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OrderSorted Algebra I: Equational Deduction for Multiple Inheritance, Overloading, Exceptions and Partial Operations
 Theoretical Computer Science
, 1992
"... This paper generalizes manysorted algebra (hereafter, MSA) to ordersorted algebra (hereafter, OSA) by allowing a partial ordering relation on the set of sorts. This supports abstract data types with multiple inheritance (in roughly the sense of objectoriented programming), several forms of pol ..."
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Cited by 208 (33 self)
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This paper generalizes manysorted algebra (hereafter, MSA) to ordersorted algebra (hereafter, OSA) by allowing a partial ordering relation on the set of sorts. This supports abstract data types with multiple inheritance (in roughly the sense of objectoriented programming), several forms of polymorphism and overloading, partial operations (as total on equationally defined subsorts), exception handling, and an operational semantics based on term rewriting. We give the basic algebraic constructions for OSA, including quotient, image, product and term algebra, and we prove their basic properties, including Quotient, Homomorphism, and Initiality Theorems. The paper's major mathematical results include a notion of OSA deduction, a Completeness Theorem for it, and an OSA Birkhoff Variety Theorem. We also develop conditional OSA, including Initiality, Completeness, and McKinseyMalcev Quasivariety Theorems, and we reduce OSA to (conditional) MSA, which allows lifting many known MSA results to OSA. Retracts, which intuitively are left inverses to subsort inclusions, provide relatively inexpensive runtime error handling. We show that it is safe to add retracts to any OSA signature, in the sense that it gives rise to a conservative extension. A final section compares and contrasts many different approaches to OSA. This paper also includes several examples demonstrating the flexibility and applicability of OSA, including some standard benchmarks like STACK and LIST, as well as a much more substantial example, the number hierarchy from the naturals up to the quaternions.
Completeness of manysorted equational logic
 Houston Journal of Mathematics
, 1985
"... ABSTRACT. Assuming that manysorted oquationallogic "goes just as for the onesorted case " has led to incorrect statements of results in manysorted universal algebra; in fact, the onesorted rules are not sound for manysortededuction. This paper gives sound and complete rules, and characterizes w ..."
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Cited by 62 (6 self)
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ABSTRACT. Assuming that manysorted oquationallogic "goes just as for the onesorted case " has led to incorrect statements of results in manysorted universal algebra; in fact, the onesorted rules are not sound for manysortededuction. This paper gives sound and complete rules, and characterizes when the onesorted rules can still be used safely; it also characterizes the related question of when manysorted algebras can be represented as onesorted algebras. The paper contains a detailed introduction to Hall's theory of clones (later developed into "algebraic theories " by Lawvere and Benabou); this allows a full algebraization of manysorted equational deduction that is not possible with the usual fully invariant congruences on the free algebra on countably many generators. 1. Introduction. The
Categories and groupoids
, 1971
"... In 1968, when this book was written, categories had been around for 20 years and groupoids for twice as long. Category theory had by then become widely accepted as an essential tool in many parts of mathematics and a number of books on the subject had appeared, or were about to appear (e.g. [13, 22, ..."
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Cited by 41 (2 self)
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In 1968, when this book was written, categories had been around for 20 years and groupoids for twice as long. Category theory had by then become widely accepted as an essential tool in many parts of mathematics and a number of books on the subject had appeared, or were about to appear (e.g. [13, 22, 37, 58, 65] 1). By contrast, the use of groupoids was confined to a small number of pioneering articles, notably by Ehresmann [12] and Mackey [57], which were largely ignored by the mathematical community. Indeed groupoids were generally considered at that time not to be a subject for serious study. It was argued by several wellknown mathematicians that group theory sufficed for all situations where groupoids might be used, since a connected groupoid could be reduced to a group and a set. Curiously, this argument, which makes no appeal to elegance, was not applied to vector spaces: it was well known that the analogous reduction in this case is not canonical, and so is not available, when there is extra structure, even such simple structure as an endomorphism. Recently, Corfield in [41] has discussed methodological issues in mathematics with this topic, the resistance to the notion of groupoids, as a prime example. My book was intended chiefly as an attempt to reverse this general assessment of the time by presenting applications of groupoids to group theory
Categorybased Semantics for Equational and Constraint Logic Programming
, 1994
"... This thesis proposes a general framework for equational logic programming, called categorybased equational logic by placing the general principles underlying the design of the programming language Eqlog and formulated by Goguen and Meseguer into an abstract form. This framework generalises equation ..."
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Cited by 24 (10 self)
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This thesis proposes a general framework for equational logic programming, called categorybased equational logic by placing the general principles underlying the design of the programming language Eqlog and formulated by Goguen and Meseguer into an abstract form. This framework generalises equational deduction to an arbitrary category satisfying certain natural conditions; completeness is proved under a hypothesis of quantifier projectivity, using a semantic treatment that regards quantifiers as models rather than variables, and regards valuations as model morphisms rather than functions. This is used as a basis for a model theoretic categorybased approach to a paramodulationbased operational semantics for equational logic programming languages. Categorybased equational logic in conjunction with the theory of institutions is used to give mathematical foundations for modularisation in equational logic programming. We study the soundness and completeness problem for module imports i...
Pseudo algebras and pseudo double categories
 J. Homotopy Relat. Struct
"... Abstract. As an example of the categorical apparatus of pseudo algebras over 2theories, we show that pseudo algebras over the 2theory of categories can be viewed as pseudo double categories with folding or as appropriate 2functors into bicategories. Foldings are equivalent to connection pairs, an ..."
