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InductiveDataType Systems
, 2002
"... In a previous work ("Abstract Data Type Systems", TCS 173(2), 1997), the leI two authors presented a combined lmbined made of a (strongl normal3zG9 alrmal rewrite system and a typed #calA#Ik enriched by patternmatching definitions folnitio a certain format,calat the "General Schema", whichgenera ..."
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Cited by 755 (22 self)
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In a previous work ("Abstract Data Type Systems", TCS 173(2), 1997), the leI two authors presented a combined lmbined made of a (strongl normal3zG9 alrmal rewrite system and a typed #calA#Ik enriched by patternmatching definitions folnitio a certain format,calat the "General Schema", whichgeneral39I theusual recursor definitions fornatural numbers and simil9 "basic inductive types". This combined lmbined was shown to bestrongl normalIk39f The purpose of this paper is toreformul33 and extend theGeneral Schema in order to make it easil extensibl3 to capture a more general cler of inductive types, cals, "strictly positive", and to ease the strong normalgAg9Ik proof of theresulGGg system. Thisresul provides a computation model for the combination of anal"DAfGI specification language based on abstract data types and of astrongl typed functional language with strictly positive inductive types.
The TPTP Problem Library
, 1999
"... This report provides a detailed description of the TPTP Problem Library for automated theorem proving systems. The library is available via Internet, and forms a common basis for development of and experimentation with automated theorem provers. This report provides: ffl the motivations for buildin ..."
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Cited by 100 (6 self)
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This report provides a detailed description of the TPTP Problem Library for automated theorem proving systems. The library is available via Internet, and forms a common basis for development of and experimentation with automated theorem provers. This report provides: ffl the motivations for building the library; ffl a discussion of the inadequacies of previous problem collections, and how these have been resolved in the TPTP; ffl a description of the library structure, including overview information; ffl descriptions of supplementary utility programs; ffl guidelines for obtaining and using the library; Contents 1 Introduction 2 1.1 Previous Problem Collections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 What is Required? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2 Inside the TPTP 6 2.1 The TPTP Domain Structure . . . . . . . . . . . . . . . . . . . . . ...
Single Identities for Lattice Theory and for Weakly Associative Lattices
 Algebra Universalis
, 1995
"... . We present a single identity for the variety of all lattices that is much simpler than those previously known to us. We also show that the variety of weakly associative lattices is onebased, and we present a generalized onebased theorem for subvarieties of weakly associative lattices that can be ..."
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Cited by 11 (10 self)
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. We present a single identity for the variety of all lattices that is much simpler than those previously known to us. We also show that the variety of weakly associative lattices is onebased, and we present a generalized onebased theorem for subvarieties of weakly associative lattices that can be defined with absorption laws. The automated theoremproving program Otter was used in a substantial way to obtain the results. 1 Introduction Equational identities are, perhaps, the simplest form of sentences expressing many basic properties of algebras. Several familiar classes of algebras, such as semigroups, groups, rings, lattices, and Boolean algebras, are defined by equational identities. Such a class of algebras is known as an equational Supported by the Office of Scientific Computing, U.S. Department of Energy, under Contract W31109Eng38. y Supported by an operating grant from NSERC of Canada (#A8215). class of algebras or a variety of algebras (for mathematical properti...
Computers, Reasoning and Mathematical Practice
"... ion in itself is not the goal: for Whitehead [117]"it is the large generalisation, limited by a happy particularity, which is the fruitful conception." As an example consider the theorem in ring theory, which states that if R is a ring, f(x) is a polynomial over R and f(r) = 0 for every element of ..."
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Cited by 6 (2 self)
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ion in itself is not the goal: for Whitehead [117]"it is the large generalisation, limited by a happy particularity, which is the fruitful conception." As an example consider the theorem in ring theory, which states that if R is a ring, f(x) is a polynomial over R and f(r) = 0 for every element of r of R then R is commutative. Special cases of this, for example f(x) is x 2 \Gamma x or x 3 \Gamma x, can be given a first order proof in a few lines of symbol manipulation. The usual proof of the general result [20] (which takes a semester's postgraduate course to develop from scratch) is a corollary of other results: we prove that rings satisfying the condition are semisimple artinian, apply a theorem which shows that all such rings are matrix rings over division rings, and eventually obtain the result by showing that all finite division rings are fields, and hence commutative. This displays von Neumann's architectural qualities: it is "deep" in a way in which the symbol manipulati...
