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Completeness (logic)
In mathematical logic and metalogic, a formal system is called complete with respect to a particular property (philosophy), property if every Well-formed formula, formula having the property can be formal proof, derived using that system, i.e. is one of its theorems; otherwise the system is said to be incomplete. The term "complete" is also used without qualification, with differing meanings depending on the context, mostly referring to the property of semantical Validity (logic), validity. Intuitively, a system is called complete in this particular sense, if it can derive every formula that is true. Other properties related to completeness The property Converse (logic)#Categorical converse, converse to completeness is called soundness: a system is sound with respect to a property (mostly semantical validity) if each of its theorems has that property. Forms of completeness Expressive completeness A formal language is expressively complete if it can express the subject matte ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Logic
Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of formal systems of logic such as their expressive or deductive power. However, it can also include uses of logic to characterize correct mathematical reasoning or to establish foundations of mathematics. Since its inception, mathematical logic has both contributed to and been motivated by the study of foundations of mathematics. This study began in the late 19th century with the development of axiomatic frameworks for geometry, arithmetic, and analysis. In the early 20th century it was shaped by David Hilbert's program to prove the consistency of foundational theories. Results of Kurt Gödel, Gerhard Gentzen, and others provided partial resolution to the program, and clarified the issues involved in proving consistency. Work in set theory sho ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Propositional Function
In propositional calculus, a propositional function or a predicate is a sentence expressed in a way that would assume the value of true or false, except that within the sentence there is a variable (''x'') that is not defined or specified (thus being a free variable), which leaves the statement undetermined. The sentence may contain several such variables (e.g. ''n'' variables, in which case the function takes ''n'' arguments). Overview As a mathematical function, ''A''(''x'') or ''A''(''x'', ''x'', ..., ''x''), the propositional function is abstracted from predicates or propositional forms. As an example, consider the predicate scheme, "x is hot". The substitution of any entity for ''x'' will produce a specific proposition that can be described as either true or false, even though "''x'' is hot" on its own has no value as either a true or false statement. However, when a value is assigned to ''x'' , such as lava, the function then has the value ''true''; while one assigns to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
A Modern Approach
A, or a, is the first letter and the first vowel of the Latin alphabet, used in the modern English alphabet, the alphabets of other western European languages and others worldwide. Its name in English is ''a'' (pronounced ), plural ''aes''. It is similar in shape to the Ancient Greek letter alpha, from which it derives. The uppercase version consists of the two slanting sides of a triangle, crossed in the middle by a horizontal bar. The lowercase version can be written in two forms: the double-storey a and single-storey ɑ. The latter is commonly used in handwriting and fonts based on it, especially fonts intended to be read by children, and is also found in italic type. In English grammar, " a", and its variant " an", are indefinite articles. History The earliest certain ancestor of "A" is aleph (also written 'aleph), the first letter of the Phoenician alphabet, which consisted entirely of consonants (for that reason, it is also called an abjad to distinguish it f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Peter Norvig
Peter Norvig (born December 14, 1956) is an American computer scientist and Distinguished Education Fellow at the Stanford Institute for Human-Centered AI. He previously served as a director of research and search quality at Google. Norvig is the co-author with Stuart J. Russell of the most popular textbook in the field of AI: '' Artificial Intelligence: A Modern Approach'' used in more than 1,500 universities in 135 countries. Education Norvig received a Bachelor of Science in applied mathematics from Brown University and a Ph.D. in computer science from the University of California, Berkeley. Career and research Norvig is a councilor of the Association for the Advancement of Artificial Intelligence and co-author, with Stuart J. Russell, of '' Artificial Intelligence: A Modern Approach'', now the leading college text in the field. He was head of the Computational Sciences Division (now the Intelligent Systems Division) at NASA Ames Research Center, where he oversaw a staff ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stuart J
