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Abstract An important tool for controlling search in an object logic is the use of structured theory presentations. In order to apply these ideas to the setting of a logical framework, the authors study the behavior of structured theory... more
Abstract An important tool for controlling search in an object logic is the use of structured theory presentations. In order to apply these ideas to the setting of a logical framework, the authors study the behavior of structured theory presentations under representation in a framework, focusing on the problem of lifting presentations, from the object logic to the metalogic of the framework. The authors also consider imposing structure on logic presentations so that logical systems may themselves be defined in a modular fashion. ...
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It has been a common belief that the standard results of universal algebra as developed since the work of Birkhoff and others in the thirties carry over without much change to the framework of many-sorted algebras. Perhaps the only... more
It has been a common belief that the standard results of universal algebra as developed since the work of Birkhoff and others in the thirties carry over without much change to the framework of many-sorted algebras. Perhaps the only exception widely noticed by the community is the care needed in the treatment of many-sorted equational logic. However, while the standard results remain valid in essence in the many-sorted frameworks, some nuances and technicalities require considerably more care in formulation and ∗This work was funded in part by by the Polish Ministry of Science and Higher Education, grant N206 493138, and by the European IST FET programme under the IST-2005-016004 SENSORIA project. proof of the results. We give some examples of this, indicating how equational calculus, Birkhoff’s variety theorem and interpolation results should be adjusted for many-sorted algebras.
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The purpose of this chapter is to present the basic definitions and results on which the following chapters rely. Most of this material is quite standard and for that reason the presentation will be concise. More detailed presentations... more
The purpose of this chapter is to present the basic definitions and results on which the following chapters rely. Most of this material is quite standard and for that reason the presentation will be concise. More detailed presentations with greater emphasis on motivation, exercises, and examples may be found in [EM85, Wir90, LEW96, ST]. The most basic assumption of work on algebraic specification is that a program is modeled as an algebra, that is, a set of data together with a number of functions over this set. The branch of mathematics which deals with algebras in a general sense (as opposed to the study of specific classes of algebras, such as groups and rings) is called universal algebra or sometimes general algebra. This chapter presents the basics of universal algebra, generalized to the many-sorted case as required to model programs which manipulate several kinds or sorts of data. Some extensions useful for modeling more complex programs are sketched at the end of the chapter.
CASL is an expressive specication language that has been designed to supersede many existing algebraic specication languages and provide a standard. CASL consists of several layers, including basic (unstructured) specications, struc-
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Different approaches to algebraic specification involve different mathematical underpinnings. These involve variations on the definitions of signature and algebra, the language of axioms used, and what it means for an algebra to satisfy... more
Different approaches to algebraic specification involve different mathematical underpinnings. These involve variations on the definitions of signature and algebra, the language of axioms used, and what it means for an algebra to satisfy an axiom. Different choices are useful for different purposes, and there appears to be no “best” choice that can be used for everything. We deal with this situation by making the theory of specification independent of this choice, using the notion of an institution which formalises the informal concept of logical system. This allows work on theories, results, and practical tools to be done just once for a wide range of logical systems, while at the same time forcing, via abstraction, deeper insight into the essence of the concepts and results. This chapter explains the basic elements of the theory of institutions on which the remaining chapters are based.
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We propose a simple framework of algebraic constructions for software specification, modular design and development. Algebraic constructions generalise (parameterised) modules by allowing on one hand a rather arbitrary collection of... more
We propose a simple framework of algebraic constructions for software specification, modular design and development. Algebraic constructions generalise (parameterised) modules by allowing on one hand a rather arbitrary collection of elements to form the parameter and on the other hand dependencies between the module elements to be spelled out explicitly. Algebraic constructions are specified in a very natural way by means of ordinary algebraic specifications. They are combined using a sum operation which captures as special cases various operations on (parameterised) modules offered by standard specification and development frameworks. We show the expected composability result for the sum of algebraic constructions and of their specifications.
