|
Occurs Check
In computer science, the occurs check is a part of algorithms for syntactic unification. It causes unification of a variable ''V'' and a structure ''S'' to fail if ''S'' contains ''V''. Application in theorem proving In theorem proving, unification without the occurs check can lead to unsound inference. For example, the Prolog goal X = f(X) will succeed, binding ''X'' to a cyclic structure which has no counterpart in the Herbrand universe. As another example, without occurs-check, a resolution proof can be found for the non-theorem (\forall x \exists y. p(x,y)) \rightarrow (\exists y \forall x. p(x,y)): the negation of that formula has the conjunctive normal form p(X,f(X)) \land \lnot p(g(Y),Y), with f and g denoting the Skolem function for the first and second existential quantifier, respectively; the literals p(X,f(X)) and p(g(Y),Y) are unifiable without occurs check, producing the refuting empty clause. Rational tree unification Prolog implementations usually omit th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Computer Science
Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to Applied science, practical disciplines (including the design and implementation of Computer architecture, hardware and Computer programming, software). Computer science is generally considered an area of research, academic research and distinct from computer programming. Algorithms and data structures are central to computer science. The theory of computation concerns abstract models of computation and general classes of computational problem, problems that can be solved using them. The fields of cryptography and computer security involve studying the means for secure communication and for preventing Vulnerability (computing), security vulnerabilities. Computer graphics (computer science), Computer graphics and computational geometry address the generation of images. Progr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conjunctive Normal Form
In Boolean logic, a formula is in conjunctive normal form (CNF) or clausal normal form if it is a conjunction of one or more clauses, where a clause is a disjunction of literals; otherwise put, it is a product of sums or an AND of ORs. As a canonical normal form, it is useful in automated theorem proving and circuit theory. All conjunctions of literals and all disjunctions of literals are in CNF, as they can be seen as conjunctions of one-literal clauses and conjunctions of a single clause, respectively. As in the disjunctive normal form (DNF), the only propositional connectives a formula in CNF can contain are and, or, and not. The not operator can only be used as part of a literal, which means that it can only precede a propositional variable or a predicate symbol. In automated theorem proving, the notion "''clausal normal form''" is often used in a narrower sense, meaning a particular representation of a CNF formula as a set of sets of literals. Examples and non-examples ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Logic Programming
Logic programming is a programming paradigm which is largely based on formal logic. Any program written in a logic programming language is a set of sentences in logical form, expressing facts and rules about some problem 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'': :H :- B1, …, Bn. and are read declaratively as logical implications: :H if B1 and … and Bn. H is called the ''head'' of the rule and B1, ..., Bn is called the ''body''. Facts are rules that have no body, and are written in the simplified form: :H. In the simplest case in which H, B1, ..., Bn are all atomic formulae, these clauses are called definite clauses or Horn clauses. However, there are many extensions of this simple case, the most important one being the case in which conditions in the body of a clause can also be negations of atomic formulas. Logic programming languag ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 major impetus for the development of computer science. Logical foundations While the roots of formalised logic go back to Aristotle, the end of the 19th and early 20th centuries saw the development of modern logic and formalised mathematics. Frege's ''Begriffsschrift'' (1879) introduced both a complete propositional calculus and what is essentially modern predicate logic. His ''Foundations of Arithmetic'', published 1884, expressed (parts of) mathematics in formal logic. This approach was continued by Russell and Whitehead in their influential ''Principia Mathematica'', first published 1910–1913, and with a revised second edition in 1927. Russell and Whitehead thought they could derive all mathematical truth using axioms and inference ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
SWI-Prolog
SWI-Prolog is a free implementation of the programming language Prolog, commonly used for teaching and semantic web applications. It has a rich set of features, libraries for constraint logic programming, multithreading, unit testing, GUI, interfacing to Java, ODBC and others, literate programming, a web server, SGML, RDF, RDFS, developer tools (including an IDE with a GUI debugger and GUI profiler), and extensive documentation. SWI-Prolog runs on Unix, Windows, Macintosh and Linux platforms. SWI-Prolog has been under continuous development since 1987. Its main author is Jan Wielemaker. The name SWI is derived from ("Social Science Informatics"), the former name of the group at the University of Amsterdam, where Wielemaker is employed. The name of this group has changed to HCS (Human-Computer Studies). Web framework SWI-Prolog installs with a web framework based on definite clause grammars. Distributed computing SWI-Prolog queries may be distributed over several se ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Strawberry Prolog
The garden strawberry (or simply strawberry; ''Fragaria × ananassa'') is a widely grown hybrid species of the genus '' Fragaria'', collectively known as the strawberries, which are cultivated worldwide for their fruit. The fruit is widely appreciated for its characteristic aroma, bright red color, juicy texture, and sweetness. It is consumed in large quantities, either fresh or in such prepared foods as jam, juice, pies, ice cream, milkshakes, and chocolates. Artificial strawberry flavorings and aromas are also widely used in products such as candy, soap, lip gloss, perfume, and many others. The garden strawberry was first bred in Brittany, France, in the 1750s via a cross of ''Fragaria virginiana'' from eastern North America and ''Fragaria chiloensis'', which was brought from Chile by Amédée-François Frézier in 1714. Cultivars of ''Fragaria'' × ''ananassa'' have replaced, in commercial production, the woodland strawberry ('' Fragaria vesca''), which was the first stra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unification (computer Science)
