Results 1  10
of
16
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...
Algebraiccoalgebraic specification in CoCasl
 J. LOGIC ALGEBRAIC PROGRAMMING
, 2006
"... We introduce CoCasl as a simple coalgebraic extension of the algebraic specification language Casl. CoCasl allows the nested combination of algebraic datatypes and coalgebraic process types. We show that the wellknown coalgebraic modal logic can be expressed in CoCasl. We present sufficient criter ..."
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Cited by 19 (8 self)
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We introduce CoCasl as a simple coalgebraic extension of the algebraic specification language Casl. CoCasl allows the nested combination of algebraic datatypes and coalgebraic process types. We show that the wellknown coalgebraic modal logic can be expressed in CoCasl. We present sufficient criteria for the existence of cofree models, also for several variants of nested cofree and free specifications. Moreover, we describe an extension of the existing proof support for Casl (in the shape of an encoding into higherorder logic) to CoCasl.
Equivalences among Various Logical Frameworks of Partial Algebras
 Computer Science Logic. 9th Workshop, CSL'95. Paderborn
, 1996
"... We examine a variety of liberal logical frameworks of partial algebras. Therefore we use simple, conjunctive and weak embeddings of institutions which preserve model categories and may map sentences to sentences, finite sets of sentences, or theory extensions using unique existential quantifiers, re ..."
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Cited by 17 (7 self)
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We examine a variety of liberal logical frameworks of partial algebras. Therefore we use simple, conjunctive and weak embeddings of institutions which preserve model categories and may map sentences to sentences, finite sets of sentences, or theory extensions using unique existential quantifiers, respectively. They faithfully represent theories, model categories, theory morphisms, colimit of theories, reducts etc. Moreover, along simple and conjunctive embeddings, theorem provers can be reused in a way that soundness and completeness is preserved. Our main result states the equivalence of all the logical frameworks with respect to weak embeddability. This gives us compilers between all frameworks. Thus it is a chance to unify the different branches of specification using liberal partial logics. This is important for reaching the goal of formal interoperability of different specification languages for software development. With formal interoperability, a specification can contain part...
Categorial Fibring of Logics with Terms and Binding Operators
 FRONTIERS OF COMBINING SYSTEMS 2, STUDIES IN LOGIC AND COMPUTATION
, 1998
"... Categorial characterizations are given of both unconstrained and constrained fibring of Hibert calculi and interpretation systems for languages with variables, terms, variable binding operators and modal like operators. Some preliminary transference results are established. A brief comparison wi ..."
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Cited by 14 (10 self)
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Categorial characterizations are given of both unconstrained and constrained fibring of Hibert calculi and interpretation systems for languages with variables, terms, variable binding operators and modal like operators. Some preliminary transference results are established. A brief comparison with model theoretic parchments is included.
Permissive Subsorted Partial Logic in CASL
, 1997
"... . This paper presents a permissive subsorted partial logic used in the CoFI Algebraic Specification Language. In contrast to other ordersorted logics, subsorting is not modeled by set inclusions, but by injective embeddings allowing for more general models in which subtypes can have different data t ..."
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Cited by 13 (8 self)
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. This paper presents a permissive subsorted partial logic used in the CoFI Algebraic Specification Language. In contrast to other ordersorted logics, subsorting is not modeled by set inclusions, but by injective embeddings allowing for more general models in which subtypes can have different data type representations. Furthermore, there are no restrictions like monotonicity, regularity or local filtration on signatures at all. Instead, the use of overloaded functions and predicates in formulae is required to be sufficiently disambiguated, such that all parses have the same semantics. An overload resolution algorithm is sketched. 1 Introduction During the past decades a large number of algebraic specification languages have been developed. The presence of so many similar specification languages with no common framework hinders the dissemination and application of research results in algebraic specification. In particular, it makes it difficult to produce educational material, to reus...
Collapsing Partial Combinatory Algebras
 HigherOrder Algebra, Logic, and Term Rewriting
, 1996
"... Partial combinatory algebras occur regularly in the literature as a framework for an abstract formulation of computation theory or recursion theory. In this paper we develop some general theory concerning homomorphic images (or collapses) of pca's, obtained by identification of elements in a pca. We ..."
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Cited by 12 (2 self)
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Partial combinatory algebras occur regularly in the literature as a framework for an abstract formulation of computation theory or recursion theory. In this paper we develop some general theory concerning homomorphic images (or collapses) of pca's, obtained by identification of elements in a pca. We establish several facts concerning final collapses (maximal identification of elements). `En passant' we find another example of a pca that cannot be extended to a total one. 1
Moving Specification Structures Between Logical Systems
 13th WADT’98
, 1998
"... The conditions under which a formal system for reasoning about structural specifications, built over one logical system could be reused for reasoning about structured specifications built over another logical system are formulated and studied. Following Goguen and Burstall, the notion of a logical s ..."
