Linear time property
In model checking, a branch of computer science, linear time properties are used to describe requirements of a model of a computer system. Example properties include "the vending machine does not dispense a drink until money has been entered" or "the computer program eventually terminates". Fairness properties can be used to rule out unrealistic paths of a model. For instance, in a model of two traffic lights, the liveness property "both traffic lights are green infinitely often" may only be true under the unconditional fairness constraint "each traffic light changes colour infinitely often".
Formally, a linear time property is an ω-language over the power set of "atomic propositions". That is, the property contains sequences of sets of propositions, each sequence known as a "word". Every property can be rewritten as "P and Q both occur" for some safety property P and liveness property Q. An invariant for a system is something that is true or false for a particular state. Invariant properties describe an invariant that every reachable state of a model must satisfy, while persistence properties are of the form "eventually forever some invariant holds".
Temporal logics such as linear temporal logic describe types of linear time properties using formulae.
Definition
Let AP be a set of atomic propositions. A word over is an infinite sequence of sets of propositions, such as . A linear time property over AP is a subset of i.e. a set of words. An example of an LT property over the set is "the set of words which contain a infinitely often". The word w is in this set, because a is contained in, which occurs infinitely often. A word not in this set is, as a only occurs once.An LT property is an ω-language over the alphabet .
We denote by pref the finite prefixes of w. The closure of an LT property P is:
Applications
Using the theory of finite-state machines, a program or computer system can be modelled by a Kripke structure. LT properties then describe restrictions on the traces of a Kripke structure. For instance, if two traffic lights at an intersection are represented by a Kripke structure then the atomic propositions may be the possible colours of each light and it may be desirable that the traces satisfy the LT property "the traffic lights cannot both be green at the same time".If every trace of the Kripke structure TS is a trace of TS then every LT property which TS satisfies is satisfied by TS. This is useful in model checking to allow abstraction: if a simplified model of the system satisfies an LT property then the actual model of the system will satisfy it as well.
Classification of linear time properties
Safety properties
A safety property is informally of the form "a bad thing does not happen". For instance, if a system models an automated teller machine then such a property is "money should not be dispensed unless a PIN has been entered". Formally, a safety property is an LT property such that any word that violates the property has a "bad prefix", for which no word with that prefix satisfies the property. That is,In the ATM example, a minimal bad prefix is a finite set of steps carried out in which money is dispensed in the last step and a PIN is not entered at any step. To verify a safety property, it is sufficient to consider only the finite traces of a Kripke structure and check whether any such trace is a bad prefix.
An LT property P is a safety property if and only if.
Invariant properties
An invariant property is a type of safety property in which the condition only refers to the current state. For instance, the ATM example is not an invariant because we cannot tell whether the property is violated by seeing that the current state is "dispense money", only by seeing that the current state is "dispense money" and no previous state was "read PIN". An example of an invariant is the traffic light condition "the traffic lights cannot both be green at the same time" above. Another is "the variable x is never negative", in a model of a computer program.Formally, an invariant is of the form:
for some propositional logic formula.
A Kripke structure satisfies an invariant if and only if every reachable state satisfies the invariant, which can be checked by a breadth-first search or depth-first search. Safety properties can be verified inductively using invariants.
Liveness properties
A liveness property is informally of the form "something good eventually happens". Formally, P is a liveness property if i.e. any finite string can be continued to a valid trace. An example of a liveness property is the previous LT property "the set of words which contain a infinitely often". No finite prefix of a word can prove that the word does not satisfy this property, as the word could continue on to have infinitely many as.In terms of computer programs, useful liveness properties include "the program eventually terminates" and, in concurrent computing, "every process must eventually be served".
Persistence properties
A persistence property is a liveness property of the form "eventually forever ". That is, a property of the form:Relation between safety and liveness properties
No LT property other than is both a safety and a liveness property. Though not every property is a safety property or a liveness property, every property is the intersection of a safety and a liveness property.In topology, the set of all words can be equipped with the metric:
Then, a safety property is a closed set and a liveness property is a dense set.
Fairness properties
Fairness properties are preconditions imposed on a system to rule out unrealistic traces. Unconditional fairness is of the form "every process gets its turn infinitely often". Strong fairness is of the form "every process gets its turn infinitely often if it is enabled infinitely often". Weak fairness is of the form "every process gets its turn infinitely often if it is continuous enabled from a particular point". A program "fairly satisfies" an LT property P with respect to a set of fairness conditions if for every trace, either the trace fails a fairness condition or it satisfies P.A fairness property is realizable for a Kripke structure if every reachable state has a fair path starting from that state. So long as a set of fairness conditions are realizable, they are irrelevant to safety properties.