HoareLogik

HoareLogik, named after Tony Hoare, is a formal system for proving the correctness of imperative computer programs. Introduced in 1969, it provides a rigorous framework to reason about what must be true before and after executing a program fragment. It has become a foundational method in formal methods and program verification.

The central notion is the Hoare triple {P} C {Q}, where P is a precondition, C a command, and Q a postcondition. The informal meaning is: if P holds before C and C terminates, then Q holds afterwards. The logic distinguishes partial correctness (termination is not assumed) from total correctness (C terminates and then Q holds).

Hoare logic is built from a small set of axioms and inference rules. Key components include the assignment axiom {P[E/x]} x := E {P}, the rule of composition for sequences, the consequence rule, and a conditional rule for if statements. The while rule uses an invariant I: from {I ∧ B} S {I} deduce {I} while B do S {I ∧ ¬B}. For total correctness, a termination argument is added.

Soundness guarantees that every derivable triple is semantically valid; relative completeness (Cook's theorem) states that, given a sufficiently expressive assertion language, every true triple can be derived if we can prove all true assertions about expressions. Hoare logic underpins many program verification techniques and has influenced automated verifiers and static analysis tools.