Hoare logic

https://en.wikipedia.org/wiki/Hoare_logic

Hoare logic, also known as Floyd-Hoare logic or Hoare rules, was proposed in 1969 by the British computer scientist and logician Tony Hoare, and subsequently refined by Hoare and other researchers.

Hoare logic is a formal system with a set of logical rules for reasoning rigorously about the correctness of computer programs.

The central feature of Hoare logic is the Hoare triple. A triple describes how the execution of a piece of code changes the state of the computation.

A Hoare triple is of the form {P}C{Q}, where P and Q are assertions and C is a command. P is a precondition and Q postcondition: when the precondition is met, executing the command establishes the postcondition. Assertions are formulae in predicate logic.

Hoare logic provides axioms and inference rules for all the constructs of a simple imperative programming language. The rules for other language constructs have also been developed, including the rules for concurrency, procedures, pointers, jumps.

Using standard Hoare logic, only partial correctness can be proven, while termination needs to be proved separately. Thus the intuitive reading of a Hoare triple is: Whenever P holds of the state before the execution of C, then Q will hold afterwards, or C does not terminate. In the latter case, there is no "after", so Q can be any statement at all. Indeed, one can choose Q to be false to express that C does not terminate.

Total correctness can also be proven with an extended version of the While rule.

In his 1969 paper, Hoare used a narrower notion of termination which also entailed absence of any run-time errors: "Failure to terminate may be due to an infinite loop; or it may be due to violation of an implementation-defined limit, for example, the range of numeric operands, the size of storage, or an operating system time limit.