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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.
KripkeStyle Models for Typed Lambda Calculus
 Annals of Pure and Applied Logic
, 1996
"... The semantics of typed lambda calculus is usually described using Henkin models, consisting of functions over some collection of sets, or concrete cartesian closed categories, which are essentially equivalent. We describe a more general class of Kripkestyle models. In categorical terms, our Kripke ..."
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Cited by 44 (3 self)
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The semantics of typed lambda calculus is usually described using Henkin models, consisting of functions over some collection of sets, or concrete cartesian closed categories, which are essentially equivalent. We describe a more general class of Kripkestyle models. In categorical terms, our Kripke lambda models are cartesian closed subcategories of the presheaves over a poset. To those familiar with Kripke models of modal or intuitionistic logics, Kripke lambda models are likely to seem adequately \semantic." However, when viewed as cartesian closed categories, they do not have the property variously referred to as concreteness, wellpointedness, or having enough points. While the traditional lambda calculus proof system is not complete for Henkin models that may have empty types, we prove strong completeness for Kripke models. In fact, every set of equations that is closed under implication is the theory of a single Kripke model. We also develop some properties of logical relations ...
Type Inferencing for Polymorphic OrderSorted Logic Programs
 PROCEEDINGS OF THE TWELFTH INTERNATIONAL CONFERENCE ON LOGIC PROGRAMMING
, 1995
"... The purpose of this paper is to study the problem of complete type inferencing for polymorphic ordersorted logic programs. We show that previous approaches are incomplete even if one does not employ the full power of the used type systems. We present a complete type inferencing algorithm that cover ..."
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Cited by 23 (0 self)
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The purpose of this paper is to study the problem of complete type inferencing for polymorphic ordersorted logic programs. We show that previous approaches are incomplete even if one does not employ the full power of the used type systems. We present a complete type inferencing algorithm that covers the polymorphic ordersorted types in PROTOSL, a logic programming language that allows for polymorphism as in ML and for hierarchically structured monomorphic types.
A Complete Calculus for the Multialgebraic and Functional Semantics of Nondeterminism
, 1995
"... : The current algebraic models for nondeterminism focus on the notion of possibility rather than necessity, and con sequently equate (nondeterministic) terms that one intuitively would not consider equal. Furthermore, existing models for nondeterminism depart radically from the standard models for ( ..."
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Cited by 22 (9 self)
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: The current algebraic models for nondeterminism focus on the notion of possibility rather than necessity, and con sequently equate (nondeterministic) terms that one intuitively would not consider equal. Furthermore, existing models for nondeterminism depart radically from the standard models for (equational) specifications of deterministic operators. One would prefer that a specification language for nondeterministic operators be based on an extension of the standard model concepts, preferably in such a way that the reasoning system for (possibly nondeterministic) operators becomes the standard equational one whenever restricted to the deterministic operators  the objective should be to minimize the departure from the standard frameworks. In this paper we define a specification language for nondeterministic operators and multialgebraic semantics. The first complete reasoning system for such specifications is introduced. We also define a transformation of specifications of nondeterm...
From Total Equational to Partial First Order Logic
, 1998
"... The focus of this chapter is the incremental presentation of partial firstorder logic, seen as a powerful framework where the specification of most data types can be directly represented in the most natural way. Both model theory and logical deduction are described in full detail. Alternatives to pa ..."
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Cited by 19 (8 self)
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The focus of this chapter is the incremental presentation of partial firstorder logic, seen as a powerful framework where the specification of most data types can be directly represented in the most natural way. Both model theory and logical deduction are described in full detail. Alternatives to partiality, like (variants of) error algebras and ordersortedness are also discussed, showing their uses and limitations. Moreover, both the total and the partial (positive) conditional fragment are investigated in detail, and in particular the existence of initial (free) models for such restricted logical paradigms is proved. Some more powerful algebraic frameworks are sketched at the end. Equational specifications introduced in last chapter, are a powerful tool to represent the most common data types used in programming languages and their semantics. Indeed, Bergstra and Tucker have shown in a series of papers (see [BT87] for a complete exposition of results) that a data type is semicompu...
Parameterized Recursion Theory  A Tool for the Systematic Classification of Specification Methods
 Proceedings of the Third International Conference on Algebraic Methodology and Software Technology, 1993, Workshops in Computing
"... We examine four specification methods with increasing expressiveness. Parameterized recursion theory allows to characterize the power of parameterization in the methods, using a computational model based on Moschovakis' search computability. The four specification methods can be characterized by fou ..."
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Cited by 3 (2 self)
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We examine four specification methods with increasing expressiveness. Parameterized recursion theory allows to characterize the power of parameterization in the methods, using a computational model based on Moschovakis' search computability. The four specification methods can be characterized by four different notions of semicomputable parameterized abstract data type, which differ in the availability of the parameter algebra and of nondeterminism. These characterizations further lead to different algebraic properties of specifiable PADTs. Together with example PADTs, they enable us to prove a hierarchy theorem. Given a sample PADT, the algebraic properties help to find out the lowest position (= most restricted method) in the hierarchy usable to specify it. This is important because the available tools may become weaker, if we choose a too general method.
Translating OBJ3 into CASL: the Institution Level
 In Recent Trends in Algebraic Development Techniques, Proc. 13th International Workshop, WADT '98
, 1998
"... We translate OBJ3 to CASL. At the level of basic specifications, we set up several institution representations between the underlying institutions. They correspond to different methodological views of OBJ3. The translations can be the basis for automated tools translating OBJ3 to CASL. ..."
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Cited by 3 (0 self)
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We translate OBJ3 to CASL. At the level of basic specifications, we set up several institution representations between the underlying institutions. They correspond to different methodological views of OBJ3. The translations can be the basis for automated tools translating OBJ3 to CASL.
1 The Typed Situation Calculus
"... ABSTRACT. We propose a theory for reasoning about actions based on ordersorted predicate logic where one can consider an elaborate taxonomy of objects. We are interested in the projection problem: whether a statement is true after executing a sequence of actions. To solve it we design a regression ..."
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ABSTRACT. We propose a theory for reasoning about actions based on ordersorted predicate logic where one can consider an elaborate taxonomy of objects. We are interested in the projection problem: whether a statement is true after executing a sequence of actions. To solve it we design a regression operator takes advantage of wellsorted unification between terms. We show that answering projection queries in our logical theories is sound and complete wrt answering similar queries in Reiterâ€™s basic action theories. This proves correctness of our approach. Moreover, we demonstrate that our regression operator based on ordersorted logic can provide significant computational advantages in comparison to Reiterâ€™s regression operator. 1
research. Object Models
"... by Robert HaringSmithAuthorization to lend and reproduce this thesis As the sole author of this thesis, I authorize Brown university to lend it to other institutions or individuals for the purpose of scholarly researclvj, ..."
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by Robert HaringSmithAuthorization to lend and reproduce this thesis As the sole author of this thesis, I authorize Brown university to lend it to other institutions or individuals for the purpose of scholarly researclvj,