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Cited by 16 (2 self)
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Abstract. As an example of the categorical apparatus of pseudo algebras over 2theories, we show that pseudo algebras over the 2theory of categories can be viewed as pseudo double categories with folding or as appropriate 2functors into bicategories. Foldings are equivalent to connection pairs, and also to thin structures if the vertical and horizontal morphisms coincide. In a sense, the squares of a double category with folding are determined in a functorial way by the 2cells of the horizontal 2category. As a special case, strict 2algebras with one object and everything invertible are crossed modules under a group.
Algebraic Processing of Programming Languages
 Theoretical Computer Science
, 1995
"... Current methodology for compiler construction evolved from the need to release programmers form the burden of writing machinelanguage programs. This methodology does not assume a formal concept of a programming language and is not based on mathematical algorithms that model the behavior of a compil ..."
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Cited by 13 (10 self)
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Current methodology for compiler construction evolved from the need to release programmers form the burden of writing machinelanguage programs. This methodology does not assume a formal concept of a programming language and is not based on mathematical algorithms that model the behavior of a compiler. The side effect is that compiler implementation is a difficult task and the correctness of a compiler usually is not proven mathematically. Moreover, a compiler may be based on assumptions about its source and target languages that are not necessarily acceptable for another compiler that has the same source and target languages. The consequence is that programs are not portable between platforms of machines and between generations of languages. In addition, while a conventional compiler freezes the notation that programmers can use to develop their programs the problem domain evolves and requires extensions that are not supported by the compiler. These problems are addressed by two direc...
Algebraic logic, varieties of algebras, and algebraic varieties
, 1995
"... Abstract. The aim of the paper is discussion of connections between the three kinds of objects named in the title. In a sense, it is a survey of such connections; however, some new directions are also considered. This relates, especially, to sections 3, 4 and 5, where we consider a field that could ..."
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Cited by 13 (5 self)
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Abstract. The aim of the paper is discussion of connections between the three kinds of objects named in the title. In a sense, it is a survey of such connections; however, some new directions are also considered. This relates, especially, to sections 3, 4 and 5, where we consider a field that could be understood as an universal algebraic geometry. This geometry is parallel to universal algebra. In the monograph [51] algebraic logic was used for building up a model of a database. Later on, the structures arising there turned out to be useful for solving several problems from algebra. This is the position which the present paper is written from.
Deductive algorithmic knowledge
 In Proc. 8th International Symposium on Artificial Intelligence and Mathematics. AI&M
, 2004
"... The framework of algorithmic knowledge assumes that agents use algorithms to compute the facts they explicitly know. In many cases of interest, a logical theory, rather than a particular algorithm, can be used to capture the formal reasoning used by the agents to compute what they explicitly know. W ..."
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Cited by 12 (0 self)
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The framework of algorithmic knowledge assumes that agents use algorithms to compute the facts they explicitly know. In many cases of interest, a logical theory, rather than a particular algorithm, can be used to capture the formal reasoning used by the agents to compute what they explicitly know. We introduce a logic for reasoning about both implicit and explicit knowledge, where the latter is given with respect to a deductive system formalizing a logical theory for agents. The highly structured nature of such logical theories leads to very natural axiomatizations of the resulting logic when interpreted over a fixed deductive system. The decision problem for the logic is NPcomplete in general, no harder than propositional logic, and moreover, it remains NPcomplete when we fix a tractable deductive system. The logic extends in a straightforward way to multiple agents, where the decision problem becomes PSPACEcomplete. 1
Polymorphic Syntax Definition
 THEOR. COMPUT. SCI
, 1997
"... Contextfree grammars are used in several algebraic specification formalisms instead of firstorder signatures for the definition of the structure of algebras, because grammars provide better notation than signatures. The rigidity of these firstorder structures enforces a choice between strongly ty ..."
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Cited by 3 (0 self)
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Contextfree grammars are used in several algebraic specification formalisms instead of firstorder signatures for the definition of the structure of algebras, because grammars provide better notation than signatures. The rigidity of these firstorder structures enforces a choice between strongly typed structures with little genericity or generic operations over untyped structures. In twolevel signatures level 1 defines the algebra of types used at level 0 providing the possibility to define polymorphic abstract data types. Twolevel grammars are the grammatical counterpart of twolevel signatures. This paper discusses the correspondence between contextfree grammars and firstorder signatures, the extension of this correspondence to twolevel grammars and signatures, examples of the usage of twolevel grammars for polymorphic syntax definition, a restriction of the class of twolevel grammars for which the parsing problem is decidable, a parsing algorithm that yields a minimal and ...
Journal of Homotopy and Related Structures, vol. 2(2), 2007, pp.119–170 PSEUDO ALGEBRAS AND PSEUDO DOUBLE CATEGORIES
"... As an example of the categorical apparatus of pseudo algebras over 2theories, we show that pseudo algebras over the 2theory of categories can be viewed as pseudo double categories with folding or as appropriate 2functors into bicategories. Foldings are equivalent to connection pairs, and also to ..."
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As an example of the categorical apparatus of pseudo algebras over 2theories, we show that pseudo algebras over the 2theory of categories can be viewed as pseudo double categories with folding or as appropriate 2functors into bicategories. Foldings are equivalent to connection pairs, and also to thin structures if the vertical and horizontal morphisms coincide. In a sense, the squares of a double category with folding are determined in a functorial way by the 2cells of the horizontal 2category. As a special case, strict 2algebras with one object and everything invertible are crossed modules under a group. 1.