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Lecture 3 Temporal Logic CS6133 Software Specification and Verification.

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1 Lecture 3 Temporal Logic CS6133 Software Specification and Verification

2 2 Temporal Logic: Overview Temporal Logic was designed for expressing the temporal ordering of events and states within a logical framework State is an assignment of values to the model’s variables. Intuitively, the system state is a snapshot of the system’s execution, in which every variable has some value Event is a trigger (e.g., signal) that can cause a system to change its state and won’t persist CS6133

3 3 Trace In temporal logic, the notion of exact time is abstracted away In temporal logic, we keep track of changes to variable values and the order in which they occur A trace σ is an infinite sequence of states that represents a particular execution of the system starts from an initial state s 0, which is determined by the initials values of all the variables σ = s 0, s 1, s 2, …… CS6133

4 4 Linear Temporal Logic Formula In linear temporal logic (LTL), a formula f is evaluated with respect to a trace σ and a particular state s j in that trace CS6133

5 5 LTL Characteristics Time is totally ordered CS6133

6 6 LTL Characteristics Time is bounded in the past and unbounded in the future CS6133

7 7 LTL Characteristics Time is discrete CS6133

8 8 Future Temporal Operators Future temporal operators are shorthand notations that quantify over states CS6133

9 9 Henceforth CS6133

10 10 Eventually CS6133

11 11 Next State CS6133

12 12 Until CS6133

13 13 Unless CS6133

14 14 Examples CS6133

15 15 LTL Properties Safety property can be expressed by a temporal formula of the form Response property can be expressed by a temporal formula of the form Precedence (a happens before b happens) CS6133

16 16 LTL Properties Precedence Chain (a before b before c) CS6133

17 17 LTL Properties P between Q and R or CS6133

18 18 Example: A Telephone System Given the predicates CS6133

19 19 Examples Using Future Operators I Formalize the following sentences in LTL A user always needs to pick up the phone before dialing After picking up the phone, the user eventually either goes back on hook or dials Whenever a user dialed a number and heard the ring tone, a connection will only result after the other user picks up the phone Immediately after the callee hangs up on a connection, the caller will hear an idle tone, then, the caller will hear a dial tone CS6133

20 20 Examples Using Future Operators II Formalize the properties of the elevator in LTL The elevator will eventually terminate, with its doors closed. The elevator shall not keep its doors open indefinitely. Pressing the button at floor 2 guarantees that the elevator will arrive at floor 2 and open its doors. Pressing the button at any floor guarantees that the elevator will arrive at that floor and open its doors. The elevator will not arrive at a floor and open its doors unless it is called. CS6133

21 21 Past Temporal Operators Past temporal operators are shorthand notations that quantify over states Past temporal operators are a symmetric counterpart to each of the future temporal operators Has-always-been Once Previous Since Back-to CS6133

22 22 Has-always-been f  T if f is true in the current and all past system states F otherwise f iff  i. 0  i  j  f S0S0 SjSj f CS6133

23 23 Once f  T if f is true in the current or some past system state F otherwise f iff  i. 0  i  j  f f f S0S0 S0S0 SjSj SjSj OR CS6133

24 24 Previous f f  T if f is true in the previous system state F otherwise f iff  i. i  j -1  f S0S0 S j-1 SjSj CS6133

25 25 Since S0S0 SkSk S k+1 SjSj f g  T if once g was true and f has been true since the last g to the present F otherwise f g iff  k. 0  k  j  g   i. k  i  j  f gf CS6133

26 26 Back-to f g  T if f has-always- been true or f since g F otherwise f g iff f g  f S0S0 SkSk S k+1 SjSj g f S0S0 SjSj f OR CS6133

27 27 Examples Using Past Operators Formalize the following sentences in LTL When a caller hears the dial tone, the caller must have picked up the phone When the callee hears the ring, a caller must dial the callee’s number and hasn’t hanged up Whenever a user dialed a number and heard the ring tone, a connection is established if the other user picks up the phone CS6133

28 28 Linear vs. Branching Views Two ways to think about the computations of reactive system Linear time: LTL Branching time: computation tree logic (CTL) A CTL formula is true/false relative to a state where as an LTL formula is true/false relative to a path CS6133

29 29 CTL There are future temporal operators of LTL There are also path quantifiers to describe the branching structure of a computation tree: A and E A means for all computation paths E means for some computation paths CS6133

30 30 CTL Syntax If p is an atomic proposition, and f1 and f2 are CTL formulae, then the set of CTL formulae consists of 1. p 2. ¬ f1, f1 ∧ f2, f1 ∨ f2, f1 ⇒ f2 3. AX f1, EX f1 4. AG f1, EG f1 5. AF f1, EF f1 6. A [f1Uf2], E [f1Uf2] Note that the path quantifiers and temporal operators are always paired together CS6133

31 31 CTL Semantics AX f if on all paths starting at state s, f holds in the next state EX f if there exists a path starting at state s on which f holds at the next state. CS6133

32 32 CTL Semantics EF f if f is reachable (i.e., if there exists a path starting at state s, on which f holds in some future state). AF f if f is inevitable (i.e., if on all paths that start at state s, f holds in some future state). CS6133

33 33 CTL Semantics EG f if there exists a path starting at state s, on which f holds globally. AG f if f is invariant (i.e., if on all paths that start at state s, f holds globally). CS6133

34 34 CTL Semantics E[g U f] if there exists a path starting at state s, on which g holds until f eventually holds. A[g U f] if on all paths that start at state s, g holds until f eventually holds. CS6133

35 35 Example of CTL Formulas “It is possible to get to a state where started holds, but ready does not hold.” “For any state, if a request occurs, then it will eventually be acknowledged.” “It is always the case that a certain process is enabled infinitely often on every computation path.” CS6133

36 36 LTL vs. CTL In LTL, we could write: FG p There is no equivalent of this formula in CTL. In CTL, we could write: AG EF p There is no equivalent of this formula in LTL. CS6133


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