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Vampire Theorem Prover
Vampire is an automatic theorem prover for first-order classical logic developed in the Department of Computer Science at the University of Manchester. Up to Version 3, it was developed by Andrei Voronkov together with Kryštof Hoder and previously with Alexandre Riazanov. Since Version 4, the development has involved a wider international team including Laura Kovacs, Giles Reger, and Martin Suda. Since 1999 it has won at least 53 trophies in the CADE ATP System Competition, the "world cup for theorem provers", including the most prestigious FOF division and the theory-reasoning TFA division. Background Vampire's kernel implements the calculi of ordered binary resolution and superposition for handling equality. The splitting rule and negative equality splitting can be simulated by the introduction of new predicate definitions and dynamic folding of such definitions. A DPLL-style algorithm splitting is also supported. A number of standard redundancy criteria and simplification ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Andrei Voronkov
Andrei Anatolievič Voronkov (born 1959) is a Professor of Formal methods in the Department of Computer Science at the University of Manchester. Education Voronkov was educated at Novosibirsk State University, graduating with a PhD in 1987. Research Voronkov is known for the Vampire automated theorem prover, the EasyChair conference management software, the Handbook of Automated Reasoning (with John Alan Robinson, 2001), and as organiser of the Alan Turing Centenary Conference 2012. Voronkov's research has been funded by the Engineering and Physical Sciences Research Council (EPSRC). Awards and honours In 2015, his contributions to the field of automated reasoning were recognized with the Herbrand Award. He has won 25 division titles in the CADE ATP System Competition The CADE ATP System Competition (CASC) is a yearly competition of fully automated theorem provers for classical logic CASC is associated with the Conference on Automated Deduction and the International ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tautology (logic)
In mathematical logic, a tautology (from el, ταυτολογία) is a formula or assertion that is true in every possible interpretation. An example is "x=y or x≠y". Similarly, "either the ball is green, or the ball is not green" is always true, regardless of the colour of the ball. The philosopher Ludwig Wittgenstein first applied the term to redundancies of propositional logic in 1921, borrowing from rhetoric, where a tautology is a repetitive statement. In logic, a formula is satisfiable if it is true under at least one interpretation, and thus a tautology is a formula whose negation is unsatisfiable. In other words, it cannot be false. It cannot be untrue. Unsatisfiable statements, both through negation and affirmation, are known formally as contradictions. A formula that is neither a tautology nor a contradiction is said to be Contingency (philosophy), logically contingent. Such a formula can be made either true or false based on the values assigned to its propositi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Executable
In computing, executable code, an executable file, or an executable program, sometimes simply referred to as an executable or binary, causes a computer "to perform indicated tasks according to encoded instruction (computer science), instructions", as opposed to a data (computing), data file that must be interpreted (parser, parsed) by a program to be meaningful. The exact interpretation depends upon the use. "Instructions" is traditionally taken to mean machine code instructions for a physical central processing unit, CPU. In some contexts, a file containing scripting instructions (such as bytecode) may also be considered executable. Generation of executable files Executable files can be hand-coded in machine language, although it is far more convenient to develop software as source code in a high-level language that can be easily understood by humans. In some cases, source code might be specified in assembly language instead, which remains human-readable while being close ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Craig Interpolation
In mathematical logic, Craig's interpolation theorem is a result about the relationship between different logical theories. Roughly stated, the theorem says that if a formula φ implies a formula ψ, and the two have at least one atomic variable symbol in common, then there is a formula ρ, called an interpolant, such that every non-logical symbol in ρ occurs both in φ and ψ, φ implies ρ, and ρ implies ψ. The theorem was first proved for first-order logic by William Craig in 1957. Variants of the theorem hold for other logics, such as propositional logic. A stronger form of Craig's interpolation theorem for first-order logic was proved by Roger Lyndon in 1959; the overall result is sometimes called the Craig–Lyndon theorem. Example In propositional logic, let ::: \varphi = \lnot(P \land Q) \to (\lnot R \land Q) ::: \psi = (S \to P) \lor (S \to \lnot R) . Then \varphi tautologically implies \psi. This can be verified by writing \varphi in conjunctive normal form: : ... [...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] |
Run-time Algorithm Specialisation
