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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.
Automated Deduction by Theory Resolution
 Journal of Automated Reasoning
, 1985
"... Theory resolution constitutes a set of complete procedures for incorporating theories into a resolution theoremproving program, thereby making it unnecessary to resolve directly upon axioms of the theory. This can greatly reduce the length of proofs and the size of the search space. Theory resoluti ..."
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Cited by 121 (1 self)
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Theory resolution constitutes a set of complete procedures for incorporating theories into a resolution theoremproving program, thereby making it unnecessary to resolve directly upon axioms of the theory. This can greatly reduce the length of proofs and the size of the search space. Theory resolution effects a beneficial division of labor, improving the performance of the theorem prover and increasing the applicability of the specialized reasoning procedures. Total theory resolution utilizes a decision procedure that is capable of determining unsatisfiability of any set of clauses using predicates in the theory. Partial theory resolution employs a weaker decision procedure that can determine potential unsatisfiability of sets of literals. Applications include the building in of both mathematical and special decision procedures, e.g., for the taxonomic information furnished by a knowledge representation system. Theory resolution is a generalization of numerous previously known resolution refinements. Its power is demonstrated by comparing solutions of "Schubert's Steamroller" challenge problem with and without building in axioms through theory resolution. 1 1
Logic Programming over Polymorphically OrderSorted Types
, 1989
"... This thesis presents the foundations for relational logic programming over polymorphically ordersorted data types. This type discipline combines the notion of parametric polymorphism, which has been developed for higherorder functional programming, with the notion of ordersorted typing, which ha ..."
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Cited by 58 (0 self)
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This thesis presents the foundations for relational logic programming over polymorphically ordersorted data types. This type discipline combines the notion of parametric polymorphism, which has been developed for higherorder functional programming, with the notion of ordersorted typing, which has been developed for equational firstorder specification and programming. Polymorphically ordersorted types are obtained as canonical models of a class of specifications in a suitable logic accommodating sort functions. Algorithms for constraint solving, type checking and type inference are given and proven correct.
The Substitutional Framework for Sorted Deduction: Fundamental Results on Hybrid Reasoning
 Artificial Intelligence
, 1990
"... Researchers in artificial intelligence have recently been taking great interest in hybrid representations, among them sorted logicslogics that link a traditional logical representation to a taxonomic (or sort) representation such as those prevalent in semantic networks. This paper introduces a ge ..."
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Cited by 50 (9 self)
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Researchers in artificial intelligence have recently been taking great interest in hybrid representations, among them sorted logicslogics that link a traditional logical representation to a taxonomic (or sort) representation such as those prevalent in semantic networks. This paper introduces a general frameworkthe substitutional frameworkfor integrating logical deduction and sortal deduction to form a deductive system for sorted logic. This paper also presents results that provide the theoretical underpinnings of the framework. A distinguishing characteristic of a deductive system that is structured according to the substitutional framework is that the sort subsystem is invoked only when the logic subsystem performs unification, and thus sort information is used only in determining what substitutions to make for variables. Unlike every other known approach to sorted deduction, the substitutional framework provides for a systematic transformation of unsorted deductive systems ...
A Survey on Temporal Reasoning in Artificial Intelligence
, 1994
"... The notion of time is ubiquitous in any activity that requires intelligence. In particular, several important notions like change, causality, action are described in terms of time. Therefore, the representation of time and reasoning about time is of crucial importance for many Artificial Intelligenc ..."
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Cited by 42 (4 self)
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The notion of time is ubiquitous in any activity that requires intelligence. In particular, several important notions like change, causality, action are described in terms of time. Therefore, the representation of time and reasoning about time is of crucial importance for many Artificial Intelligence systems. Specifically during the last 10 years, it has been attracting the attention of many AI researchers. In this survey, the results of this work are analysed. Firstly, Temporal Reasoning is defined. Then, the most important representational issues which determine a Temporal Reasoning approach are introduced: the logical form on which the approach is based, the ontology (the units taken as primitives, the temporal relations, the algorithms that have been developed,. . . ) and the concepts related with reasoning about action (the representation of change, causality, action,. . . ). For each issue the different choices in the literature are discussed. 1 Introduction The notion of time i...
