NKU CSC 685 Fall 2006 Temporal Propositional Logic

1. NKU CSC 685 Fall 2006 Temporal Propositional Logic NKU CSC 686 Kirby

2. Preview: Modal Logic • This is still informal. • To specify a logic, we need • a proof theory () • a model theory () • (HR Chapter 5) p p is true p p is possibly true []p p is necessarily true  it is raining [] all bachelors are unmarried Duality p  []p p  []p NKU CSC 686 Kirby

3. Preview: Linear Temporal Logic p p is true now Xp p will be true next Fp p will be true in the future* Gp p will be true from now on p p p p Duality Xp  Xp Fp  Gp Gp  Fp Distributive Properties F(pq)  FpFq G(pq)  GpGq Why not these as well? G(pq)  GpGq F(pq)  FpFq * the "future" always includes "now" NKU CSC 686 Kirby

4. Preview: Linear Temporal Logic pWq p is true at least until q is true p q q includes these special cases note change! p Other temporal logic operators: pUq := pWq  Fq pRq := (pUq) pWq  pUq  Gq NKU CSC 686 Kirby

5. a b happy sleepy happy chilly c ... 3 2 Semantics for Linear Temporal Logic (LTL) • A modelM in LTL consist of: • Q = finite set of states • A = finite set of atoms // true/false propositions • a situation function L:QPow(A)// what's true at each state • a transition relation  QQ // what can happen next A path is a denumerable sequence of states connected by  A = {happy,sleepy,chilly} Q = {a,b,c}  = {(a,b), (b,a),(b,c),(c,)} L: L(a)= {happy} L(b)= {happy,sleepy} L(c)= {chilly}  = abcccc ... NKU CSC 686 Kirby

6. a b happy sleepy happy chilly c ... 3 2 Semantics for Linear Temporal Logic (LTL) Let  be an LTL wff and let  be a path in a model M.   means:  is true at the first state (="now") in the path  M,  happy   sleepy M,  chilly M,  Xsleepy M,  XXchilly M,  Fchilly M,  XXGchilly M,  XX sleepy M,  G (happy  chilly) M,  Fchilly XXGchilly M,  happyWchilly M,  happyUchilly  = abcccc ... Semantic equivalence:  if for all M and all  in M: M,  iff M, NKU CSC 686 Kirby

7. a b happy sleepy happy chilly c Semantics for Linear Temporal Logic (LTL) Let  be an LTL wff and let  be a path in a model M. M,q  means:  is true at the first state on all paths in M starting at q let's check these... yes yes yes NO! NO! NO! yes yes NO! yes NO! M,a  happy   sleepy M,a  chilly M,a  Xsleepy M,a  XXchilly M,a  Fchilly M,a  XXGchilly M,a  XX sleepy M,a  G (happy  chilly) M,a  Fchilly XXGchilly M,a  happyWchilly M,a  happyUchilly NKU CSC 686 Kirby

8. How do we prove semantic equivalence? If we had a sound and complete proof theory, we could prove  using . We don't. So we prove  directly. We can translate LTL formulas into first-order logic. How? Example: Prove U  (U())  F

9. Computation Tree Logic (CTL) Future = a branching structure now LTL is limited: Cannot quantify over paths within the logic. CTL models are the same as LTL (= transition systems). AX EX X holds along all paths / some paths AG EG G holds along all paths / some paths AF EF F holds along all paths / some paths AU EU U holds along all paths / some paths The textbook writes A[U] for AU, etc.