In computer science, run-time algorithm specialization is a methodology for creating efficient algorithms for costly computation tasks of certain kinds. The methodology originates in the field of automated theorem proving and, more specifically, in the Vampire theorem prover project. The idea is inspired by the use of partial evaluation in optimising program translation. Many core operations in theorem provers exhibit the following pattern. Suppose that we need to execute some algorithm \mathit(A,B) in a situation where a value of A ''is fixed for potentially many different values of'' B. In order to do this efficiently, we can try to find a specialization of \mathit for every fixed A, i.e., such an algorithm \mathit_A, that executing \mathit_A(B) is equivalent to executing \mathit(A,B). The specialized algorithm may be more efficient than the generic one, since it can ''exploit some particular properties'' of the fixed value A. Typically, \mathit_A(B) can avoid some operations tha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Term Indexing
In computer science, a term index is a data structure to facilitate fast lookup of terms and clauses in a logic program, deductive database, or automated theorem prover. Overview Many operations in automatic theorem provers require search in huge collections of terms and clauses. Such operations typically fall into the following scheme. Given a collection S of terms (clauses) and a query term (clause) q, find in S some/all terms t related to q according to a certain retrieval condition. Most interesting retrieval conditions are formulated as existence of a substitution that relates in a special way the query and the retrieved objects t. Here is a list of retrieval conditions frequently used in provers: * term q is unifiable with term t, i.e., there exists a substitution \theta , such that q\theta = t\theta * term t is an instance of q, i.e., there exists a substitution \theta, such that q\theta = t * term t is a generalisation of q, i.e., there exists a substitution \theta, suc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Knuth–Bendix Completion Algorithm
The Knuth–Bendix completion algorithm (named after Donald Knuth and Peter Bendix) is a semi-decision algorithm for transforming a set of equations (over terms) into a confluent term rewriting system. When the algorithm succeeds, it effectively solves the word problem for the specified algebra. Buchberger's algorithm for computing Gröbner bases is a very similar algorithm. Although developed independently, it may also be seen as the instantiation of Knuth–Bendix algorithm in the theory of polynomial rings. Introduction For a set ''E'' of equations, its deductive closure () is the set of all equations that can be derived by applying equations from ''E'' in any order. Formally, ''E'' is considered a binary relation, () is its rewrite closure, and () is the equivalence closure of (). For a set ''R'' of rewrite rules, its deductive closure ( ∘ ) is the set of all equations that can be confirmed by applying rules from ''R'' left-to-right to both sides until they are literal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hierarchy
A hierarchy (from Greek: , from , 'president of sacred rites') is an arrangement of items (objects, names, values, categories, etc.) that are represented as being "above", "below", or "at the same level as" one another. Hierarchy is an important concept in a wide variety of fields, such as architecture, philosophy, design, mathematics, computer science, organizational theory, systems theory, systematic biology, and the social sciences (especially political philosophy). A hierarchy can link entities either directly or indirectly, and either vertically or diagonally. The only direct links in a hierarchy, insofar as they are hierarchical, are to one's immediate superior or to one of one's subordinates, although a system that is largely hierarchical can also incorporate alternative hierarchies. Hierarchical links can extend "vertically" upwards or downwards via multiple links in the same direction, following a path. All parts of the hierarchy that are not linked vertically to one ano ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
DPLL Algorithm
In logic and computer science, the Davis–Putnam–Logemann–Loveland (DPLL) algorithm is a complete, backtracking-based search algorithm for deciding the satisfiability of propositional logic formulae in conjunctive normal form, i.e. for solving the CNF-SAT problem. It was introduced in 1961 by Martin Davis, George Logemann and Donald W. Loveland and is a refinement of the earlier Davis–Putnam algorithm, which is a resolution-based procedure developed by Davis and Hilary Putnam in 1960. Especially in older publications, the Davis–Logemann–Loveland algorithm is often referred to as the "Davis–Putnam method" or the "DP algorithm". Other common names that maintain the distinction are DLL and DPLL. Implementations and applications The SAT problem is important both from theoretical and practical points of view. In complexity theory it was the first problem proved to be NP-complete, and can appear in a broad variety of applications such as ''model checking'', automate ... [...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] |
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>(off ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
