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:
Theoretical foundations and analysis Use of computer technology to aid logicians Use of concepts from logic for computer applications
Theoretical foundations and analysis Logic plays a fundamental role in computer science. Some of the key areas of logic that are particularly significant are computability theory (formerly called recursion theory), modal logic and category theory. The theory of computation is based on concepts defined by logicians and mathematicians such as Alonzo Church and Alan Turing. Church first showed the existence of algorithmically unsolvable problems using his notion of lambda-definability. Turing gave the first compelling analysis of what can be called a mechanical procedure and Kurt Gödel asserted that he found Turing's analysis "perfect." In addition some other major areas of theoretical overlap between logic and computer science are:
Gödel's incompleteness theorem proves that any logical system powerful enough to characterize arithmetic will contain statements that can neither be proved nor disproved within that system. This has direct application to theoretical issues relating to the feasibility of proving the completeness and correctness of software. The frame problem is a basic problem that must be overcome when using first-order logic to represent the goals of an artificial intelligence agent and the state of its environment. The Curry–Howard correspondence is a relation between logical systems and programming languages. This theory established a precise correspondence between proofs and programs. In particular it showed that terms in the simply typed lambda calculus correspond to proofs of intuitionistic propositional logic. Category theory represents a view of mathematics that emphasizes the relations between structures. It is intimately tied to many aspects of computer science: type systems for programming languages, the theory of transition systems, models of programming languages and the theory of programming language semantics. Logic programming is a programming, database and knowledge representation paradigm that is based on formal logic. A logic program is a set of sentences about some problem domain. Computation is performed by applying logical reasoning to solve problems in the domain. Major logic programming language families include Prolog, Answer Set Programming (ASP) and Datalog.
Computers to assist logicians One of the first applications to use the term artificial intelligence was the Logic Theorist system developed by Allen Newell, Cliff Shaw, and Herbert Simon in 1956. One of the things that a logician does is to take a set of statements in logic and deduce the conclusions (additional statements) that must be true by the laws of logic. For example, if given the statements "All humans are mortal" and "Socrates is human" a valid conclusion is "Socrates is mortal". Of course this is a trivial example. In actual logical systems the statements can be numerous and complex. It was realized early on that this kind of analysis could be significantly aided by the use of computers. Logic Theorist validated the theoretical work of Bertrand Russell and Alfred North Whitehead in their influential work on mathematical logic called Principia Mathematica. In addition, subsequent systems have been utilized by logicians to validate and discover new mathematical theorems and proofs.
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