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In mathematics and computer science, the Entscheidungsproblem is a challenge posed by David Hilbert and Wilhelm Ackermann in 1928. The problem asks for an algorithm that considers, as input, a statement and answers "yes" or "no" according to whether the statement is universally valid, i.e., valid in every structure.
Alonzo Church was an American mathematician, computer scientist, logician, and philosopher who made major contributions to mathematical logic and the foundations of theoretical computer science. He is best known for the lambda calculus, the Church–Turing thesis, proving the unsolvability of the Entscheidungsproblem, the Frege–Church ontology, and the Church–Rosser theorem. Alongside his doctoral student Alan Turing, Church is considered one of the founders of computer science.
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.
Metamathematics is the study of mathematics itself using mathematical methods. This study produces metatheories, which are mathematical theories about other mathematical theories. Emphasis on metamathematics owes itself to David Hilbert's attempt to secure the foundations of mathematics in the early part of the 20th century. Metamathematics provides "a rigorous mathematical technique for investigating a great variety of foundation problems for mathematics and logic". An important feature of metamathematics is its emphasis on differentiating between reasoning from inside a system and from outside a system. An informal illustration of this is categorizing the proposition "2+2=4" as belonging to mathematics while categorizing the proposition "'2+2=4' is valid" as belonging to metamathematics.
In the context of hardware and software systems, formal verification is the act of proving or disproving the correctness of a system with respect to a certain formal specification or property, using formal methods of mathematics. Formal verification is a key incentive for formal specification of systems, and is at the core of formal methods. It represents an important dimension of analysis and verification in electronic design automation and is one approach to software verification. The use of formal verification enables the highest Evaluation Assurance Level (EAL7) in the framework of common criteria for computer security certification.
In computer science, model checking or property checking is a method for checking whether a finite-state model of a system meets a given specification. This is typically associated with hardware or software systems, where the specification contains liveness requirements as well as safety requirements.
In computer science, program synthesis is the task to construct a program that provably satisfies a given high-level formal specification. In contrast to program verification, the program is to be constructed rather than given; however, both fields make use of formal proof techniques, and both comprise approaches of different degrees of automation. In contrast to automatic programming techniques, specifications in program synthesis are usually non-algorithmic statements in an appropriate logical calculus.
Zohar Manna was an Israeli-American computer scientist who was a professor of computer science at Stanford University.
In logic, linear temporal logic or linear-time temporal logic (LTL) is a modal temporal logic with modalities referring to time. In LTL, one can encode formulae about the future of paths, e.g., a condition will eventually be true, a condition will be true until another fact becomes true, etc. It is a fragment of the more complex CTL*, which additionally allows branching time and quantifiers. LTL is sometimes called propositional temporal logic, abbreviated PTL. In terms of expressive power, linear temporal logic (LTL) is a fragment of first-order logic.
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 sugaring. It is widely used in the hardware design and verification industry, where formal verification tools and/or logic simulation tools are used to prove or refute that a given PSL formula holds on a given design.
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 realized. It is used in formal verification of software or hardware artifacts, typically by software applications known as model checkers, which determine if a given artifact possesses safety or liveness properties. For example, CTL can specify that when some initial condition is satisfied, then all possible executions of a program avoid some undesirable condition. In this example, the safety property could be verified by a model checker that explores all possible transitions out of program states satisfying the initial condition and ensures that all such executions satisfy the property. Computation tree logic belongs to a class of temporal logics that includes linear temporal logic (LTL). Although there are properties expressible only in CTL and properties expressible only in LTL, all properties expressible in either logic can also be expressed in CTL*.
In computer science, concurrency is the ability of different parts or units of a program, algorithm, or problem to be executed out-of-order or in partial order, without affecting the outcome. This allows for parallel execution of the concurrent units, which can significantly improve overall speed of the execution in multi-processor and multi-core systems. In more technical terms, concurrency refers to the decomposability of a program, algorithm, or problem into order-independent or partially-ordered components or units of computation.
Formal equivalence checking process is a part of electronic design automation (EDA), commonly used during the development of digital integrated circuits, to formally prove that two representations of a circuit design exhibit exactly the same behavior.
Logic in computer science covers the overlap between the field of logic and that of computer science. The topic can essentially be divided into three main areas:
Rajeev Alur is an American professor of computer science at the University of Pennsylvania who has made contributions to formal methods, programming languages, and automata theory, including notably the introduction of timed automata and nested words.
Ernest Allen Emerson II, better known as E. Allen Emerson, is an American computer scientist and winner of the 2007 Turing Award. He is Professor and Regents Chair Emeritus at the University of Texas at Austin, United States.
TLA+ is a formal specification language developed by Leslie Lamport. It is used for designing, modelling, documentation, and verification of programs, especially concurrent systems and distributed systems. TLA+ is considered to be exhaustively-testable pseudocode, and its use likened to drawing blueprints for software systems; TLA is an acronym for Temporal Logic of Actions.
Structural synthesis of programs (SSP) is a special form of (automatic) program synthesis that is based on propositional calculus. More precisely, it uses intuitionistic logic for describing the structure of a program in such a detail that the program can be automatically composed from pieces like subroutines or even computer commands. It is assumed that these pieces have been implemented correctly, hence no correctness verification of these pieces is needed. SSP is well suited for automatic composition of services for service-oriented architectures and for synthesis of large simulation programs.
Action model learning is an area of machine learning concerned with creation and modification of software agent's knowledge about effects and preconditions of the actions that can be executed within its environment. This knowledge is usually represented in logic-based action description language and used as the input for automated planners.
References
- ↑ Church, Alonzo (1962). "Logic, arithmetic, and automata". International Congress of Mathematicians . pp. 23–35.