Properties of an execution of a computer program—particularly for
concurrent and distributed
systems —have long been
formulated by giving safety properties ("bad things don't happen") and liveness properties
("good things do happen").¹

A program is totally correct
with respect to a precondition
{\displaystyle P} and postcondition {\displaystyle Q} if any execution started in a state satisfying {\displaystyle P} terminates in a state satisfying {\displaystyle Q}. Total correctness is a conjunction of a
safety property and a liveness property:²

• The safety property prohibits these "bad things": executions that start in a state satisfying
  {\displaystyle P} and terminate in a final state that does not satisfy {\displaystyle Q}. For a
  program {\displaystyle C}, this safety property is usually written using the Hoare
  triple {\displaystyle \{P\}C\{Q\}}.
• The liveness property, the "good thing", is that execution that starts in a state satisfying
  {\displaystyle P} terminates.

Note that a bad thing is discrete,³ since it happens at a particular place during execution. A
"good thing" need not be discrete, but the liveness property of termination is discrete.

Formal definitions that were ultimately proposed for safety properties ⁴ and liveness properties
⁵ demonstrated that this decomposition is not only intuitively appealing but is also complete:
all properties of an execution are a conjunction of safety and liveness properties.⁵ Moreover,
undertaking the decomposition can be helpful, because the formal definitions enable a proof that
different methods must be used for verifying safety properties versus for verifying liveness
properties.^{6 7}

Safety

A safety property proscribes discrete bad things from occurring during an execution.¹ A safety
property thus characterizes what is permitted by stating what is prohibited. The requirement that
the bad thing be discrete means that a bad thing occurring during execution necessarily occurs
at some identifiable point.⁵

Examples of a discrete bad thing that could be used to define a safety property include:⁵

• An execution that starts in a state satisfying a given precondition terminates, but the final
  state does not satisfy the required postcondition;
• An execution of two concurrent processes, where the program counters for both processes designate
  statements within a critical section;
• An execution of two concurrent processes where each process is waiting for another to change
  state (known as deadlock).

An execution of a program can be described formally by giving the infinite sequence of program
states that results as execution proceeds, where the last state for a terminating program is
repeated infinitely. For a program of interest, let {\displaystyle S} denote the set of possible
program states, {\displaystyle S^{*}} denote the set of finite sequences of program states, and
{\displaystyle S^{\omega }} denote the set of infinite sequences of program states. The relation
{\displaystyle \sigma \leq \tau } holds for sequences {\displaystyle \sigma } and
{\displaystyle \tau } iff {\displaystyle \sigma } is a
prefix of {\displaystyle \tau } or
{\displaystyle \sigma } equals {\displaystyle \tau }.⁵

A property of a program is the set of allowed executions.

The essential characteristic of a safety property {\displaystyle SP} is: If some execution
{\displaystyle \sigma } does not satisfy {\displaystyle SP} then the defining bad thing for
that safety property occurs at some point in {\displaystyle \sigma }. Notice that after such a
bad thing, if further execution results in an execution {\displaystyle \sigma ^{\prime }}, then
{\displaystyle \sigma ^{\prime }} also does not satisfy {\displaystyle SP}, since the bad
thing in {\displaystyle \sigma } also occurs in {\displaystyle \sigma ^{\prime }}. We take
this inference about the irremediability of bad things to be the defining characteristic for
{\displaystyle SP} to be a safety property. Formalizing this in predicate logic gives a formal
definition for {\displaystyle SP} being a safety property.⁵

{\displaystyle \forall \sigma \in S^{\omega }:\sigma \notin SP\implies (\exists \beta \leq \sigma )}

This formal definition for safety properties implies that if an execution {\displaystyle \sigma }
satisfies a safety property {\displaystyle SP} then every prefix of {\displaystyle \sigma }
(with the last state repeated) also satisfies {\displaystyle SP}.

Liveness

A liveness property prescribes good things for every execution or, equivalently, describes
something that must happen during an execution.¹ The good thing need not be discrete—it
might involve an infinite number of steps. Examples of a good thing used to define a liveness
property include:⁵

• Termination of an execution that is started in a suitable state;
• Non-termination of an execution that is started in a suitable state;
• Guaranteed eventual entry to a critical section whenever entry is attempted;
• Fair access to a resource in the presence of contention.

The good thing in the first example is discrete but not in the others.

Producing an answer within a specified real-time bound is a safety property rather than a liveness
property. This is because a discrete bad thing is being proscribed: a partial execution that
reaches a state where the answer still has not been produced and the value of the clock (a state
variable) violates the bound. Deadlock freedom is a safety property: the "bad thing" is a
deadlock (which is discrete).

Most of the time, knowing that a program eventually does some "good thing" is not satisfactory; we
want to know that the program performs the "good thing" within some number of steps or before some
deadline. A property that gives a specific bound to the "good thing" is a safety property (as noted
above), whereas the weaker property that merely asserts the bound exists is a liveness property.
Proving such a liveness property is likely to be easier than proving the tighter safety property
because proving the liveness property doesn't require the kind of detailed accounting that is
required for proving the safety property.

To differ from a safety property, a liveness property {\displaystyle LP} cannot rule out any
finite prefix {\displaystyle \alpha \in S^{*}} ⁸ of an execution (since such an
{\displaystyle \alpha } would be a "bad thing" and, thus, would be defining a safety property).
That leads to defining a liveness property {\displaystyle LP} to be a property that does not rule
out any finite prefix.⁵

{\displaystyle \forall \alpha \in S^{*}:(\exists \tau \in S^{\omega }:\alpha \tau \in LP)}

This definition does not restrict a good thing to being discrete—the good thing can involve
all of {\displaystyle \tau }, which is an infinite-length execution.

History

Lamport used the terms safety
property and liveness property in his 1977 paper ¹ on proving the correctness of
multiprocess (concurrent) programs. He borrowed the terms from Petri net theory, which was using the terms liveness and boundedness for describing how the
assignment of a Petri net's "tokens" to its "places" could evolve; Petri net safety was a
specific form of boundedness. Lamport subsequently developed a formal definition of safety for a
NATO short course on distributed systems in Munich.⁹ It assumed that properties are invariant
under stuttering.
The formal definition of safety given above appears in a paper by Alpern and Schneider;⁵ the
connection between the two formalizations of safety properties appears in a paper by Alpern,
Demers, and Schneider.¹⁰

Alpern and Schneider ⁵ gives the formal definition for liveness, accompanied by a proof that all
properties can be constructed using safety properties and liveness properties. That proof was
inspired by Gordon Plotkin 's
insight that safety properties correspond to closed sets and liveness properties correspond to dense
sets in a natural
topology on the set {\displaystyle S^{\omega }} of infinite sequences of program states.¹¹ Subsequently, Alpern and Schneider ¹² not only
gave a Büchi automaton
characterization for the formal definitions of safety properties and liveness properties but used
these automata formulations to show that verification of safety properties would require an
invariant and verification of liveness properties would require a
well-foundedness argument. The correspondence between the kind of property (safety vs liveness) with kind of proof
(invariance vs well-foundedness) was a strong argument that the decomposition of properties into
safety and liveness (as opposed to some other partitioning) was a useful one—knowing the type of
property to be proved dictated the type of proof that is required.

References