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Model Elimination
Model Elimination is the name attached to a pair of proof procedures invented by Donald W. Loveland, the first of which was published in 1968 in the Journal of the ACM. Their primary purpose is to carry out automated theorem proving, though they can readily be extended to logic programming, including the more general disjunctive logic programming. Model Elimination is closely related to resolution while also bearing characteristics of a Tableaux method. It is a progenitor of the SLD resolution procedure used in the Prolog Prolog is a logic programming language associated with artificial intelligence and computational linguistics. Prolog has its roots in first-order logic, a formal logic, and unlike many other programming languages, Prolog is intended primarily ... logic programming language. While somewhat eclipsed by attention to, and progress in, Resolution theorem provers, Model Elimination has continued to attract the attention of researchers and software developers. Tod ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Proof Procedure
In logic, and in particular proof theory, a proof procedure for a given logic is a systematic method for producing proofs in some proof calculus of (provable) statements. Types of proof calculi used There are several types of proof calculi. The most popular are natural deduction, sequent calculi (i.e., Gentzen type systems), Hilbert systems, and semantic tableaux or trees. A given proof procedure will target a specific proof calculus, but can often be reformulated so as to produce proofs in other proof styles. Completeness A proof procedure for a logic is ''complete'' if it produces a proof for each provable statement. The theorems of logical systems are typically recursively enumerable, which implies the existence of a complete but extremely inefficient proof procedure; however, a proof procedure is only of interest if it is reasonably efficient. Faced with an unprovable statement, a complete proof procedure may sometimes succeed in detecting and signalling its unprovability. In ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Donald W
Donald is a masculine given name derived from the Gaelic name ''Dòmhnall''.. This comes from the Proto-Celtic *''Dumno-ualos'' ("world-ruler" or "world-wielder"). The final -''d'' in ''Donald'' is partly derived from a misinterpretation of the Gaelic pronunciation by English speakers, and partly associated with the spelling of similar-sounding Germanic names, such as '' Ronald''. A short form of ''Donald'' is '' Don''. Pet forms of ''Donald'' include ''Donnie'' and ''Donny''. The feminine given name ''Donella'' is derived from ''Donald''. ''Donald'' has cognates in other Celtic languages: Modern Irish ''Dónal'' (anglicised as ''Donal'' and ''Donall'');. Scottish Gaelic ''Dòmhnall'', ''Domhnull'' and ''Dòmhnull''; Welsh '' Dyfnwal'' and Cumbric ''Dumnagual''. Although the feminine given name '' Donna'' is sometimes used as a feminine form of ''Donald'', the names are not etymologically related. Variations Kings and noblemen Domnall or Domhnall is the name of many ... [...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 an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Logic Programming
Logic programming is a programming paradigm which is largely based on formal logic Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from premis .... 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 ''Clause (logic), 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 formu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Disjunctive Logic Programming
Answer set programming (ASP) is a form of declarative programming oriented towards difficult (primarily NP-hard) search problems. It is based on the stable model (answer set) semantics of logic programming. In ASP, search problems are reduced to computing stable models, and ''answer set solvers''—programs for generating stable models—are used to perform search. The computational process employed in the design of many answer set solvers is an enhancement of the DPLL algorithm and, in principle, it always terminates (unlike Prolog query evaluation, which may lead to an infinite loop). In a more general sense, ASP includes all applications of answer sets to knowledge representation and the use of Prolog-style query evaluation for solving problems arising in these applications. History An early example of answer set programming was the planning method proposed in 1997 by Dimopoulos, Nebel and Köhler. Their approach is based on the relationship between plans and stable m ... [...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 need ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Method Of Analytic Tableaux
In proof theory, the semantic tableau (; plural: tableaux, also called truth tree) is a decision procedure for sentential and related logics, and a proof procedure for formulae of first-order logic. An analytic tableau is a tree structure computed for a logical formula, having at each node a subformula of the original formula to be proved or refuted. Computation constructs this tree and uses it to prove or refute the whole formula. The tableau method can also determine the satisfiability of finite sets of formulas of various logics. It is the most popular proof procedure for modal logics (Girle 2000). Introduction For refutation tableaux, the objective is to show that the negation of a formula cannot be satisfied. There are rules for handling each of the usual connectives, starting with the main connective. In many cases, applying these rules causes the subtableau to divide into two. Quantifiers are instantiated. If any branch of a tableau leads to an evident contradiction, t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
SLD Resolution
SLD resolution (''Selective Linear Definite'' clause resolution) is the basic inference rule used in logic programming. It is a refinement of resolution, which is both sound and refutation complete for Horn clauses. The SLD inference rule Given a goal clause, represented as the negation of a problem to be solved : \neg L_1 \lor \cdots \lor \neg L_i \lor \cdots \lor \neg L_n with selected literal \neg L_i , and an input definite clause: L \lor \neg K_1 \lor \cdots \lor \neg K_m whose positive literal (atom) L\, unifies with the atom L_i \, of the selected literal \neg L_i \, , SLD resolution derives another goal clause, in which the selected literal is replaced by the negative literals of the input clause and the unifying substitution \theta \, is applied: (\neg L_1 \lor \cdots \lor \neg K_1 \lor \cdots \lor \neg K_m\ \lor \cdots \lor \neg L_n)\theta In the simplest case, in propositional logic, the atoms L_i \, and L \, are identical, and the unifying ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prolog
Prolog is a logic programming language associated with artificial intelligence and computational linguistics. Prolog has its roots in first-order logic, a formal logic, and unlike many other programming languages, Prolog is intended primarily as a declarative programming language: the program logic is expressed in terms of Finitary relation, relations, represented as facts and Rule of inference, rules. A computation is initiated by running a ''query'' over these relations. The language was developed and implemented in Marseille, France, in 1972 by Alain Colmerauer with Philippe Roussel, based on Robert Kowalski's procedural interpretation of Horn clauses at University of Edinburgh. Prolog was one of the first logic programming languages and remains the most popular such language today, with several free and commercial implementations available. The language has been used for automated theorem proving, theorem proving, expert systems, term rewriting, type systems, and automated ... [...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 an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Logical Calculi
Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from premises in a topic-neutral way. When used as a countable noun, the term "a logic" refers to a logical formal system that articulates a proof system. Formal logic contrasts with informal logic, which is associated with informal fallacies, critical thinking, and argumentation theory. While there is no general agreement on how formal and informal logic are to be distinguished, one prominent approach associates their difference with whether the studied arguments are expressed in formal or informal languages. Logic plays a central role in multiple fields, such as philosophy, mathematics, computer science, and linguistics. Logic studies arguments, which consist of a set of premises together with a conclusion. Premises and conclusions are usually unde ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
