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19
Algebraic Approaches to Nondeterminism  an Overview
 ACM Computing Surveys
, 1997
"... this paper was published as Walicki, M.A. and Meldal, S., 1995, Nondeterministic Operators in Algebraic Frameworks, Tehnical Report No. CSLTR95664, Stanford University ..."
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Cited by 23 (3 self)
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this paper was published as Walicki, M.A. and Meldal, S., 1995, Nondeterministic Operators in Algebraic Frameworks, Tehnical Report No. CSLTR95664, Stanford University
A Functorial Semantics for MultiAlgebras and Partial Algebras, With Applications to Syntax
, 2000
"... Multialgebras allow for the modeling of nondeterminism in an algebraic framework by interpreting operators as functions from individual arguments to sets of possible results. We propose a functorial presentation of various categories of multialgebras and partial algebras, analogous to the classica ..."
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Cited by 14 (7 self)
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Multialgebras allow for the modeling of nondeterminism in an algebraic framework by interpreting operators as functions from individual arguments to sets of possible results. We propose a functorial presentation of various categories of multialgebras and partial algebras, analogous to the classical presentation of algebras over a signature as cartesian functors from the algebraic theory over to Set. We introduce two dierent notions of theory over a signature, both having a structure weaker than cartesian, and we consider functors from them to Rel or Pfn, the categories of sets and relations or partial functions, respectively. Next we discuss how the functorial presentation provides guidelines for the choice of syntactical notions for a class of algebras, and as an application we argue that the natural generalization of usual terms are \conditioned terms" for partial algebras, and \term graphs" for multialgebras. Contents 1 Introduction 2 2 A short recap on multialgebras 4 3...
Structured Specifications and Implementation of Nondeterministic Data Types
, 1995
"... The use of nondeterminism in specifications, as distinct from underspecification, is motivated by an example in the context of data refinement. A simple formalism for specifying nondeterministic data types is introduced. Its semantics is given in terms of the existing formalisms of relations, multia ..."
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Cited by 8 (5 self)
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The use of nondeterminism in specifications, as distinct from underspecification, is motivated by an example in the context of data refinement. A simple formalism for specifying nondeterministic data types is introduced. Its semantics is given in terms of the existing formalisms of relations, multialgebras, sets of functions and oracles by means of appropriate translation rules. Nondeterministic data refinement is studied from the syntactic and semantic perspective, and the correctness of the suggested proof obligations is proved. More general, the implementation relation and parameterisation of nondeterministic data types are discussed and the standard theorems of vertical and horizontal composition are generalized to the nondeterministic case.
Reasoning with Nondeterministic Specifications
 Polish Academy of Sciences, Institute of CS
, 1995
"... this paper we concentrate on a specification language that can be of use in both cases. We present a sound and complete Gentzenstyle deduction system for a logic that can be best (though rather informally) described as first order logic with inclusion and a let construct binding variables in nondet ..."
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Cited by 7 (2 self)
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this paper we concentrate on a specification language that can be of use in both cases. We present a sound and complete Gentzenstyle deduction system for a logic that can be best (though rather informally) described as first order logic with inclusion and a let construct binding variables in nondeterministic terms. We should also remark here that this paper presents a nontrivial firstorder extension of a calculus presented in [BK 95a]. It contains a detailed analysis of the phenomena related to empty carriers in multisorted environment. Also, we do not place any restrictions on the models, and allow for both empty carriers and partial functions. One may observe at this point that instead of multialgebras we might have used relational algebras. However, we prefer functions for the same reason that functions are present in first order logic  they are more intuitive in applications and have properties that make their use easier. Moreover, as it will be explained in the last section, we need "ordinary" algebras to be a special case of the formalism we develop.
The Institution of Multialgebras  a general framework for algebraic software development
, 2002
"... this technicality ..."
Term Graph Syntax for MultiAlgebras
, 2000
"... Multialgebras allow to model nondeterminism in an algebraic framework by interpreting operators as functions from individual arguments to sets of possible results. Starting from a functorial presentation of multialgebras based on gsmonoidal theories, we argue that speci cations for multialgebras ..."
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Cited by 5 (4 self)
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Multialgebras allow to model nondeterminism in an algebraic framework by interpreting operators as functions from individual arguments to sets of possible results. Starting from a functorial presentation of multialgebras based on gsmonoidal theories, we argue that speci cations for multialgebras should be based on the notion of term graphs instead of on standard terms. We consider the simplest case of (term graph) equational specification, showing that it enjoys an unrestricted form of substitutivity. We discuss the expressive power of equational specification for multialgebras, and we sketch possible extensions of the calculus.
