In the context of hardware and software systems, formal verification is the act of proving or disproving the correctness of intended algorithms underlying a system with respect to a certain formal specification or property, using formal methods of mathematics.^{[1]}
Formal verification can be helpful in proving the correctness of systems such as: cryptographic protocols, combinational circuits, digital circuits with internal memory, and software expressed as source code.
The verification of these systems is done by providing a formal proof 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, timed automata, hybrid automata, process algebra, formal semantics of programming languages such as operational semantics, denotational semantics, axiomatic semantics and Hoare logic.
Approaches to formal verification
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) or computational tree logic (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 interactive theorem provers (such as HOL, ACL2, Isabelle, or Coq), automatic theorem provers, or SMT solvers. This approach has the disadvantage that it typically requires 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 an sequence of theorems to be proved or in the form of specifications of system components (e.g. functions or procedures) and perhaps subcomponents (such as loops or data structures).
A slightly different (and complementary) approach is program derivation, in which efficient code is produced from functional specifications by a series of correctness-preserving steps. An example of this approach is the Bird-Meertens Formalism, and this approach can be seen as another form of correctness by construction.
Formal verification for software
Logical inference for the formal verification of software can be further divided into:
- the more traditional 1970s approach in which code is first written in the usual way, and subsequently proven correct in a separate step;
- 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 the first approach as a special case.
A slightly different (and complementary) approach is program derivation, in which efficient code is produced from functional specifications by a series of correctness-preserving steps. An example of this approach is the Bird-Meertens Formalism, and this approach can be seen as another form of correctness by construction.
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 cases?").
Industry use
The growth in complexity of designs increases the importance of formal verification techniques in the hardware industry.^{[2]}^{[3]} 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.^{[4]}
As of 2011^{[update]}, several operating systems have been formally verified:
NICTA's Secure Embedded L4 microkernel, sold commercially as seL4 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.^{[5]}^{[6]}
See also
References
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