In logic and computer science, unification is an algorithmic process of solving equations between symbolic expressions. Depending on which expressions (also called ''terms'') are allowed to occur in an equation set (also called ''unification problem''), and which expressions are considered equal, several frameworks of unification are distinguished. If higher-order variables, that is, variables representing functions, are allowed in an expression, the process is called higher-order unification, otherwise first-order unification. If a solution is required to make both sides of each equation literally equal, the process is called syntactic or free unification, otherwise semantic or equational unification, or E-unification, or unification modulo theory. A ''solution'' of a unification problem is denoted as a substitution, that is, a mapping assigning a symbolic value to each variable of the problem's expressions. A unification algorithm should compute for a given problem a ''complete ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Example For Syntactic Unification Without Occurs Check Leading To Infinite Tree Svg
Example may refer to: * '' exempli gratia'' (e.g.), usually read out in English as "for example" * .example, reserved as a domain name that may not be installed as a top-level domain of the Internet ** example.com, example.net, example.org, example.edu, second-level domain names reserved for use in documentation as examples * HMS ''Example'' (P165), an Archer-class patrol and training vessel of the Royal Navy Arts * ''The Example'', a 1634 play by James Shirley * ''The Example'' (comics), a 2009 graphic novel by Tom Taylor and Colin Wilson * Example (musician), the British dance musician Elliot John Gleave (born 1982) * ''Example'' (album), a 1995 album by American rock band For Squirrels See also * * Exemplar (other), a prototype or model which others can use to understand a topic better * Exemplum, medieval collections of short stories to be told in sermons * Eixample The Eixample (; ) is a district of Barcelona between the old city ( Ciutat Vella) an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Skolem Function
In mathematical logic, a formula of first-order logic is in Skolem normal form if it is in prenex normal form with only universal first-order quantifiers. Every first-order formula may be converted into Skolem normal form while not changing its satisfiability via a process called Skolemization (sometimes spelled Skolemnization). The resulting formula is not necessarily equivalent to the original one, but is equisatisfiable with it: it is satisfiable if and only if the original one is satisfiable. Reduction to Skolem normal form is a method for removing existential quantifiers from formal logic statements, often performed as the first step in an automated theorem prover. Examples The simplest form of Skolemization is for existentially quantified variables that are not inside the scope of a universal quantifier. These may be replaced simply by creating new constants. For example, \exists x P(x) may be changed to P(c), where c is a new constant (does not occur anywhere else i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Resolution (logic)
In mathematical logic and automated theorem proving, resolution is a rule of inference leading to a refutation complete theorem-proving technique for sentences in propositional logic and first-order logic. For propositional logic, systematically applying the resolution rule acts as a decision procedure for formula unsatisfiability, solving the (complement of the) Boolean satisfiability problem. For first-order logic, resolution can be used as the basis for a semi-algorithm for the unsatisfiability problem of first-order logic, providing a more practical method than one following from Gödel's completeness theorem. The resolution rule can be traced back to Davis and Putnam (1960); however, their algorithm required trying all ground instances of the given formula. This source of combinatorial explosion was eliminated in 1965 by John Alan Robinson's syntactical unification algorithm, which allowed one to instantiate the formula during the proof "on demand" just as far as needed to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algorithm
In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific Computational problem, problems or to perform a computation. Algorithms are used as specifications for performing calculations and data processing. More advanced algorithms can perform automated deductions (referred to as automated reasoning) and use mathematical and logical tests to divert the code execution through various routes (referred to as automated decision-making). Using human characteristics as descriptors of machines in metaphorical ways was already practiced by Alan Turing with terms such as "memory", "search" and "stimulus". In contrast, a Heuristic (computer science), heuristic is an approach to problem solving that may not be fully specified or may not guarantee correct or optimal results, especially in problem domains where there is no well-defined correct or optimal result. As an effective method, an algorithm ca ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Herbrand Universe
In first-order logic, a Herbrand structure ''S'' is a structure over a vocabulary σ that is defined solely by the syntactical properties of σ. The idea is to take the symbols of terms as their values, e.g. the denotation of a constant symbol ''c'' is just "''c''" (the symbol). It is named after Jacques Herbrand. Herbrand structures play an important role in the foundations of logic programming. Herbrand universe Definition The ''Herbrand universe'' serves as the universe in the ''Herbrand structure''. Example Let , be a first-order language with the vocabulary * constant symbols: ''c'' * function symbols: ''f''(·), ''g''(·) then the Herbrand universe of (or ) is . Notice that the relation symbols are not relevant for a Herbrand universe. Herbrand structure A ''Herbrand structure'' interprets terms on top of a ''Herbrand universe''. Definition Let ''S'' be a structure, with vocabulary σ and universe ''U''. Let ''W'' be the set of all terms over σ and ''W' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
