In algorithmic information theory, the Kolmogorov complexity of an object, such as a piece of text, is the length of a shortest computer program that produces the object as output. It is a measure of the computational resources needed to specify the object, and is also known as algorithmic complexity, Solomonoff–Kolmogorov–Chaitin complexity, program-size complexity, descriptive complexity, or algorithmic entropy. It is named after Andrey Kolmogorov, who first published on the subject in 1963 and is a generalization of classical information theory.
First-order logic—also called predicate logic, predicate calculus, quantificational logic—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantified variables over non-logical objects, and allows the use of sentences that contain variables, so that rather than propositions such as "Socrates is a man", one can have expressions in the form "there exists x such that x is Socrates and x is a man", where "there exists" is a quantifier, while x is a variable. This distinguishes it from propositional logic, which does not use quantifiers or relations; in this sense, propositional logic is the foundation of first-order logic.
Logic programming is a programming, database and knowledge representation paradigm based on formal logic. A logic program is a set of sentences in logical form, representing knowledge about some problem domain. Computation is performed by applying logical reasoning to that knowledge, to solve problems in the domain. Major logic programming language families include Prolog, Answer Set Programming (ASP) and Datalog. In all of these languages, rules are written in the form of clauses:
Prolog is a logic programming language that has its origins in artificial intelligence, automated theorem proving and computational linguistics.
Presburger arithmetic is the first-order theory of the natural numbers with addition, named in honor of Mojżesz Presburger, who introduced it in 1929. The signature of Presburger arithmetic contains only the addition operation and equality, omitting the multiplication operation entirely. The theory is computably axiomatizable; the axioms include a schema of induction.
In logic and computer science, specifically automated reasoning, unification is an algorithmic process of solving equations between symbolic expressions, each of the form Left-hand side = Right-hand side. For example, using x,y,z as variables, and taking f to be an uninterpreted function, the singleton equation set { f(1,y) = f(x,2) } is a syntactic first-order unification problem that has the substitution { x ↦ 1, y ↦ 2 } as its only solution.
In computer science, bogosort is a sorting algorithm based on the generate and test paradigm. The function successively generates permutations of its input until it finds one that is sorted. It is not considered useful for sorting, but may be used for educational purposes, to contrast it with more efficient algorithms.
In computer science and automata theory, a deterministic Büchi automaton is a theoretical machine which either accepts or rejects infinite inputs. Such a machine has a set of states and a transition function, which determines which state the machine should move to from its current state when it reads the next input character. Some states are accepting states and one state is the start state. The machine accepts an input if and only if it will pass through an accepting state infinitely many times as it reads the input.
Default logic is a non-monotonic logic proposed by Raymond Reiter to formalize reasoning with default assumptions.
Descriptive complexity is a branch of computational complexity theory and of finite model theory that characterizes complexity classes by the type of logic needed to express the languages in them. For example, PH, the union of all complexity classes in the polynomial hierarchy, is precisely the class of languages expressible by statements of second-order logic. This connection between complexity and the logic of finite structures allows results to be transferred easily from one area to the other, facilitating new proof methods and providing additional evidence that the main complexity classes are somehow "natural" and not tied to the specific abstract machines used to define them.
Constraint Handling Rules (CHR) is a declarative, rule-based programming language, introduced in 1991 by Thom Frühwirth at the time with European Computer-Industry Research Centre (ECRC) in Munich, Germany. Originally intended for constraint programming, CHR finds applications in grammar induction, type systems, abductive reasoning, multi-agent systems, natural language processing, compilation, scheduling, spatial-temporal reasoning, testing, and verification.
Finite model theory is a subarea of model theory. Model theory is the branch of logic which deals with the relation between a formal language (syntax) and its interpretations (semantics). Finite model theory is a restriction of model theory to interpretations on finite structures, which have a finite universe.
The simply typed lambda calculus, a form of type theory, is a typed interpretation of the lambda calculus with only one type constructor that builds function types. It is the canonical and simplest example of a typed lambda calculus. The simply typed lambda calculus was originally introduced by Alonzo Church in 1940 as an attempt to avoid paradoxical use of the untyped lambda calculus.
Negation as failure is a non-monotonic inference rule in logic programming, used to derive from failure to derive . Note that can be different from the statement of the logical negation of , depending on the completeness of the inference algorithm and thus also on the formal logic system.
A definite clause grammar (DCG) is a way of expressing grammar, either for natural or formal languages, in a logic programming language such as Prolog. It is closely related to the concept of attribute grammars / affix grammars. DCGs are usually associated with Prolog, but similar languages such as Mercury also include DCGs. They are called definite clause grammars because they represent a grammar as a set of definite clauses in first-order logic.
In computer science, recursion is a method of solving a computational problem where the solution depends on solutions to smaller instances of the same problem. Recursion solves such recursive problems by using functions that call themselves from within their own code. The approach can be applied to many types of problems, and recursion is one of the central ideas of computer science.
The power of recursion evidently lies in the possibility of defining an infinite set of objects by a finite statement. In the same manner, an infinite number of computations can be described by a finite recursive program, even if this program contains no explicit repetitions.
In combinatorics, a square-free word is a word that does not contain any squares. A square is a word of the form XX, where X is not empty. Thus, a square-free word can also be defined as a word that avoids the pattern XX.
In descriptive set theory, within mathematics, Wadge degrees are levels of complexity for sets of reals. Sets are compared by continuous reductions. The Wadge hierarchy is the structure of Wadge degrees. These concepts are named after William W. Wadge.
The study of facility location problems (FLP), also known as location analysis, is a branch of operations research and computational geometry concerned with the optimal placement of facilities to minimize transportation costs while considering factors like avoiding placing hazardous materials near housing, and competitors' facilities. The techniques also apply to cluster analysis.
Dis-unification, in computer science and logic, is an algorithmic process of solving inequations between symbolic expressions.