Reasoning and Rewriting with SetRelations I: Ground Completeness
, 1994
"... . The paper investigates reasoning with setrelations: intersection, inclusion and identity of 1element sets. A language is introduced which, interpreted in a multialgebraic semantics, allows one to specify such relations. An inference system is given and shown sound and refutationally groundcompl ..."
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Cited by 4 (2 self)
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. The paper investigates reasoning with setrelations: intersection, inclusion and identity of 1element sets. A language is introduced which, interpreted in a multialgebraic semantics, allows one to specify such relations. An inference system is given and shown sound and refutationally groundcomplete for a particular proof strategy which selects only maximal literals from the premise clauses. Each of the introduced setrelations satisfies only two among the three properties of the equivalence relations  we study rewriting with such nonequivalence relations and point out differences from the equational case. As a corollary of the main groundcompleteness theorem we obtain groundcompleteness of the introduced rewriting technique. 1 Introduction Reasoning with sets becomes an important issue in different areas of computer science. Its relevance can be noticed in constraint and logic programming e.g. [SD86, DO92, Jay92, Sto93], in algebraic approach to nondeterminism e.g. [Hus93, He...
On Specialization of Derivations in Axiomatic Equality Theories
 in Proc. of LFCS'94, LNCS
, 1994
"... Walicki and Meldal have defined a calculus DEQ ("Disjunctive EQuational calculus") for reasoning about nondeterministic operators when specifying nondeterministic systems in an equationoriented style. A variant of DEQ, the calculus DEQ for axiomatic equality theories with cutlike rules introduc ..."
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Cited by 3 (1 self)
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Walicki and Meldal have defined a calculus DEQ ("Disjunctive EQuational calculus") for reasoning about nondeterministic operators when specifying nondeterministic systems in an equationoriented style. A variant of DEQ, the calculus DEQ for axiomatic equality theories with cutlike rules introducing as cut formulas only the negative equalities of a specific axiom, is constructed. For pure positive specific axioms (i.e. with empty antecedents) and for so called noncontrary equality theories DEQ does not contain cutlike rules at all. The variant of the calculus DEQ without structural rules of contraction and exchange is constructed. A simple cutelimination procedure for axiomatic equality theories is presented. 1. Introduction The motivation for introducing DEQ was the need for the disjunctive formulae when specifying and reasoning about nondeterministic operations [9]. The notion of nondeterminism arises naturally in describing concurrent systems. Nondeterminism is also a n...
RasiowaSikorski Deduction Systems: a Handy Tool for Computer Science Logics
 Recent Trends in Algebraic Specification Techniques, volume 1589 of LNCS
, 1998
"... . A RasiowaSikorski system is a sequencetype formalization of logics based on building decomposition trees of formulae labelled with sequences of formulae. Proofs are nite decomposition trees with leaves having \fundamental", valid labels. The system is dual to the tableau system. The author gives ..."
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Cited by 3 (1 self)
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. A RasiowaSikorski system is a sequencetype formalization of logics based on building decomposition trees of formulae labelled with sequences of formulae. Proofs are nite decomposition trees with leaves having \fundamental", valid labels. The system is dual to the tableau system. The author gives examples of applying the RS formalism to various C.S and A.I. logic, including a logic for reasoning about relative similarity, a threevalued software specication logic with McCarthy's connectives, and a logic for nondeterministic specications. As a new result, an RS system for manysorted rst order logic with possibly empty carriers of some sorts is developed. 1 Introduction An issue in computer science logics that has gained much popularity lately are the socalled labelled deductive systems [5]. The predecessors of this type of deductive systems were Beth's tableau systems [1] and RasiowaSikorski (RS) deduction systems [12], both developed over thirty years ago. Their important...
Quantifierfree logic for multialgebraic theories
, 2002
"... We develop a new logic for deriving consequences of multialgebraic theories (specifications). Multilagebras are used as models for nondeterminism in the context of algebraic specifications. They are many sorted algebras with set valued operations. Atomic formulae are set inclusion t ≺ t ′ –the inter ..."
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Cited by 2 (1 self)
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We develop a new logic for deriving consequences of multialgebraic theories (specifications). Multilagebras are used as models for nondeterminism in the context of algebraic specifications. They are many sorted algebras with set valued operations. Atomic formulae are set inclusion t ≺ t ′ –the interpretation of t is included in the interpretation of t ′ , and element equality t. = t ′ – t and t ′ denote the same element of the carrier. We introduce the RasiowaSikorski logic RS for proving multilagebraic tautologies and show its soundness and completeness. We then extend this system for proving consequences of specifications based on translation of theories into logical formulae. Finally, we show how such a translation may be avoided –introduction of specific cut rules leads to a sound and complete Gentzen system for proving directly consequences of specifications.