FORMAL METHODS LECTURE III: LINEAR TEMPORAL LOGIC

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1 Alessandro Artale (FM First Semester 2007/2008) p. 1/39 FORMAL METHODS LECTURE III: LINEAR TEMPORAL LOGIC Alessandro Artale Faculty of Computer Science Free University of Bolzano artale/ Some material (text, figures) displayed in these slides is courtesy of: M. Benerecetti, A. Cimatti, M. Fisher, F. Giunchiglia, M. Pistore, M. Roveri, R.Sebastiani.

2 Alessandro Artale (FM First Semester 2007/2008) p. 2/39 Summary of Lecture III Introducing Temporal Logics. Intuitions beyond Linear Temporal Logic. LTL: Syntax and Semantics. LTL in Computer Science. LTL Interpreted over Kripke Models. LTL and Model Checking: Intuitions.

3 Alessandro Artale (FM First Semester 2007/2008) p. 3/39 An Introduction to Temporal Logics In classical logic, formulae are evaluated within a single fixed world. For example, a proposition such as it is Monday must be either true or false. Propositions are then combined using constructs such as,, etc. But, most (not just computational) systems are dynamic. In temporal logics, evaluation takes place within a set of worlds. Thus, it is Monday may be satisfied in some worlds, but not in others.

4 Alessandro Artale (FM First Semester 2007/2008) p. 4/39 An Introduction to Temporal Logics (Cont.) The set of worlds correspond to moments in time. How we navigate between these worlds depends on our particular view of time. The particular model of time is captured by a temporal accessibility relation between worlds. Essentially, temporal logic extends classical propositional logic with a set of temporal operators that navigate between worlds using this accessibility relation.

5 Typical Models of Time Alessandro Artale (FM First Semester 2007/2008) p. 5/39

6 Alessandro Artale (FM First Semester 2007/2008) p. 6/39 Summary Introducing Temporal Logics. Intuitions beyond Linear Temporal Logic. LTL: Syntax and Semantics. LTL in Computer Science. LTL Interpreted over Kripke Models. LTL and Model Checking: Intuitions.

7 Alessandro Artale (FM First Semester 2007/2008) p. 7/39 Linear Temporal Logic (LTL): Intuitions Consider a simple temporal logic (LTL) where the accessibility relation characterises a discrete, linear model isomorphic to the Natural Numbers. Typical temporal operators used are ϕ ϕ ϕ ϕu ψ Examples: ϕ is true in the next moment in time ϕ is true in all future moments ϕ is true in some future moment ϕ is true until ψ is true (( passport ticket) board_ f light)

8 Alessandro Artale (FM First Semester 2007/2008) p. 8/39 Computational Example (requested received) (received processed) (processed done) From the above we should be able to infer that it is not the case that the system continually re-sends a request, but never sees it completed ( done); i.e. the statement requested done should be inconsistent.

9 Alessandro Artale (FM First Semester 2007/2008) p. 9/39 Summary Introducing Temporal Logics. Intuitions beyond Linear Temporal Logic. LTL: Syntax and Semantics. LTL in Computer Science. LTL Interpreted over Kripke Models. LTL and Model Checking: Intuitions.

10 Alessandro Artale (FM First Semester 2007/2008) p. 10/39 LTL: Syntax Countable set Σ of atomic propositions: p,q,... the set FORM of formulas is: ϕ, ψ p (atomic proposition) (true) (false) ϕ (complement) ϕ ψ (conjunction) ϕ ψ (disjunction) ϕ (next time) ϕ (always) ϕ (sometime) (until) ϕu ψ

11 Alessandro Artale (FM First Semester 2007/2008) p. 11/39 Temporal Semantics We interpret our temporal formulae in a discrete, linear model of time. Formally, this structure is represented by M = N,I where I : N 2 Σ maps each Natural number (representing a moment in time) to a set of propositions. The semantics of a temporal formula is provided by the satisfaction relation: =: (M N FORM) {true,false}

