Modal, Dynamic and Temporal Logics

1. Modal, Dynamic and Temporal Logics SWE 623 Duminda Wijesekera

2. Modal Logic • Logic of Necessity and Possibility • Has a philosophical background • Syntax has two extra symbols • [] read as necessity ([] X is “necessarily X”) • Also called “box X” • <> read as possibility (<> X “possibly X”) • Also called “diamond X” • See http://turing.wins.uva.nl/~mdr/AiML/background.html Duminda Wijesekera

3. Kripke Semantics of Modal Logic W4 W1 • The “universe” seen as a collection of worlds. • Truth defined “in each world”. • Say U is the universe. • I.e. each w e U is a prepositional or predicate model. W2 W3 Duminda Wijesekera

4. Kripke Semantics of Modal Logic W4 W1 • W1 satisfies [] X if X is satisfied in each world accessible from W1. • If W3 and W4 satisfy X. • Notation: • W1 |= [] X if and only if • W3 |= X and W4 |= X • W1 W1 satisfies <> X if X is satisfied in at least one world accessible from W1. W2 W3 • Notation: • W1 |= <> X if and only if • W3 |= X or W4 |= X Duminda Wijesekera

5. Proof Rules for Modal Logic • Modal Generalization A [] A • Monotonicity of  A  B  A  B • Monotonicity of  A  B [] A []B Duminda Wijesekera

6. An Axiom System for Prepositional Logic • (A  (B  C))  (A  B)  (A  C) • A  (B  A) • (( A  false )  false ) A • Modus Ponens A, A -> B  B Duminda Wijesekera

7. An Axiom System for Predicate Logic • x (A(x)  B(x))  (xA(x) xB(x)) • x A(x)  A[t/x] provided t is free for x in A • A x A(x) provided x is not free in A • Modus Ponens A, A -> B B • Generalization A x A(x) Duminda Wijesekera

8. Some Facts About Modal Logic • A couple of Valid Modal Formulas: •  (A  B ) <-> ( A)  ( B) • [](A  B ) <-> ([] A)  ([] B) •  (false) (false) • ( A)  ([]B)   (A  B ) • Counter-examples to invalid modal formulas • ( A)  ( [] A ) Duminda Wijesekera

9. Proving Modal Formulas Duminda Wijesekera

10. A counter-example in Modal Logic Duminda Wijesekera

11. Dynamic Logic • A special kind of Modal Logic where each world is a system state. • Definition of State • The set of variables x1, … xn. • x1= a1, … xn= an. is a state, where each variable takes a value. • Accessibility is state change perhaps due to executing code. • x1= a1, … xn= an is changed to x1= b1, … xn= an by the program (x1 := b1). Duminda Wijesekera

12. Dynamic Logic • Issues: • What kind of program constructs result in what type of state change • What is the logic • Two Levels • Prepositional: • Only deals with state change at (abstract) symbolic level • Predicate: • Details of variables, values and programming operators • Deals well with non-determinism, concurrency etc. Duminda Wijesekera

13. Prepositional Dynamic LogicSyntax • If A, B propositions and a, b programs, • Following are formulas • A /\ B, A  B,  A, A B, [a]A, < a>A are formulas. • Following are programs • U b = non-deterministic choice a; b = sequential composition (A?) a = test. a* = non-deterministic iteration Duminda Wijesekera

14. Prepositional Dynamic LogicSemantics • A collection of states: S = {si : i >= 0}. • For each state si a notion of satisfiability of atomic prepositions. I.e. si |= A for each A. • For each each atomic program a, a relation Ra on SxS. • Raub = Ra u Rb • R(A?) = { (s,s) : s |= A } • Ra;b = Ra ; Rb ={ (s1,s3) :  s2 (s1,s2) e Ra and(s2,s3) e Rb } • Ra* = U {Rai: i >=0 }. WhereRaiis defined inductively as Ra(i+1) = Rai ; RaandRa0 = Identity. Duminda Wijesekera

15. PDL Semantics - Satisfaction • Prepositional connectives as usual: • I.e. si |= A /\ B if si |= A and si |= B • I.e. si |= A  B if si |= A or si |= B • Modal Connectives as in Modal Logic • I.e. si |= [a]A, if for all states sj such that (si , sj) e Ra sj |= A • I.e. si |= <a>A, there is a state sj with (si , sj) e Ra andsj |= A Duminda Wijesekera

