraint that –V(w,v) = 1 iff w ∈ D. I.e., we use v to denote all sanction or punishment worlds. R: a binary relation on W, s.t. –wRw’ means w’ is a possible world that the agent may choose to become the real world from w.

31. Slide-31 Truth conditions for PA,OA, &FA – M,w |= FA iff M,w |= [] (A  v) ie., for all w’, if wRw’ & M,w|=A then M,w |= v. –M,w |= OA iff M,w |= F~A iff M,w |= [](~A  v) –M,w |= PA iff M,w |=~FA iff M,w |= ⃟ (A/\ ~v) I.e., there is a world w’ s.t. wRw’ & M,w |= A /\ ~v.

32. Slide-32 Properties of the deontic logic: By definition: –FA = [] (A  v) ; –OA = F~A = [](~A  v); –PA = ~FA =  (A /\ ~v); All KD axioms(K, D) Desirable property: OA => PA: not valid in K but valid in KD (I.e., R must be serial)

33. Slide-33 Temporal interpretation of modal logic real history now possible future real past possible past real future Taxonomy of temporal structures: linear v.s. branch-time, past time v.s. future time v.s. past&future continuous v.s. discrete

34. Slide-34 Linear discrete time temporal logic Temporal operators: –FA means A is eventually true –GA means A is always true –A U B means A is true until B becomes true –0A: A is true at the next time.

35. Slide-35 Meaning of temporal formulas 0 1 2 3..... n n+1m Fp p Gq q q q q.... q..... q 0r r AUB A A A A B Linear discrete-time temporal structure: initial world

36. Slide-36 Meaning of temporal formulas linear discrete temporal logic: W = N = {0,1,2,3,...} :time point set V:NXPV -> {0,1} Truth conditions: –M,n |= 0A iff M,n+1 |= A. –M,n |= FA iff there is m  n s.t., M,m |= A –M,n |= GA iff for all m  n, M,m |= A. –M,n |= A U B iff there is m  n s.t., M,m|= B & for all m > s  n, M,s |= A.

37. Slide-37 Logic of programs and actions Modal logic of programs (Dynamic Logic) PDL: propositional version of DL The language: –Primitive programs: a,b,c,... –Primitive propositions: p,q,r... –program constructs: “ ;”, “|”,”*”,”?”. –logic connectives: /\,~, [A] for each program A.

38. Slide-38 – (Compound) Programs A ::= a | any primitive program is a program (x++ in C) A;B | doing A and then doing B A+B | doing A or doing B nondeterministically A* | iterate A a nondeterminstic number of times A* = t + A + A;A + A;A;A +... P? | test if P is true. Syntax of Programs

39. Slide-39 Syntax of Formulas –Formulas(assertions): P ::= –p any primitive proposition is a formula –P /\ Q both P and Q are true –~P P is not true –[A]P After A terminates, P will be true. – P = ~[A]~P means P holds at some execution of A.

40. Slide-40 An Example: integer x,y,z –x := 3 ; –y := (1,4); –z := x+1 | y := x Problems: –Is it true that z > 0 or y  x-2 after executing the program, suppose initially the program state is (4,3,2) ?

41. Slide-41 Formalization of the problem: two primitive propositions: – p = “z > 0” ; q = “z  x-2” four primitive programs: – a = “x := 3”, b = “y :=(1,4)”, – c = “z := x+1”, d = “y := x”. The program : A = a;b; (c | d) The problem: is [A] (p \/ q) true ?

42. Slide-42 Analysis: A program state is triple (I,j,k) of integers, – which denote the possible simultaneous values of variables (x,y,z). Let W = {(i,j,k) | i,j,k are integers} be the set of all possible program states.

43. Slide-43 a = “x := 3”, b = “y :=(1,4)”, c = “z := x+1”, d = “y := x”. p = “z > 3”, q = “z >= x+1” (3,1,4) (3,3,2) (3,4,2) (3,1,2) (4,3,2) a b b (3,3,2) (3,4,4) c c d d p ~p p q ~q q p\/q ~(p\/q) p\/q ~(p\/q) a;b c+d a;b;(c+d) initial program state

44. Slide-44 (i,j,k) (3,j,k) a: x:=3 (i,1,k)(i,j,i+1) b: y:=(1,4) (i,4,k) b d: y := x c: z:= x+1

45. Slide-45 The Semantic rules 0. Let W = the set of all possible program states 1. Each primitive proposition has a truth value in a program state: –denoted by a function: V: W x PV  {1,0} s.t. –V(w,p) = 1 iff p is true at state w. 2. Each primitive program a is a state transformer, denoted by a binary relation R(a): WxW s.t., w R(a) w’ means the program state can become w’ from w by executing a. M= is called a (program) structure.

46. Slide-46 Composition rule for programs: R(A;B) = R(A)R(B) = {(w,w’’) | there is w’ s.t., w R w’ and w R w’’. R(A+B) = R(A) U R(B); R(A)* = I UR(A) UR(A)R(A) U... = R(A)* I.e., ref. and trans closure of R(A). R(P?) = {(w,w) | P is true at w}. Define classical program constructs: –if P then A else B –if P then A else B  P?;A + ~P?;B –while P do A –while P do A  (P?;A)* ; (~P?) –Repeat A until P –Repeat A until P  A;(~P?;A)*;P?

47. Slide-47 Truth conditions for Formulas –M,w |= p iff V(w,p)=1 –M,w |= P /\ Q iff M,w|=P and M,w|=Q. –M,w|=~P iff not M,w|=P. –M,w|= [A]P iff for all w’, w RA w’ then M,w’|=P. –M,w|= P iff there is w s.t. wRAw’ & M,w’|=p. A formula is valid iff it is true at every world of every program structure. A formula is satisfiable if it is true at some world of some program structure. Subsume Hoare logic: P {A} Q  (P  [A] Q)

48. Slide-48 Variants of PDL [Harel84] DPDL –atomic programs are deterministic SPDL (structure PDL) –remove + and * –add “if then else” and “while do”. SDPDL (structure DPDL): –atomic programs are deterministic –replace + and * by “if then else” and “while do”.

49. Slide-49 PDL as a logic of actions Too strong part: –The *-operator may not be necessary –The +-operator is not very natural Too weak part: –need a notion of not doing something (I.e., A: an action => -A : an action (not doing A) –need a notion of concurrent/parallel execution of actions. A,B: actions => A&B means (doing A and B in parallel)) A \/ B means A;B + B;A + A&B –Need internal free choice: A  B

50. Slide-50 Axiomatize PDL The following formulas are valid in PDL 1. CPL: all tautologies of propositonal logic 2. K: [A](P  Q) /\ [A]P  [A]Q 3. cmp: [A;B]P [A][B]P 4. union: [A+B]P ([A]P /\ [B]P) 5. test: [P?]Q (P  Q) 6. mix: [A*]P -> (P /\[A]P /\ [A][A]P /\ …) ∴ [A*]P -> (P /\ [A][A*]P) 7. induction: (P /\ [A*](P  [A]P))  [A*]P

51. Slide-51 PDL Valid inference rules in PDL: –MP: From P and P  Q infer Q –Gen: From P infer [A]P Theorem:Theorem: –1. P is valid in PDL iff P can be proved from the above calculus. (In symbols, |= PDL P  |- PDL P) –2. The set {A | A is a valid in PDL} is EXPTIME- complete