12 Alessandro Artale (FM First Semester 2007/2008) p. 12/39 Semantics: The Propositional Aspect We start by defining when an atomic proposition is true at a time point i M, i = p iff p I(i) (for p Σ) The semantics for the classical operators is as expected: M, i = ϕ iff M, i = ϕ M, i = ϕ ψ iff M, i = ϕ and M, i = ψ M, i = ϕ ψ iff M, i = ϕ or M, i = ψ M, i = ϕ ψ iff if M, i = ϕ then M, i = ψ M, i = M, i =

13 Alessandro Artale (FM First Semester 2007/2008) p. 13/39 Temporal Operators: next M, i = ϕ iff M, i+1 = ϕ This operator provides a constraint on the next moment in time. Examples: (sad rich) sad ((x = 0) add3) (x = 3)

14 Alessandro Artale (FM First Semester 2007/2008) p. 14/39 Temporal Operators: sometime M, i = ϕ iff there exists j. ( j i) M, j = ϕ N.B. while we can be sure that ϕ will be true either now or in the future, we can not be sure exactly when it will be true. Examples: ( resigned sad) famous sad happy send receive

15 Alessandro Artale (FM First Semester 2007/2008) p. 15/39 Temporal Operators: always M, i = ϕ iff for all j. if ( j i) then M, j = ϕ This can represent invariant properties. Examples: lottery-win rich

16 Alessandro Artale (FM First Semester 2007/2008) p. 16/39 Temporal Operators: until M, i = ϕu ψ iff there exists j. ( j i) M, j = ψ for all k. (i k < j) M, k = ϕ Examples: start_lecture talk U end_lecture born alive U dead request reply U acknowledgement

17 Alessandro Artale (FM First Semester 2007/2008) p. 17/39 Satisfiability and Validity A structurem = N,I is a model of φ, if M, i = φ, for some i N. Similarly as in classical logic, an LTL formula φ can be satisfiable, unsatisfiable or valid. A formula φ is: Satisfiable, if there is model for φ. Unsatisfiable, if φ is not satisfiable. Valid (i.e., a Tautology): = φ iff M, i N. M, i = φ.

18 Alessandro Artale (FM First Semester 2007/2008) p. 18/39 Entailment and Equivalence Similarly as in classical logic we can define the notions of entailment and equivalence between two LTL formulas Entailment. φ = ψ iff M, i N. M, i = φ M, i = ψ Equivalence. φ ψ iff M, i N. M, i = φ M, i = ψ

19 Alessandro Artale (FM First Semester 2007/2008) p. 19/39 Equivalences in LTL The temporal operators and are duals ϕ ϕ (and then ) can be rewritten in terms ofu ϕ U ϕ All the temporal operators can be rewritten using the Until and Next operators

20 Alessandro Artale (FM First Semester 2007/2008) p. 20/39 Equivalences in LTL (Cont.) distributes over while distributes over (ϕ ψ) ϕ ψ (ϕ ψ) ϕ ψ The following equivalences are useful for generating formulas in Negated Normal Form. ϕ ϕ (ϕu ψ) ( ψu ( ϕ ψ)) ψ

21 Alessandro Artale (FM First Semester 2007/2008) p. 21/39 LTL Vs. FOL Linear Temporal Logic can be thought of as a specific decidable (PSPACE-complete) fragment of classical first-order logic We just map each proposition to a unary predicate in FOL. In general, the following satisfiability preserving mapping ( ) holds: p p(t) p p(t + 1) p t. (t t) p(t ) p t. (t t) p(t )

22 Alessandro Artale (FM First Semester 2007/2008) p. 22/39 Summary Introducing Temporal Logics. Intuitions beyond Linear Temporal Logic. LTL: Syntax and Semantics. LTL in Computer Science. LTL Interpreted over Kripke Models. LTL and Model Checking: Intuitions.

