In the context of hardware and software systems, formal verification is the act of proving or disproving the correctness of intended

algorithm In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific problems or to perform a computation. Algorithms are used as specifications for performing ...

s underlying a system with respect to a certain formal specification or property, using formal methods of

mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

. Formal verification can be helpful in proving the correctness of systems such as:

cryptographic protocol A security protocol (cryptographic protocol or encryption protocol) is an abstract or concrete protocol that performs a security-related function and applies cryptographic methods, often as sequences of cryptographic primitives. A protocol descri ...

s, combinational circuits, digital circuits with internal memory, and software expressed as

source code In computing, source code, or simply code, is any collection of code, with or without comments, written using a human-readable programming language, usually as plain text. The source code of a program is specially designed to facilitate the ...

. The verification of these systems is done by providing a

formal proof In logic and mathematics, a formal proof or derivation is a finite sequence of sentences (called well-formed formulas in the case of a formal language), each of which is an axiom, an assumption, or follows from the preceding sentences in the seq ...

on an abstract mathematical model of the system, the correspondence between the mathematical model and the nature of the system being otherwise known by construction. Examples of mathematical objects often used to model systems are: finite-state machines, labelled transition systems, Petri nets, vector addition systems, timed automata,

hybrid automata In automata theory, a hybrid automaton (plural: ''hybrid automata'' or ''hybrid automatons'') is a mathematical model for precisely describing hybrid systems, for instance systems in which digital computational processes interact with analog physica ...

process algebra In computer science, the process calculi (or process algebras) are a diverse family of related approaches for formally modelling concurrent systems. Process calculi provide a tool for the high-level description of interactions, communications, and ...

, formal semantics of programming languages such as operational semantics, denotational semantics, axiomatic semantics and Hoare logic.

Approaches

One approach and formation is model checking, which consists of a systematically exhaustive exploration of the mathematical model (this is possible for finite models, but also for some infinite models where infinite sets of states can be effectively represented finitely by using abstraction or taking advantage of symmetry). Usually, this consists of exploring all states and transitions in the model, by using smart and domain-specific abstraction techniques to consider whole groups of states in a single operation and reduce computing time. Implementation techniques include state space enumeration, symbolic state space enumeration, abstract interpretation, symbolic simulation, abstraction refinement. The properties to be verified are often described in temporal logics, such as linear temporal logic (LTL),

Property Specification Language Property Specification Language (PSL) is a temporal logic extending linear temporal logic with a range of operators for both ease of expression and enhancement of expressive power. PSL makes an extensive use of regular expressions and syntactic suga ...

(PSL), SystemVerilog Assertions (SVA), or

computational tree logic Computation tree logic (CTL) is a branching-time logic, meaning that its model of time is a tree-like structure in which the future is not determined; there are different paths in the future, any one of which might be an actual path that is realiz ...

(CTL). The great advantage of model checking is that it is often fully automatic; its primary disadvantage is that it does not in general scale to large systems; symbolic models are typically limited to a few hundred bits of state, while explicit state enumeration requires the state space being explored to be relatively small. Another approach is deductive verification. It consists of generating from the system and its specifications (and possibly other annotations) a collection of mathematical ''proof obligations'', the truth of which imply conformance of the system to its specification, and discharging these obligations using either

proof assistant In computer science and mathematical logic, a proof assistant or interactive theorem prover is a software tool to assist with the development of formal proofs by human-machine collaboration. This involves some sort of interactive proof edi ...

s (interactive theorem provers) (such as

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ACL2 ACL2 ("A Computational Logic for Applicative Common Lisp") is a software system consisting of a programming language, created by Timothy Still it was an extensible theory in a first-order logic, and an automated theorem prover. ACL2 is designed ...

, Isabelle, Coq or PVS), or automatic theorem provers, including in particular satisfiability modulo theories (SMT) solvers. This approach has the disadvantage that it may require the user to understand in detail why the system works correctly, and to convey this information to the verification system, either in the form of a sequence of theorems to be proved or in the form of specifications (invariants, preconditions, postconditions) of system components (e.g. functions or procedures) and perhaps subcomponents (such as loops or data structures).

Software

Formal verification of software programs involves proving that a program satisfies a formal specification of its behavior. Subareas of formal verification include deductive verification (see above), abstract interpretation,

automated theorem proving Automated theorem proving (also known as ATP or automated deduction) is a subfield of automated reasoning and mathematical logic dealing with proving mathematical theorems by computer programs. Automated reasoning over mathematical proof was a ...

type system In computer programming, a type system is a logical system comprising a set of rules that assigns a property called a type to every "term" (a word, phrase, or other set of symbols). Usually the terms are various constructs of a computer progra ...

s, and lightweight formal methods. A promising type-based verification approach is dependently typed programming, in which the types of functions include (at least part of) those functions' specifications, and type-checking the code establishes its correctness against those specifications. Fully featured dependently typed languages support deductive verification as a special case. Another complementary approach is program derivation, in which efficient code is produced from

functional Functional may refer to: * Movements in architecture: ** Functionalism (architecture) ** Form follows function * Functional group, combination of atoms within molecules * Medical conditions without currently visible organic basis: ** Functional sy ...

specifications by a series of correctness-preserving steps. An example of this approach is the

Bird–Meertens formalism The Bird–Meertens formalism (BMF) is a calculus for deriving programs from specifications (in a functional-programming setting) by a process of equational reasoning. It was devised by Richard Bird and Lambert Meertens as part of their work wit ...