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Cited by 10 (1 self)
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The conditions under which a formal system for reasoning about structural specifications, built over one logical system could be reused for reasoning about structured specifications built over another logical system are formulated and studied. Following Goguen and Burstall, the notion of a logical system is formalized as an institution and extended to a Dinstitution. A new function between classes of specifications, inspired by a similar function from [HST 94], is defined as a natural extension of institution representations to structured specifications. 1
The HasCasl prologue: categorical syntax and semantics of the partial λcalculus
 COMPUT. SCI
, 2006
"... We develop the semantic foundations of the specification language HasCasl, which combines algebraic specification and functional programming on the basis of Moggi’s partial λcalculus. Generalizing Lambek’s classical equivalence between the simply typed λcalculus and cartesian closed categories, we ..."
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Cited by 6 (4 self)
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We develop the semantic foundations of the specification language HasCasl, which combines algebraic specification and functional programming on the basis of Moggi’s partial λcalculus. Generalizing Lambek’s classical equivalence between the simply typed λcalculus and cartesian closed categories, we establish an equivalence between partial cartesian closed categories (pccc’s) and partial λtheories. Building on these results, we define (settheoretic) notions of intensional Henkin model and syntactic λalgebra for Moggi’s partial λcalculus. These models are shown to be equivalent to the originally described categorical models in pccc’s via the global element construction. The semantics of HasCasl is defined in terms of syntactic λalgebras. Correlations between logics and classes of categories facilitate reasoning both on the logical and on the categorical side; as an application, we pinpoint unique choice as the distinctive feature of topos logic (in comparison to intuitionistic higherorder logic of partial functions, which by our results is the logic of pccc’s with equality). Finally, we give some applications of the modeltheoretic equivalence result to the semantics of HasCasl and its relation to firstorder Casl.
Representations, Hierarchies, and Graphs of Institutions
, 1996
"... For the specification of abstract data types, quite a number of logical systems have been developed. In this work, we will try to give an overview over this variety. As a prerequisite, we first study notions of {\em representation} and embedding between logical systems, which are formalized as {\em ..."
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Cited by 5 (4 self)
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For the specification of abstract data types, quite a number of logical systems have been developed. In this work, we will try to give an overview over this variety. As a prerequisite, we first study notions of {\em representation} and embedding between logical systems, which are formalized as {\em institutions} here. Different kinds of representations will lead to a looser or tighter connection of the institutions, with more or less good possibilities of faithfully embedding the semantics and of reusing proof support. In the second part, we then perform a detailed ``empirical'' study of the relations among various wellknown institutions of total, ordersorted and partial algebras and firstorder structures (all with Horn style, i.e.\ universally quantified conditional, axioms). We thus obtain a {\em graph} of institutions, with different kinds of edges according to the different kinds of representations between institutions studied in the first part. We also prove some separation results, leading to a {\em hierarchy} of institutions, which in turn naturally leads to five subgraphs of the above graph of institutions. They correspond to five different levels of expressiveness in the hierarchy, which can be characterized by different kinds of conditional generation principles. We introduce a systematic notation for institutions of total, ordersorted and partial algebras and firstorder structures. The notation closely follows the combination of features that are present in the respective institution. This raises the question whether these combinations of features can be made mathematically precise in some way. In the third part, we therefore study the combination of institutions with the help of socalled parchments (which are certain algebraic presentations of institutions) and parchment morphisms. The present book is a revised version of the author's thesis, where a number of mathematical problems (pointed out by Andrzej Tarlecki) and a number of misuses of the English language (pointed out by Bernd KriegBr\"uckner) have been corrected. Also, the syntax of specifications has been adopted to that of the recently developed Common Algebraic Specification Language {\sc Casl} \cite{CASL/Summary,Mosses97TAPSOFT}.
Classifying Categories for Partial Equational Logic
, 2002
"... Along the lines of classical categorical type theory for total functions, we establish equivalence results between certain classes of partial equational theories on the one hand and corresponding classes of categories on the other hand, staying close to standard categorical notions. ..."
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Cited by 5 (3 self)
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Along the lines of classical categorical type theory for total functions, we establish equivalence results between certain classes of partial equational theories on the one hand and corresponding classes of categories on the other hand, staying close to standard categorical notions.