23 Alessandro Artale (FM First Semester 2007/2008) p. 23/39 Temporal Logic in Computer Science Temporal logic was originally developed in order to represent tense in natural language. Within Computer Science, it has achieved a significant role in the formal specification and verification of concurrent reactive systems. Much of this popularity has been achieved as a number of useful concepts can be formally, and concisely, specified using temporal logics, e.g. safety properties liveness properties fairness properties

24 Alessandro Artale (FM First Semester 2007/2008) p. 24/39 Safety Properties Safety: something bad will not happen Typical examples: (reactor_temp > 1000) (one_way other_way) ((x = 0) (y = z/x)) and so on... Usually:...

25 Alessandro Artale (FM First Semester 2007/2008) p. 25/39 Liveness Properties Liveness: something good will happen Typical examples: rich (x > 5) (start terminate) and so on... Usually:...

26 Alessandro Artale (FM First Semester 2007/2008) p. 26/39 Fairness Properties Often only really useful when scheduling processes, responding to messages, etc. Strong Fairness: if something is attempted/requested infinitely often, then it will be successful/allocated infinitely often Typical example: ready run

27 Alessandro Artale (FM First Semester 2007/2008) p. 27/39 Summary Introducing Temporal Logics. Intuitions beyond Linear Temporal Logic. LTL: Syntax and Semantics. LTL in Computer Science. LTL Interpreted over Kripke Models. LTL and Model Checking: Intuitions.

28 Alessandro Artale (FM First Semester 2007/2008) p. 28/39 Kripke Models and Linear Structures Consider the following Kripke structure:!done done Its paths/computations can be seen as a set of linear structures (computation tree):!done!done!done!done!done!done!done!done done...!done done!done!done done done!done done done!done done done done!done done done done

29 Alessandro Artale (FM First Semester 2007/2008) p. 29/39 Path-Semantics for LTL LTL formulae are evaluated over the set N of Natural Numbers. Paths in Kripke structures are infinite and linear sequences of states. Thus, they are isomorphic to the Natural Numbers: π = s 0 s 1 s i s i+1 We want to interpret LTL formulas over Kripke structures. Given a Kripke structure,km = (S,I,R,AP,L), a path π inkm, a state s S, and an LTL formula φ, we define: 1. KM,π = φ, and then 2. KM,s = φ Based on the LTL semantics over the Natural Numbers.

30 Alessandro Artale (FM First Semester 2007/2008) p. 30/39 Path-Semantics for LTL (Cont.) We first extract an LTL model,m π = (π,i π ), from the Kripke structurekm.m π = (π,i π ) is such that: π is a path inkm I π is the restriction of L to states in π: s π and p AP, p I π (s) iff p L(s) Given a Kripke structure,km = (S,I,R,AP,L), a path π inkm, a state s S, and an LTL formula φ: 1. KM,π = φ iff M π,s 0 = φ with s 0 initial state of π 2. KM,s = φ iff KM,π = φ for all paths π starting at s.

31 Alessandro Artale (FM First Semester 2007/2008) p. 31/39 LTL Model Checking Definition Given a Kripke structure,km = (S,I,R,AP,L), the LTL model checking problemkm = φ: Check if KM,s 0 = φ, for every s 0 I initial state of the Kripke structurekm.

32 Alessandro Artale (FM First Semester 2007/2008) p. 32/39 Summary Introducing Temporal Logics. Intuitions beyond Linear Temporal Logic. LTL: Syntax and Semantics. LTL in Computer Science. LTL Interpreted over Kripke Models. LTL and Model Checking: Intuitions.

33 Alessandro Artale (FM First Semester 2007/2008) p. 33/39 Example 1: mutual exclusion (safety) N = noncritical, T = trying, C = critical N1, N2 User 1 User 2 turn=0 T1, N2 N1, T2 C1, N2 T1, T2 T1, T2 N1, C2 C1, T2 T1, C2 KM = (C 1 C 2 )?

34 Alessandro Artale (FM First Semester 2007/2008) p. 33/39 Example 1: mutual exclusion (safety) N = noncritical, T = trying, C = critical N1, N2 User 1 User 2 turn=0 T1, N2 N1, T2 C1, N2 T1, T2 T1, T2 N1, C2 C1, T2 T1, C2 KM = (C 1 C 2 )? YES: There is no reachable state in which (C 1 C 2 ) holds!

