In computer science, formal methods are mathematically rigorous techniques for the specification, development, analysis, and verification of software and hardware systems. The use of formal methods for software and hardware design is motivated by the expectation that, as in other engineering disciplines, performing appropriate mathematical analysis can contribute to the reliability and robustness of a design.

Formal methods employ a variety of theoretical computer science fundamentals, including logic calculi, formal languages, automata theory, control theory, program semantics, type systems, and type theory.

Formal methods can be applied at various points through the development process.

Formal methods may be used to give a formal description of the system to be developed, at whatever level of detail desired. Further formal methods may depend on this specification to synthesize a program or to verify the correctness of a system.

Alternatively, specification may be the only stage in which formal methods is used. By writing a specification, ambiguities in the informal requirements can be discovered and resolved. Additionally, engineers can use a formal specification as a reference to guide their development processes.

The need for formal specification systems has been noted for years. In the ALGOL 58 report, John Backus presented a formal notation for describing programming language syntax, later named Backus normal form then renamed Backus–Naur form (BNF). Backus also wrote that a formal description of the meaning of syntactically valid ALGOL programs was not completed in time for inclusion in the report, stating that it "will be included in a subsequent paper." However, no paper describing the formal semantics was ever released.

Program synthesis is the process of automatically creating a program that conforms to a specification. Deductive synthesis approaches rely on a complete formal specification of the program, whereas inductive approaches infer the specification from examples. Synthesizers perform a search over the space of possible programs to find a program consistent with the specification. Because of the size of this search space, developing efficient search algorithms is one of the major challenges in program synthesis.

Formal verification is the use of software tools to prove properties of a formal specification, or to prove that a formal model of a system implementation satisfies its specification.

Once a formal specification has been developed, the specification may be used as the basis for proving properties of the specification, and by inference, properties of the system implementation.

Sign-off verificationSign-off verification is the use of a formal verification tool that is highly trusted. Such a tool can replace traditional verification methods (the tool may even be certified).

Human-directed proofSometimes, the motivation for proving the correctness of a system is not the obvious need for reassurance of the correctness of the system, but a desire to understand the system better. Consequently, some proofs of correctness are produced in the style of mathematical proof: handwritten (or typeset) using natural language, using a level of informality common to such proofs. A "good" proof is one that is readable and understandable by other human readers.

Critics of such approaches point out that the ambiguity inherent in natural language allows errors to be undetected in such proofs; often, subtle errors can be present in the low-level details typically overlooked by such proofs. Additionally, the work involved in producing such a good proof requires a high level of mathematical sophistication and expertise.

Automated proofIn contrast, there is increasing interest in producing proofs of correctness of such systems by automated means. Automated techniques fall into three general categories:

- Automated theorem proving, in which a system attempts to produce a formal proof from scratch, given a description of the system, a set of logical axioms, and a set of inference rules.
- Model checking, in which a system verifies certain properties by means of an exhaustive search of all possible states that a system could enter during its execution.
- Abstract interpretation, in which a system verifies an over-approximation of a behavioural property of the program, using a fixpoint computation over a (possibly complete) lattice representing it.

Some automated theorem provers require guidance as to which properties are "interesting" enough to pursue, while others work without human intervention. Model checkers can quickly get bogged down in checking millions of uninteresting states if not given a sufficiently abstract model.

Proponents of such systems argue that the results have greater mathematical certainty than human-produced proofs, since all the tedious details have been algorithmically verified. The training required to use such systems is also less than that required to produce good mathematical proofs by hand, making the techniques accessible to a wider variety of practitioners.

Critics note that some of those systems are like oracles: they make a pronouncement of truth, yet give no explanation of that truth. There is also the problem of "verifying the verifier"; if the program that aids in the verification is itself unproven, there may be reason to doubt the soundness of the produced results. Some modern model checking tools produce a "proof log" detailing each step in their proof, making it possible to perform, given suitable tools, independent verification.

The main feature of the abstract interpretation approach is that it provides a sound analysis, i.e. no false negatives are returned. Moreover, it is efficiently scalable, by tuning the abstract domain representing the property to be analyzed, and by applying widening operators to get fast convergence.

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Formal methods includes a number of different techniques.

The design of a computing system can be expressed using a specification language, which is a formal language that includes a proof system. Using this proof system, formal verification tools can reason about the specification and establish that a system adheres to the specification.

A binary decision diagram is a data structure that represents a Boolean function. If a Boolean formula P {\displaystyle {\mathcal {P}}} expresses that an execution of a program conforms to the specification, a binary decision diagram can be used to determine if P {\displaystyle {\mathcal {P}}} is a tautology; that is, it always evaluates to TRUE. If this is the case, then the program always conforms to the specification.

A SAT solver is a program that can solve the Boolean satisfiability problem, the problem of finding an assignment of variables that makes a given propositional formula evaluate to true. If a Boolean formula P {\displaystyle {\mathcal {P}}} expresses that a specific execution of a program conforms to the specification, then determining that ¬ P {\displaystyle \neg {\mathcal {P}}} is unsatisfiable is equivalent to determining that all executions conform to the specification. SAT solvers are often used in bounded model checking, but can also be used in unbounded model checking.

Formal methods are applied in different areas of hardware and software, including routers, Ethernet switches, routing protocols, security applications, and operating system microkernels such as seL4. There are several examples in which they have been used to verify the functionality of the hardware and software used in data centres. IBM used ACL2, a theorem prover, in the AMD x86 processor development process. Intel uses such methods to verify its hardware and firmware (permanent software programmed into a read-only memory). Dansk Datamatik Center used formal methods in the 1980s to develop a compiler system for the Ada programming language that went on to become a long-lived commercial product.

