Formal methods is the term applied to the analysis of
software (and
computer 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. By a straightforward reduction to the
halting problem, it is possible to prove that (for any
Turing complete language), finding all possible run-time errors in an arbitrary program (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 an arbitrary program may or may not exhibit runtime errors. This result dates from the works of
Church,
Gödel and
Turing in the 1930s (see:
Halting problem and
Rice's theorem). As with many undecidable questions, one can still attempt to give useful approximate solutions. Some of the implementation techniques of formal static analysis include: •
Abstract interpretation, to model the effect that every statement has on the state of an abstract machine (i.e., it 'executes' the software based on the mathematical properties of each statement and declaration). This abstract machine over-approximates 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). •
Data-flow analysis, a lattice-based technique for gathering information about the possible set of values; •
Hoare logic, a
formal system with a set of logical rules for reasoning rigorously about the
correctness of computer programs. 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, Frama-C WP (
weakest precondition) plugin for the C language extended with ACSL (
ANSI/ISO C Specification Language) ). •
Model checking, considers systems that have
finite state or may be reduced to finite state by
abstraction; •
Symbolic execution, as used to derive mathematical expressions representing the value of mutated variables at particular points in the code. •
Nullable reference analysis == Data-driven static analysis ==