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**CS6133 Software Specification and Verification**

Lecture 3 Temporal Logic

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**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

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**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 s0, which is determined by the initials values of all the variables σ = s0, s1, s2, …… CS6133

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**Linear Temporal Logic Formula**

In linear temporal logic (LTL), a formula f is evaluated with respect to a trace σ and a particular state sj in that trace CS6133

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LTL Characteristics Time is totally ordered CS6133

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

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LTL Characteristics Time is discrete CS6133

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**Future Temporal Operators**

Future temporal operators are shorthand notations that quantify over states CS6133

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Henceforth CS6133

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Eventually CS6133

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Next State CS6133

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Until CS6133

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Unless CS6133

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Examples CS6133

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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

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

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LTL Properties P between Q and R or CS6133

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**Example: A Telephone System**

Given the predicates CS6133

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**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

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**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

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**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

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**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 f S0 Sj CS6133

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**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 OR S0 Sj f S0 Sj CS6133

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**Previous f T if f is true in the previous system state F otherwise**

f iff i. i j -1 f f S0 Sj-1 Sj CS6133

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**Since 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 g f S0 Sk Sk+1 Sj CS6133

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**Back-to f g T if f has-always- been true or f since g F otherwise**

f g iff f g f f OR S0 Sj f g S0 Sk Sk+1 Sj CS6133

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**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

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**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

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**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

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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

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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

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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

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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

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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

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**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

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**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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