16. PDL Axiom System • Axioms of prepositional logic • [a] (A B) ([a]A [a]B) • [a] (A /\B) <-> ([a]A /\ [a]B) • [a U b]A <-> ([a] A /\ [b] A) • [a ; b]A <-> [a] [b] A • [B?]A <-> (B /\ A) • B /\ [a] [a*] A <-> [a*] A • B /\ [a*]( A [a]A)  [a*] A Duminda Wijesekera

17. PDL Axiom System: Rules • Modus Ponens A, A -> B B • Modal Generalization A [a] A Duminda Wijesekera

18. Some Derived Rules for PDL • Monotonicity of <a> A -> B <a>A -> <a>B • Monotonicity of [a] A -> B [a]A -> [a]B Duminda Wijesekera

19. Some Provable Properties • [a] (A /\B) ([a]A /\[a]B) • <a> (A \/B) <-> (<a>A \/ <a>B) • (<a>A /\ [a] B)  <a>(A /\ B) • [a ]A <-> ( <a>( A)) • <a>false <-> false • <a><b>A <-> <a;b>A, • [a][b]A <-> [a;b] A • < a U b>A <-> (<a>A \/ <b>B) • [ a U b]A <-> ([a]A /\ [b]B) Duminda Wijesekera

20. Translating Gires’s Style Pre/Post Conditions to PDL • Skip == True? • Fail == false? • If A then a else b == (A?;a) U (A?;b) • While A do a == (A?;a)*; (A?) Duminda Wijesekera

21. First-Order Dynamic Logic • Syntax: • The same definition as predicate logic except for the additions • If A is a formula and a is a program, then [a]A, <a>A are formulas. • If A is a formula, then A? is a test. (I.e. a program) • If A is quantifier free then its said to be a basic test, and otherwise a rich test. Duminda Wijesekera

22. First-Order Dynamic Logic • Semantics: Transitions between states defined as • R(X :=a) = { (S, S’) : if S’(x) = S(a) and S’(y) = S(y) for Y != X } • R(A?) = {(S,S) : S |= A } • Definitions of U, ; are same as in the prepositional case. Duminda Wijesekera

23. Axiomatization • Axioms • All axioms for predicate logic • All axioms for PDL • A[t/x] <-> < x:= t>A(x) • A <-> A’, A’ is obtained by replacing any program a by z:=x; a’; x:=z, where a’ is a with all occurrences of x replaced by z, and z does not appear in a Duminda Wijesekera

24. Axiomatization: Rules • modus ponens A, A -> B B • Generalization A A [a] A  x A(x) • Infinitary convergence A -> [an]B for all n B -> [a*]B Duminda Wijesekera

25. Some Example Reductions I • Reduce: X:=X+1; ((X:=a) U (X:=b))  A(X) • Step1: X=X+1; (X=a)  (X=b)  A(X) • Step2: X=X+1  (X=a)  A(X)  <X=X+1  (X=b)  A(X) • Step3: X=X+1  A • Step4: A(a)  A(b) Duminda Wijesekera

26. Some Example Reductions II • Reduce: [x:=x+1;(x:=a U x:=b)] B(X) • Step1: [x:=a+1 U x:=b+1]B(x) • Step 2: [x:=a+1]B(x) /\ [x:=b+1]B(x) • Step 3: B(a+1) /\ B(b+1) Duminda Wijesekera

27. Temporal Logic • Special kind of modal logic to reason about time. • There are many kinds of Temporal Logics • Linear and Branching Time • Future and Past times • Discrete and Continuous time • Operators in Temporal Logics (MacMillan’s Notation) • O = next time F • [] = always G •  = some times X •  = until U Duminda Wijesekera

28. Prepositional Syntax • Atomic Proposition letters p, q etc. • If p, q are propositions then so are. • MeaningLogical NotationModel Checking • Next Time p: Op Xp • All ways p: []p Gp • In the future p: p Fp • p until q: p  q pUq Duminda Wijesekera

29. Prepositional Semantics • A collection of Kripke Worlds including the current one. • Accessibility relation is evolution of time. Duminda Wijesekera

30. Prepositional Semantics II • |= Op if some world accessible from the current satisfies p. • |= []p if every world accessible from the current satisfies p. • |=  p if some world in the future from the current satisfies p. Duminda Wijesekera

31. PTL Axioms and Rules I • Axioms • [](A ->B) ->([]A -> []B) • O(A ->B) -> (OA -> OB) • (O  A) <-> (OA) • []A -> (A /\ O[]A) • [](A -> OA) -> (A -> []A) • A  B -> B • A  B <-> B \/ (A /\ O(A  B )) Duminda Wijesekera

32. PTL Axioms and Rules II • Rules • modus ponens • generalization A [] A A O A Duminda Wijesekera