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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 Schem ..."
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Cited by 821 (23 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.
Introducing OBJ
, 1993
"... This is an introduction to the philosophy and use of OBJ, emphasizing its operational semantics, with aspects of its history and its logical semantics. Release 2 of OBJ3 is described in detail, with many examples. OBJ is a wide spectrum firstorder functional language that is rigorously based on ..."
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Cited by 136 (30 self)
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This is an introduction to the philosophy and use of OBJ, emphasizing its operational semantics, with aspects of its history and its logical semantics. Release 2 of OBJ3 is described in detail, with many examples. OBJ is a wide spectrum firstorder functional language that is rigorously based on (order sorted) equational logic and parameterized programming, supporting a declarative style that facilitates verification and allows OBJ to be used as a theorem prover.
An Overview of Rewrite Rule Laboratory (RRL)
 J. of Computer and Mathematics with Applications
, 1995
"... RRL (Rewrite Rule Laboratory) was originally developed as an environment for experimenting with automated reasoning algorithms for equational logic based on rewrite techniques. It has now matured into a fullfledged theorem prover which has been used to solve hard and challenging mathematical proble ..."
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Cited by 70 (25 self)
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RRL (Rewrite Rule Laboratory) was originally developed as an environment for experimenting with automated reasoning algorithms for equational logic based on rewrite techniques. It has now matured into a fullfledged theorem prover which has been used to solve hard and challenging mathematical problems in automated reasoning literature as well as a research tool for investigating the use of formal methods in hardware and software design. We provide a brief historical account of development of RRL and its descendants, give an overview of the main capabilities of RRL and conclude with a discussion of applications of RRL. Key words. RRL, rewrite techniques, equational logic, discrimination nets 1 Introduction The theorem prover RRL (Rewrite Rule Laboratory) is an automated reasoning program based on rewrite techniques. The theorem prover has implementations of completion procedures for generating a complete set of rewrite rules from an equational axiomatization, associativecommutative mat...
Equational Inference, Canonical Proofs, And Proof Orderings
 Journal of the ACM
, 1992
"... We describe the application of proof orderingsa technique for reasoning about inference systemsto various rewritebased theoremproving methods, including re#nements of the standard KnuthBendix completion procedure based on critical pair criteria; Huet's procedure for rewriting modulo a ..."
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Cited by 30 (10 self)
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We describe the application of proof orderingsa technique for reasoning about inference systemsto various rewritebased theoremproving methods, including re#nements of the standard KnuthBendix completion procedure based on critical pair criteria; Huet's procedure for rewriting modulo a congruence; ordered completion #a refutationally complete extension of standard completion#; and a proof by consistency procedure for proving inductive theorems. # This is a substantially revised version of the paper, #Orderings for equational proofs," coauthored with J. Hsiang and presented at the Symp. on Logic in Computer Science #Boston, Massachusetts, June 1986#. It includes material from the paper #Proof by consistency in equational theories," by the #rst author, presented at the ThirdAnnual Symp. on Logic in Computer Science #Edinburgh, Scotland, July 1988#. This researchwas supported in part by the National Science Foundation under grants CCR8901322, CCR9007195, and CCR9024271. 1 ...
EQUIVALENCES AND TRANSFORMATIONS OF REGULAR SYSTEMS  APPLICATIONS TO RECURSIVE PROGRAM SCHEMES AND GRAMMARS
, 1986
"... This work presents a unified theory of recursive program schemes, contextfree grammars, grammars on arbitrary algebraic structures and, in fact, recursive definitions of all kind by means of regular systems. The equivalences of regular systems associated with either all their solutions or their le ..."
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Cited by 29 (5 self)
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This work presents a unified theory of recursive program schemes, contextfree grammars, grammars on arbitrary algebraic structures and, in fact, recursive definitions of all kind by means of regular systems. The equivalences of regular systems associated with either all their solutions or their least solutions (in all domains of appropriate type satisfying a set of algebraic laws expressed by equations) are systematically investigated and characterized (in some cases) in terms of system transformations by folding, unfolding and rewriting according to the equational algebraic laws. Grammars are better characterized in terms of polynomial systems which are regular systems involving the operation of set union, and the same questions are raised for them. We also examine conditions insuring the uniqueness of the solution of a regular or of a polynomial system. This theory applies to grammars of many kinds which generate trees, graphs, etc. We formulate some classical transformations of contextfree grammars in terms of correct transformations which only use folding, unfolding and algebraic laws and we immediately obtain their correctness.
Automating Inductionless Induction using Test Sets
 Journal of Symbolic Computation
, 1991
"... The inductionless induction (also called proof by consistency) approach for proving equations by induction from an equational theory, requires a consistency check for equational theories. A new method using test sets for checking consistency of an equational theory is proposed. Using this method, ..."
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Cited by 20 (3 self)
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The inductionless induction (also called proof by consistency) approach for proving equations by induction from an equational theory, requires a consistency check for equational theories. A new method using test sets for checking consistency of an equational theory is proposed. Using this method, a variation of the KnuthBendix completion procedure can be used for automatically proving equations by induction. The method does not suffer from limitations imposed by the methods proposed by Musser as well as by Huet and Hullot, and is as powerful as Jouannaud and Kounalis' method based on groundreducibility. A theoretical comparison of the test set method with Jouannaud and Kounalis' method is given showing that the test set method is generally much better. Both the methods have been implemented in RRL, Rewrite Rule Laboratory, a theorem proving environment based on rewriting techniques and completion. In practice also, the test set method is faster than Jouannaud and Kounalis' ...
Software Specification: A Comparison of Formal Methods
, 2001
"... Data Types and Software Validation ," Communications of the ACM, Vol. 21, No. 12, 1978, pp. 10481064. ..."
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Cited by 15 (0 self)
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Data Types and Software Validation ," Communications of the ACM, Vol. 21, No. 12, 1978, pp. 10481064.
Tossing Algebraic Flowers down the Great Divide
 In People and Ideas in Theoretical Computer Science
, 1999
"... Data Types and Algebraic Semantics The history of programming languages, and to a large extent of software engineering as a whole, can be seen as a succession of ever more powerful abstraction mechanisms. The first stored program computers were programmed in binary, which soon gave way to assembly l ..."
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Data Types and Algebraic Semantics The history of programming languages, and to a large extent of software engineering as a whole, can be seen as a succession of ever more powerful abstraction mechanisms. The first stored program computers were programmed in binary, which soon gave way to assembly languages that allowed symbolic codes for operations and addresses. fortran began the spread of "high level" programming languages, though at the time it was strongly opposed by many assembly programmers; important features that developed later include blocks, recursive procedures, flexible types, classes, inheritance, modules, and genericity. Without going into the philosophical problems raised by abstraction (which in view of the discussion of realism in Section 4 may be considerable), it seems clear that the mathematics used to describe programming concepts should in general get more abstract as the programming concepts get more abstract. Nevertheless, there has been great resistance to u...