35 Alessandro Artale (FM First Semester 2007/2008) p. 34/39 Example 2: mutual exclusion (liveness) N = noncritical, T = trying, C = critical N1, N2 User 1 User 2 turn=0 T1, N2 N1, T2 C1, N2 T1, T2 T1, T2 N1, C2 C1, T2 T1, C2 KM = C 1?

36 Alessandro Artale (FM First Semester 2007/2008) p. 34/39 Example 2: mutual exclusion (liveness) N = noncritical, T = trying, C = critical N1, N2 User 1 User 2 turn=0 T1, N2 N1, T2 C1, N2 T1, T2 T1, T2 N1, C2 C1, T2 T1, C2 KM = C 1? NO: the blue cyclic path is a counterexample!

37 Alessandro Artale (FM First Semester 2007/2008) p. 35/39 Example 3: mutual exclusion (liveness) N = noncritical, T = trying, C = critical N1, N2 User 1 User 2 turn=0 T1, N2 N1, T2 C1, N2 T1, T2 T1, T2 N1, C2 C1, T2 T1, C2 KM = (T 1 C 1 )?

38 Alessandro Artale (FM First Semester 2007/2008) p. 35/39 Example 3: mutual exclusion (liveness) N = noncritical, T = trying, C = critical N1, N2 User 1 User 2 turn=0 T1, N2 N1, T2 C1, N2 T1, T2 T1, T2 N1, C2 C1, T2 T1, C2 KM = (T 1 C 1 )? YES: in every path if T 1 holds afterwards C 1 holds!

39 Alessandro Artale (FM First Semester 2007/2008) p. 36/39 Example 4: mutual exclusion (fairness) N = noncritical, T = trying, C = critical N1, N2 User 1 User 2 turn=0 T1, N2 N1, T2 C1, N2 T1, T2 T1, T2 N1, C2 C1, T2 T1, C2 KM = C 1?

40 Alessandro Artale (FM First Semester 2007/2008) p. 36/39 Example 4: mutual exclusion (fairness) N = noncritical, T = trying, C = critical N1, N2 User 1 User 2 turn=0 T1, N2 N1, T2 C1, N2 T1, T2 T1, T2 N1, C2 C1, T2 T1, C2 KM = C 1? NO: the blue cyclic path is a counterexample!

41 Alessandro Artale (FM First Semester 2007/2008) p. 37/39 Example 4: mutual exclusion (strong fairness) N = noncritical, T = trying, C = critical N1, N2 User 1 User 2 turn=0 T1, N2 N1, T2 C1, N2 T1, T2 T1, T2 N1, C2 C1, T2 KM = T 1 C 1? T1, C2

42 Alessandro Artale (FM First Semester 2007/2008) p. 37/39 Example 4: mutual exclusion (strong fairness) N = noncritical, T = trying, C = critical N1, N2 User 1 User 2 turn=0 T1, N2 N1, T2 C1, N2 T1, T2 T1, T2 N1, C2 C1, T2 KM = T 1 C 1? T1, C2 YES: every path which visits T 1 infinitely often also visits C 1 infinitely often!

43 Alessandro Artale (FM First Semester 2007/2008) p. 38/39 LTL Alternative Notation Alternative notations are used for temporal operators. F sometime in the Future G Globally in the future X nextime

44 Alessandro Artale (FM First Semester 2007/2008) p. 39/39 Summary of Lecture III Introducing Temporal Logics. Intuitions beyond Linear Temporal Logic. LTL: Syntax and Semantics. LTL in Computer Science. LTL Interpreted over Kripke Models. LTL and Model Checking: Intuitions.

FORMAL METHODS LECTURE IV: COMPUTATION TREE LOGIC (CTL)

Alessandro Artale (FM First Semester 2007/2008) p. 1/37 FORMAL METHODS LECTURE IV: COMPUTATION TREE LOGIC (CTL) Alessandro Artale Faculty of Computer Science Free University of Bolzano artale@inf.unibz.it