Lecture 3 Temporal Logic

CS6133Software Specification and

Verification

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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

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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, ……

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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

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LTL Characteristics Time is totally ordered

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

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LTL Characteristics Time is discrete

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

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Henceforth

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Eventually

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Next State

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Until

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Unless

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Examples

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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)

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LTL Properties Precedence Chain (a before b before c)

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LTL Properties P between Q and R

or

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Example: A Telephone System

Given the predicates

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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

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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.

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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

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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

S0 Sj

f

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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

f

S0

S0

Sj

Sj

OR

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Previous

f

f T if f is true in the previous system state

F otherwise

f iff i. i j -1 f

S0 Sj-1 Sj

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Since

S0 Sk Sk+1 Sj

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

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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

S0 Sk Sk+1 Sj

g fS0 Sj

f

OR

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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

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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

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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

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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

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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.

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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).

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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).

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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.

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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.”

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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.

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