Stuart may refer to: Names *Stuart (name), a given name and surname (and list of people with the name) Automobile * Stuart (automobile) Places Australia Generally * Stuart Highway, connecting South Australia and the Northern Territory Northern Territory *Stuart, the former name for Alice Springs (changed 1933) * Stuart Park, an inner city suburb of Darwin *Central Mount Stuart, a mountain peak Queensland * Stuart, Queensland, a suburb of Townsville *Mount Stuart, Queensland, a suburb of Townsville *Mount Stuart (Queensland), a mountain South Australia * Stuart, South Australia, a locality in the Mid Murray Council *Electoral district of Stuart, a state electoral district * Hundred of Stuart, a cadastral unit Canada *Stuart Channel, a strait in the Gulf of Georgia region of British Columbia United Kingdom *Castle Stuart United States * Stuart, Florida *Stuart, Iowa *Stuart, Nebraska * Stuart, Oklahoma *Stuart, Virginia *Stuart Township, Holt County, Nebras ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Clause (logic)
In logic, a clause is a propositional formula formed from a finite collection of literals (atoms or their negations) and logical connectives. A clause is true either whenever at least one of the literals that form it is true (a disjunctive clause, the most common use of the term), or when all of the literals that form it are true (a conjunctive clause, a less common use of the term). That is, it is a finite disjunction or conjunction of literals, depending on the context. Clauses are usually written as follows, where the symbols l_i are literals: :l_1 \vee \cdots \vee l_n Empty clauses A clause can be empty (defined from an empty set of literals). The empty clause is denoted by various symbols such as \empty, \bot, or \Box. The truth evaluation of an empty disjunctive clause is always false. This is justified by considering that false is the neutral element of the monoid (\, \vee). The truth evaluation of an empty conjunctive clause is always true. This is related to the concept ... [...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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Superposition Calculus
The superposition calculus is a calculus for reasoning in equational first-order logic. It was developed in the early 1990s and combines concepts from first-order resolution with ordering-based equality handling as developed in the context of (unfailing) Knuth–Bendix completion. It can be seen as a generalization of either resolution (to equational logic) or unfailing completion (to full clausal logic). As most first-order calculi, superposition tries to show the ''unsatisfiability'' of a set of first-order clauses, i.e. it performs proofs by refutation. Superposition is refutation-complete—given unlimited resources and a ''fair'' derivation strategy, from any unsatisfiable clause set a contradiction will eventually be derived. , most of the (state-of-the-art) theorem provers for first-order logic are based on superposition (e.g. the E equational theorem prover), although only a few implement the pure calculus. Implementations * E * SPASS * Vampire * Waldmeisterbr>( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Horn Clause
In mathematical logic and logic programming, a Horn clause is a logical formula of a particular rule-like form which gives it useful properties for use in logic programming, formal specification, and model theory. Horn clauses are named for the logician Alfred Horn, who first pointed out their significance in 1951. Definition A Horn clause is a clause (a disjunction of literals) with at most one positive, i.e. unnegated, literal. Conversely, a disjunction of literals with at most one negated literal is called a dual-Horn clause. A Horn clause with exactly one positive literal is a definite clause or a strict Horn clause; a definite clause with no negative literals is a unit clause, and a unit clause without variables is a fact;. A Horn clause without a positive literal is a goal clause. Note that the empty clause, consisting of no literals (which is equivalent to ''false'') is a goal clause. These three kinds of Horn clauses are illustrated in the following propositional ... [...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] |
False (logic)
In logic, false or untrue is the state of possessing negative truth value or a nullary logical connective. In a truth-functional system of propositional logic, it is one of two postulated truth values, along with its negation, truth. Usual notations of the false are 0 (especially in Boolean logic and computer science), O (in prefix notation, O''pq''), and the up tack symbol \bot. Another approach is used for several formal theories (e.g., intuitionistic propositional calculus), where a propositional constant (i.e. a nullary connective), \bot, is introduced, the truth value of which being always false in the sense above. It can be treated as an absurd proposition, and is often called absurdity. In classical logic and Boolean logic In Boolean logic, each variable denotes a truth value which can be either true (1), or false (0). In a classical propositional calculus, each proposition will be assigned a truth value of either true or false. Some systems of c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
First-order Logic
First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—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. A theory about a topic is usually a first-order logic together with a specified domain of discourse (over which the quantified variables range), finitely many functions from that domain to itself, finitely many predicates defined on that domain, and a set of a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