There are several other projects of NASA in which formal methods are applied, such as Next Generation Air Transportation System, Unmanned Aircraft System integration in National Airspace System, and Airborne Coordinated Conflict Resolution and Detection (ACCoRD). B-Method with Atelier B, is used to develop safety automatisms for the various subways installed throughout the world by Alstom and Siemens, and also for Common Criteria certification and the development of system models by ATMEL and STMicroelectronics.

Formal verification has been frequently used in hardware by most of the well-known hardware vendors, such as IBM, Intel, and AMD. There are many areas of hardware, where Intel have used formal methods to verify the working of the products, such as parameterized verification of cache-coherent protocol, Intel Core i7 processor execution engine validation (using theorem proving, BDDs, and symbolic evaluation), optimization for Intel IA-64 architecture using HOL light theorem prover, and verification of high-performance dual-port gigabit Ethernet controller with support for PCI express protocol and Intel advance management technology using Cadence. Similarly, IBM has used formal methods in the verification of power gates, registers, and functional verification of the IBM Power7 microprocessor.

In software development, formal methods are mathematical approaches to solving software (and hardware) problems at the requirements, specification, and design levels. Formal methods are most likely to be applied to safety-critical or security-critical software and systems, such as avionics software. Software safety assurance standards, such as DO-178C allows the usage of formal methods through supplementation, and Common Criteria mandates formal methods at the highest levels of categorization.

For sequential software, examples of formal methods include the B-Method, the specification languages used in automated theorem proving, RAISE, and the Z notation.

In functional programming, property-based testing has allowed the mathematical specification and testing (if not exhaustive testing) of the expected behaviour of individual functions.

The Object Constraint Language (and specializations such as Java Modeling Language) has allowed object-oriented systems to be formally specified, if not necessarily formally verified.

For concurrent software and systems, Petri nets, process algebra, and finite state machines (which are based on automata theory; see also virtual finite state machine or event driven finite state machine) allow executable software specification and can be used to build up and validate application behaviour.

Another approach to formal methods in software development is to write a specification in some form of logic—usually a variation of first-order logic—and then to directly execute the logic as though it were a program. The OWL language, based on description logic, is an example. There is also work on mapping some version of English (or another natural language) automatically to and from logic, as well as executing the logic directly. Examples are Attempto Controlled English, and Internet Business Logic, which do not seek to control the vocabulary or syntax. A feature of systems that support bidirectional English–logic mapping and direct execution of the logic is that they can be made to explain their results, in English, at the business or scientific level.

Semi-formal methods are formalisms and languages that are not considered fully "formal". It defers the task of completing the semantics to a later stage, which is then done either by human interpretation or by interpretation through software like code or test case generators.

Some practitioners believe that the formal methods community has overemphasized full formalization of a specification or design. They contend that the expressiveness of the languages involved, as well as the complexity of the systems being modelled, make full formalization a difficult and expensive task. As an alternative, various lightweight formal methods, which emphasize partial specification and focused application, have been proposed. Examples of this lightweight approach to formal methods include the Alloy object modelling notation, Denney's synthesis of some aspects of the Z notation with use case driven development, and the CSK VDM Tools.

There are a variety of formal methods and notations available.

- Abstract State Machines (ASMs)
- A Computational Logic for Applicative Common Lisp (ACL2)
- Actor model
- Alloy
- ANSI/ISO C Specification Language (ACSL)
- Autonomic System Specification Language (ASSL)
- B-Method
- CADP
- Common Algebraic Specification Language (CASL)
- Esterel
- Java Modeling Language (JML)
- Knowledge Based Software Assistant (KBSA)
- Lustre
- mCRL2
- Perfect Developer
- Petri nets
- Predicative programming
- Process calculi
- RAISE
- Rebeca Modeling Language
- SPARK Ada
- Specification and Description Language
- TLA+
- USL
- VDM
- VDM-SL
- VDM++

- Z notation

- ESBMC
- MALPAS Software Static Analysis Toolset – an industrial-strength model checker used for formal proof of safety-critical systems
- PAT – a free model checker, simulator and refinement checker for concurrent systems and CSP extensions (e.g., shared variables, arrays, fairness)
- SPIN
- UPPAAL

Many problems in formal methods are NP-hard, but can be solved in cases arising in practice. For example, the Boolean satisfiability problem is NP-complete by the Cook–Levin theorem, but SAT solvers can solve a variety of large instances. There are "solvers" for a variety of problems that arise in formal methods, and there are many periodic competitions to evaluate the state-of-the-art in solving such problems.

- The SAT competition is a yearly competition that compares SAT solvers. SAT solvers are used in formal methods tools such as Alloy.
- CASC is a yearly competition of automated theorem provers.
- SMT-COMP is a yearly competition of SMT solvers, which are applied to formal verification.
- CHC-COMP is a yearly competition of solvers of constrained Horn clauses, which have applications to formal verification.
- QBFEVAL is a biennial competition of solvers for true quantified Boolean formulas, which have applications to model checking.
- SV-COMP is an annual competition for software verification tools.
- SyGuS-COMP is an annual competition for program synthesis tools.

- Abstract interpretation
- Automated theorem proving
- Design by contract
- Formal methods people
- Formal science
- Formal specification
- Formal verification
- Formal system
- Methodism
- Methodology
- Model checking
- Scientific method
- Software engineering
- Specification language

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