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This paper builds on the theory of institutions, a version of abstract model theory that emerged in computer science studies of software specification and semantics. To handle proof theory, our institutions use an extension of traditional... more
This paper builds on the theory of institutions, a version of abstract model theory that emerged in computer science studies of software specification and semantics. To handle proof theory, our institutions use an extension of traditional categorical logic with sets of sentences as objects instead of single sentences, and with morphisms representing proofs as usual. A natural equivalence relation on institutions is defined such that its equivalence classes are logics. Several invariants are defined for this equivalence, including a Lindenbaum algebra construction, its generalization to a Lindenbaum category construction that includes proofs, and model cardinality spectra; these are used in some examples to show logics inequivalent. Generalizations of familiar results from first order to arbitrary logics are also discussed, including Craig interpolation and Beth definability. Mathematics Subject Classification (2000). Primary 03C95; Secondary 18A15, 03G30, 18C10, 03B22.
The properties of a simple and natural notion of observational equivalence of algebras and the corresponding specification-building operation are studied. We begin with a definition of observational equivalence which is adequate to handle... more
The properties of a simple and natural notion of observational equivalence of algebras and the corresponding specification-building operation are studied. We begin with a definition of observational equivalence which is adequate to handle reachable algebras only, and show how to extend it to cope with unreachable algebras and also how it may be generalised to make sense under an arbitrary institution. Behavioural equivalence is treated as an important special case of observational equivalence, and its central role in program development is shown by means of an example.
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... Patricia Bouyer Folker den Braber Laura Brandan Briones Phil Brooke Roberto Bruni Hans Bruun Gyrd Braendeland Bettina Buth Jens ... 269 End-to-End Integrated Security and Performance Analysis on the DEGAS Choreographer Platform Mikael... more
... Patricia Bouyer Folker den Braber Laura Brandan Briones Phil Brooke Roberto Bruni Hans Bruun Gyrd Braendeland Bettina Buth Jens ... 269 End-to-End Integrated Security and Performance Analysis on the DEGAS Choreographer Platform Mikael Buchholtz, Stephen Gilmore ...
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... Scenario for Bytecode Verification of Resource Bounds Roberto M. Amadio, Solange Coupet-Grimal, Silvano Dal Zilio, Line Jakubiec ... Total Primitive Recursive Functions: Theories with Induction Andres Cordon-Franco, Alejandro... more
... Scenario for Bytecode Verification of Resource Bounds Roberto M. Amadio, Solange Coupet-Grimal, Silvano Dal Zilio, Line Jakubiec ... Total Primitive Recursive Functions: Theories with Induction Andres Cordon-Franco, Alejandro Fernandez-Margarit, F. Felix Lara-Martin 355 ...
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... Maura Cerioli, Rocco De Nicola, R˘azvan Diaconescu, Luis Dominguez, Hans-Dieter Ehrich, Hartmut Ehrig, John Fitzgerald, Michael Fourman, Harald ... Preface xi John Power, Horst Reichel, Grigore Rosu, David Rydeheard, Oliver Schoett,... more
... Maura Cerioli, Rocco De Nicola, R˘azvan Diaconescu, Luis Dominguez, Hans-Dieter Ehrich, Hartmut Ehrig, John Fitzgerald, Michael Fourman, Harald ... Preface xi John Power, Horst Reichel, Grigore Rosu, David Rydeheard, Oliver Schoett, Lutz Schr¨oder, Douglas Smith, Stefan ...
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. We use the well-known framework of concrete categories toshow how much of standard universal algebra may be done in an abstractand still rather intuitive way. This is used to recast the unifying view ofbehavioural semantics of... more
. We use the well-known framework of concrete categories toshow how much of standard universal algebra may be done in an abstractand still rather intuitive way. This is used to recast the unifying view ofbehavioural semantics of specifications based on behavioural satisfactionand, respectively, on behavioural equivalence of models abstracting awayfrom many particular features of standard algebras. We also give an