Yet another single law for lattices
"... Abstract. In this note we show that the equational theory of all lattices is defined by the single absorption law (((y∨x)∧x)∨(((z∧(x∨x))∨(u∧x))∧v))∧(w∨((s∨x)∧(x∨t))) = x. This identity of length 29 with 8 variables is shorter than previously known such equations defining lattices. 1. ..."
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Cited by 3 (0 self)
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Abstract. In this note we show that the equational theory of all lattices is defined by the single absorption law (((y∨x)∧x)∨(((z∧(x∨x))∨(u∧x))∧v))∧(w∨((s∨x)∧(x∨t))) = x. This identity of length 29 with 8 variables is shorter than previously known such equations defining lattices. 1.
Computer and Human Reasoning: Single Implicative Axioms for Groups and for Abelian Groups
, 1996
"... single axiom, but then \Delta is not product, and \Gamma1 is not inverse. The same situation holds for the Abelian case. Another curious fact is that there is no single equational axiom for groups or for Abelian groups in terms of the three standard operations of product, inverse, and Supported ..."
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Cited by 3 (1 self)
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single axiom, but then \Delta is not product, and \Gamma1 is not inverse. The same situation holds for the Abelian case. Another curious fact is that there is no single equational axiom for groups or for Abelian groups in terms of the three standard operations of product, inverse, and Supported by the Mathematical, Information, and Computational Sciences Division subprogram of the Office of Computational and Technology Research, U.S. Department of Energy, under Contract W31109Eng38. the identity element [8]. Single equational axioms in terms of product and inverse have been reported by Neumann [5] and others [3, 2]. In this note we consider single implicative axioms, that is, axioms of the form ff = fi ) fl = ffi. For Abelian groups, an axiom of this type with five variables was given by Sholander [6]. If we allow one of f
Equations in Algebra and Topology
, 1997
"... of the talk: In the early 1930's, Garrett Birkhoff introduced the notion a variety V , i.e. the class of all algebras (in the general sense) that model a fixed set \Sigma of equations, and proved his famous theorem characterizing such equational classes being classes closed under the formation o ..."
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of the talk: In the early 1930's, Garrett Birkhoff introduced the notion a variety V , i.e. the class of all algebras (in the general sense) that model a fixed set \Sigma of equations, and proved his famous theorem characterizing such equational classes being classes closed under the formation of subalgebras, homomorphic images and products. Ever since then (and more intensely since 1970), such classes have been actively studied  both in general and in many particular examples. In this talk we will survey some of this work, mentioning important results and problems of Tarski, Jonsson, McKenzie, Baker. We will also include some recent material on the modeling of the equations \Sigma by continuous operations on a topological space A. (Birkhoff was interested in this question as well.) Some recent results of the author say that many simple spaces A, e.g. a 2sphere or surface of genus 2, cannot continuously model any except the most trivial of equations \Sigma. 0.1 Garr...
Single Implicative Axioms for Groups and for Abelian Groups
, 1996
"... ) Supported by the Mathematical, Information, and Computational Sciences Division subprogram of the Office of Computational and Technology Research, U.S. Department of Energy, under Contract W31109Eng38. for abelian groups [3], and ((z \Delta (x \Delta y) \Gamma1 ) \Gamma1 \Delta (z \Delta ..."
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) Supported by the Mathematical, Information, and Computational Sciences Division subprogram of the Office of Computational and Technology Research, U.S. Department of Energy, under Contract W31109Eng38. for abelian groups [3], and ((z \Delta (x \Delta y) \Gamma1 ) \Gamma1 \Delta (z \Delta y \Gamma1 )) \Delta (y \Gamma1 \Delta y) \Gamma1 = x (4) for ordinary groups [2]. One might think it trivial, given (2), to obtain a single axiom in terms of product and inverse, by simply rewriting ff=fi to ff \Delta fi \Gamma1 . Doing so gives a single axiom, but then \Delta is not product, and \Gamma1
SINGLE AXIOMS: WITH AND WITHOUT COMPUTERS
"... This note is an (incomplete) summary of results on single equational axioms for algebraic theories. Pioneering results were obtained decades ago (without the use of computers) by logicians such asTarski, Higman, Neumann, and Padmanabhan. Use of today's highspeed computers and sophisticated software ..."
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This note is an (incomplete) summary of results on single equational axioms for algebraic theories. Pioneering results were obtained decades ago (without the use of computers) by logicians such asTarski, Higman, Neumann, and Padmanabhan. Use of today's highspeed computers and sophisticated software