, and this approach can be seen as another form of correctness by construction. These techniques can be ''

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'', meaning that the verified properties can be logically deduced from the semantics, or ''unsound'', meaning that there is no such guarantee. A sound technique yields a result only once it has covered the entire space of possibilities. An example of an unsound technique is one that covers only a subset of the possibilities, for instance only integers up to a certain number, and give a "good-enough" result. Techniques can also be '' decidable'', meaning that their algorithmic implementations are guaranteed to terminate with an answer, or undecidable, meaning that they may never terminate. By bounding the scope of possibilities, unsound techniques that are decidable might be able to be constructed when no decidable sound techniques are available.

Verification and validation

Verification is one aspect of testing a product's fitness for purpose. Validation is the complementary aspect. Often one refers to the overall checking process as V & V. * Validation: "Are we trying to make the right thing?", i.e., is the product specified to the user's actual needs? * Verification: "Have we made what we were trying to make?", i.e., does the product conform to the specifications? The verification process consists of static/structural and dynamic/behavioral aspects. E.g., for a software product one can inspect the source code (static) and run against specific test cases (dynamic). Validation usually can be done only dynamically, i.e., the product is tested by putting it through typical and atypical usages ("Does it satisfactorily meet all

use case In software and systems engineering, the phrase use case is a polyseme with two senses: # A usage scenario for a piece of software; often used in the plural to suggest situations where a piece of software may be useful. # A potential scenario ...

s?").

Automated program repair

Program repair is performed with respect to an oracle, encompassing the desired functionality of the program which is used for validation of the generated fix. A simple example is a test-suite—the input/output pairs specify the functionality of the program. A variety of techniques are employed, most notably using satisfiability modulo theories (SMT) solvers, and genetic programming, using evolutionary computing to generate and evaluate possible candidates for fixes. The former method is deterministic, while the latter is randomized. Program repair combines techniques from formal verification and program synthesis. Fault-localization techniques in formal verification are used to compute program points which might be possible bug-locations, which can be targeted by the synthesis modules. Repair systems often focus on a small pre-defined class of bugs in order to reduce the search space. Industrial use is limited owing to the computational cost of existing techniques.

Industry use

The growth in complexity of designs increases the importance of formal verification techniques in the

hardware industry Hardware may refer to: Technology Computing and electronics * Electronic hardware, interconnected electronic components which perform analog or logic operations ** Digital electronics, electronics that operate on digital signals *** Computer hardw ...

. At present, formal verification is used by most or all leading hardware companies, but its use in the software industry is still languishing. This could be attributed to the greater need in the hardware industry, where errors have greater commercial significance. Because of the potential subtle interactions between components, it is increasingly difficult to exercise a realistic set of possibilities by simulation. Important aspects of hardware design are amenable to automated proof methods, making formal verification easier to introduce and more productive. , several operating systems have been formally verified: NICTA's Secure Embedded L4 microkernel, sold commercially as

seL4 L4 is a family of second-generation microkernels, used to implement a variety of types of operating systems (OS), though mostly for Unix-like, ''Portable Operating System Interface'' (POSIX) compliant types. L4, like its predecessor microkernel ...

by OK Labs; OSEK/VDX based real-time operating system ORIENTAIS by East China Normal University; Green Hills Software's Integrity operating system; and SYSGO's

PikeOS PikeOS is a commercial, hard real-time operating system (RTOS) that offers a separation kernel based hypervisor with multiple logical partition types for many other operating systems (OS), each called a GuestOS, and applications. It enables use ...

. In 2016, a team led by Zhong Shao at Yale developed a formally verified operating system kernel called CertiKOS. As of 2017, formal verification has been applied to the design of large computer networks through a mathematical model of the network, and as part of a new network technology category, intent-based networking. Network software vendors that offer formal verification solutions include

Cisco Cisco Systems, Inc., commonly known as Cisco, is an American-based multinational digital communications technology conglomerate corporation headquartered in San Jose, California. Cisco develops, manufactures, and sells networking hardware, ...

Forward Networks and Veriflow Systems. The SPARK programming language provides a toolset which enables software development with formal verification and is used in several high-integrity systems. The CompCert C compiler is a formally verified C compiler implementing the majority of ISO C.

References