SemanticsBased Translation Methods for Modal Logics
, 1991
"... A general framework for translating logical formulae from one logic into another logic is presented. The framework is instantiated with two different approaches to translating modal logic formulae into predicate logic. The first one, the well known ‘relational’ translation makes the modal logic’s po ..."
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Cited by 40 (1 self)
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A general framework for translating logical formulae from one logic into another logic is presented. The framework is instantiated with two different approaches to translating modal logic formulae into predicate logic. The first one, the well known ‘relational’ translation makes the modal logic’s possible worlds structure explicit by introducing a distinguished predicate symbol to represent the accessibility relation. In the second approach, the ‘functional ’ translation method, paths in the possible worlds structure are represented by compositions of functions which map worlds to accessible worlds. On the syntactic level this means that every flexible symbol is parametrized with particular terms denoting whole paths from the initial world to the actual world. The ‘target logic’ for the translation is a firstorder manysorted logic with built in equality. Therefore the ‘source logic’ may also be firstorder manysorted with built in equality. Furthermore flexible function symbols are allowed. The modal operators may be parametrized with arbitrary terms and particular properties of the accessibility relation may be specified within the
A NonReified Temporal Logic
, 1989
"... A temporal logic is presented for reasoning about propositions whose truth values might change as a function of time. The temporal propositions consist of formulae in a sorted firstorder logic, with each atomic predicate taking some set of temporal arguments as well as a set of nontemporal argument ..."
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Cited by 36 (1 self)
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A temporal logic is presented for reasoning about propositions whose truth values might change as a function of time. The temporal propositions consist of formulae in a sorted firstorder logic, with each atomic predicate taking some set of temporal arguments as well as a set of nontemporal arguments. The temporal arguments serve to specify the predicate's dependence on time. By partitioning the terms of the language into two sorts, temporal and nontemporal, time is given a special syntactic and semantic status without having to resort to reification. The benefits of this logic are that it has a clear semantics and a well studied prooftheory. Unlike the firstorder logic presented by Shoham, propositions can be expressed and interpreted with respect to any number of temporal arguments, not just with respect to a pair of time points (an interval). We demonstrate the advantages of this flexibility. In addition, nothing is lost by this added flexibility and more standard and useable syn...
(ML)²: A formal language for KADS models of expertise
, 1993
"... This paper reports on an investigation into a formal language for specifying kads models of expertise. After arguing the need for and the use of such formal representations, we discuss each of the layers of a kads model of expertise in the subsequent sections, and define the formal constructions tha ..."
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Cited by 35 (9 self)
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This paper reports on an investigation into a formal language for specifying kads models of expertise. After arguing the need for and the use of such formal representations, we discuss each of the layers of a kads model of expertise in the subsequent sections, and define the formal constructions that we use to represent the kads entities at every layer: ordersorted logic at the domain layer, metalogic at the inference layer, and dynamiclogic at the task layer. All these constructions together make up (ml) 2 , the language that we use to represent models of expertise. We illustrate the use of (ml) 2 in a small example model. We conclude by describing our experience to date with constructing such formal models in (ml) 2 , and by discussing some open problems that remain for future work. 1 Introduction One of the central concerns of "knowledge engineering" is the construction of a model of some problem solving behaviour. This model should eventually lead to the construction of a...
Colouring Terms to Control Equational Reasoning
 Journal of Automated Reasoning
, 1997
"... . In this paper we present an approach to prove the equality between terms in a goaldirected way developed in the field of inductive theorem proving. The two terms to be equated are syntactically split into expressions which are common to both and those which occur only in one term. According to the ..."
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Cited by 25 (13 self)
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. In this paper we present an approach to prove the equality between terms in a goaldirected way developed in the field of inductive theorem proving. The two terms to be equated are syntactically split into expressions which are common to both and those which occur only in one term. According to the computed differences we apply appropriate equations to the terms in order to reduce the differences in a goaldirected way. Although this approach was developed for purposes of inductive theorem proving  we use this technique to manipulate the conclusion of an induction step to enable the use of the hypothesis  it is a powerful method for the control of equational reasoning in general. 1. Introduction The automation of equational reasoning is one of the most important obstacles in the field of automating deductions. Even small equational problems result in a huge search space, and finding a proof often fails due to the combinatorial explosion. Proving (conditional) equations by inductio...