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Equational Prover
{{about, automated theorem proving, the complexity class named EQP, EQP (complexity) EQP, an abbreviation for equational prover, is an automated theorem proving program for equational logic, developed by the Mathematics and Computer Science Division of the Argonne National Laboratory. It was one of the provers used for solving a longstanding problem posed by Herbert Robbins, namely, whether all Robbins algebras are Boolean algebra In mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variables are the truth values ''true'' and ''false'', usually denoted 1 and 0, whereas in e ...s. External links EQP project Robbins Algebras Are Boolean Argonne National Laboratory, Mathematics and Computer Science Division Theorem proving software systems ... [...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] |
Equational Logic
First-order equational logic consists of quantifier-free terms of ordinary first-order logic, with equality as the only predicate symbol. The model theory of this logic was developed into universal algebra by Birkhoff, Grätzer, and Cohn. It was later made into a branch of category theory by Lawvere ("algebraic theories").equational logic. (n.d.). The Free On-line Dictionary of Computing. Retrieved October 24, 2011, from Dictionary.com website: http://dictionary.reference.com/browse/equational+logic The terms of equational logic are built up from variables and constants using function symbols (or operations). Syllogism Here are the four inference rules of logic. P := E/math> denotes textual substitution of expression E for variable x in expression P. Next, b = c denotes equality, for b and c of the same type, while b \equiv c, or equivalence, is defined only for b and c of type boolean. For b and c of type boolean, b = c and b \equiv c have the same meaning. Gries, D. (2010 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Argonne National Laboratory
Argonne National Laboratory is a science and engineering research United States Department of Energy National Labs, national laboratory operated by University of Chicago, UChicago Argonne LLC for the United States Department of Energy. The facility is located in Lemont, Illinois, outside of Chicago, and is the largest national laboratory by size and scope in the Midwest. Argonne had its beginnings in the Metallurgical Laboratory of the University of Chicago, formed in part to carry out Enrico Fermi's work on nuclear reactors for the Manhattan Project during World War II. After the war, it was designated as the first national laboratory in the United States on July 1, 1946. In the post-war era the lab focused primarily on non-weapon related nuclear physics, designing and building the first power-producing nuclear reactors, helping design the reactors used by the United States' nuclear navy, and a wide variety of similar projects. In 1994, the lab's nuclear mission ended, and today ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Herbert Robbins
Herbert Ellis Robbins (January 12, 1915 – February 12, 2001) was an American mathematician and statistician. He did research in topology, measure theory, statistics, and a variety of other fields. He was the co-author, with Richard Courant, of '' What is Mathematics?'', a popularization that is still () in print. The Robbins lemma, used in empirical Bayes methods, is named after him. Robbins algebras are named after him because of a conjecture (since proved) that he posed concerning Boolean algebras. The Robbins theorem, in graph theory, is also named after him, as is the Whitney–Robbins synthesis, a tool he introduced to prove this theorem. The well-known unsolved problem of minimizing in sequential selection the expected rank of the selected item under full information, sometimes referred to as the fourth secretary problem, also bears his name: Robbins' problem (of optimal stopping). Biography Robbins was born in New Castle, Pennsylvania. As an undergraduate, Rob ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Robbins Algebra
In abstract algebra, a Robbins algebra is an algebra containing a single binary operation, usually denoted by \lor, and a single unary operation usually denoted by \neg. These operations satisfy the following axioms: For all elements ''a'', ''b'', and ''c'': # Associativity: a \lor \left(b \lor c \right) = \left(a \lor b \right) \lor c # Commutativity: a \lor b = b \lor a # ''Robbins equation'': \neg \left( \neg \left(a \lor b \right) \lor \neg \left(a \lor \neg b \right) \right) = a For many years, it was conjectured, but unproven, that all Robbins algebras are Boolean algebras. This was proved in 1996, so the term "Robbins algebra" is now simply a synonym for "Boolean algebra". History In 1933, Edward Huntington proposed a new set of axioms for Boolean algebras, consisting of (1) and (2) above, plus: *''Huntington's equation'': \neg(\neg a \lor b) \lor \neg(\neg a \lor \neg b) = a. From these axioms, Huntington derived the usual axioms of Boolean algebra. Very soon thereafte ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Boolean Algebra (structure)
In abstract algebra, a Boolean algebra or Boolean lattice is a complemented distributive lattice. This type of algebraic structure captures essential properties of both set operations and logic operations. A Boolean algebra can be seen as a generalization of a power set algebra or a field of sets, or its elements can be viewed as generalized truth values. It is also a special case of a De Morgan algebra and a Kleene algebra (with involution). Every Boolean algebra gives rise to a Boolean ring, and vice versa, with ring multiplication corresponding to conjunction or meet ∧, and ring addition to exclusive disjunction or symmetric difference (not disjunction ∨). However, the theory of Boolean rings has an inherent asymmetry between the two operators, while the axioms and theorems of Boolean algebra express the symmetry of the theory described by the duality principle. __TOC__ History The term "Boolean algebra" honors George Boole (1815–1864), a self-educated English ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
