Formal methods is the term applied to the analysis of software (and hardware) whose results are obtained purely through the use of rigorous mathematical methods. The mathematical techniques used include denotational semantics, axiomatic semantics, operational semantics, and abstract interpretation.
It has been proven that, barring some hypothesis that the state space of programs is finite and small, finding possible run-time errors, or more generally any kind of violation of a specification on the final result of a program, is undecidable: there is no mechanical method that can always answer truthfully whether a given program may or may not exhibit runtime errors. This result dates from the works of Church, Gödel and Turing in the 1930s.
As with most undecidable questions, one can still attempt to give useful approximate solutions.Some of the implementation techniques of formal static analysis include:Model checking considers systems that have finite state or may be reduced to finite state by abstraction;
Abstract interpretation models the effect that every statement has on the state of an abstract machine (ie, it 'executes' the software based on the mathematical properties of each statement and declaration). This abstract machine overapproximates the behaviours of the system: the abstract system is thus made simpler to analyze, at the expense of incompleteness (not every property true of the original system is true of the abstract system). If properly done, though, abstract interpretation is sound (every property true of the abstract system can be mapped to a true property of the original system).
Use of assertions in program code as first suggested by Hoare logic. There is tool support for some programming languages (e.g., the SPARK programming language (a subset of Ada) and the Java Modeling Language — JML — using ESC/Java and ESC/Java2).