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Testing can be formal, too
, 1995
"... Abstract. The paper presents a theory of program testing based on formal specifications. The formal semantics of the specifications is the basis for a notion of an exhaustive test set. Under some minimal hypotheses on the program under test, the success of this test set is equivalent to the satisfac ..."
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Cited by 77 (1 self)
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Abstract. The paper presents a theory of program testing based on formal specifications. The formal semantics of the specifications is the basis for a notion of an exhaustive test set. Under some minimal hypotheses on the program under test, the success of this test set is equivalent to the satisfaction of the specification. The selection of a finite subset of the exhaustive test set can be seen as the introduction of more hypotheses on the program, called selection hypotheses. Several examples of commonly used selection hypotheses are presented. Another problem is the observability of the results of a program with respect to its specification: contrary to some common belief, the use of a formal specification is not always sufficient to decide whether a test execution is a success. As soon as the specification deals with more abstract entities than the program, program results may appear in a form which is not obviously equivalent to the specificied results. A solution to this problem is proposed in the case of algebraic specifications. 1
Automating Recursive Type Definitions in Higher Order Logic
 Current Trends in Hardware Verification and Automated Theorem Proving
, 1988
"... : The expressive power of higher order logic makes it possible to define a wide variety of types within the logic and to prove theorems that state the properties of these types concisely and abstractly. This paper contains a tutorial introduction to the logical basis for such type definitions. Examp ..."
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Cited by 74 (6 self)
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: The expressive power of higher order logic makes it possible to define a wide variety of types within the logic and to prove theorems that state the properties of these types concisely and abstractly. This paper contains a tutorial introduction to the logical basis for such type definitions. Examples are given of the formal definitions in logic of several simple types. A method is then described for systematically defining any instance of a certain class of commonlyused recursive types. The automation of this method in HOL, an interactive system for generating proofs in higher order logic, is also discussed. 1 To appear in Current Trends in Hardware Verification and Automated Theorem Proving, proceedings of the 1988 Banff Workshop on Hardware Verification, edited by G. Birtwistle and P. Subrahmanyam (SpringerVerlag, 1988). Revised 28 January Contents Introduction 5 1 Introduction to Higher Order Logic 6 1.1 Notation : : : : : : : : : : : : : : : : : : : : : : : : : : : : : ...
GeoRelational Algebra: A Model and Query Language for Geometric Database Systems
 Int. Conf. on Extending Database Technology
, 1988
"... : The user's conceptual model of a database system for geometric data should be simple and precise: easy to learn and understand, with clearly defined semantics, expressive: allow to express with ease all desired query and data manipulation tasks, efficiently implementable. To achieve these goals we ..."
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Cited by 69 (7 self)
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: The user's conceptual model of a database system for geometric data should be simple and precise: easy to learn and understand, with clearly defined semantics, expressive: allow to express with ease all desired query and data manipulation tasks, efficiently implementable. To achieve these goals we propose to extend relational database management systems by integrating geometry at all levels: At the conceptual level, relational algebra is extended to include geometric data types and operators. At the implementation level, the wealth of algorithms and data structures for geometric problems developed in the past decade in the field of Computational Geometry is exploited.  The paper starts from a view of relational algebra as a manysorted algebra which allows to easily embed geometric data types and operators. A concrete algebra for twodimensional applications is developed. It can be used as a highly expressive retrieval and data manipulation language for geometric as well as standard...
Structural Induction and Coinduction in a Fibrational Setting
 Information and Computation
, 1997
"... . We present a categorical logic formulation of induction and coinduction principles for reasoning about inductively and coinductively defined types. Our main results provide sufficient criteria for the validity of such principles: in the presence of comprehension, the induction principle for in ..."
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Cited by 67 (14 self)
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. We present a categorical logic formulation of induction and coinduction principles for reasoning about inductively and coinductively defined types. Our main results provide sufficient criteria for the validity of such principles: in the presence of comprehension, the induction principle for initial algebras is admissible, and dually, in the presence of quotient types, the coinduction principle for terminal coalgebras is admissible. After giving an alternative formulation of induction in terms of binary relations, we combine both principles and obtain a mixed induction/coinduction principle which allows us to reason about minimal solutions X = oe(X) where X may occur both positively and negatively in the type constructor oe. We further strengthen these logical principles to deal with contexts and prove that such strengthening is valid when the (abstract) logic we consider is contextually/functionally complete. All the main results follow from a basic result about adjunc...
A Categorical Programming Language
, 1987
"... A theory of data types and a programming language based on category theory are presented. Data types play a crucial role in programming. They enable us to write programs easily and elegantly. Various programming languages have been developed, each of which may use different kinds of data types. Ther ..."
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Cited by 67 (0 self)
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A theory of data types and a programming language based on category theory are presented. Data types play a crucial role in programming. They enable us to write programs easily and elegantly. Various programming languages have been developed, each of which may use different kinds of data types. Therefore, it becomes important to organize data types systematically so that we can understand the relationship between one data type and another and investigate future directions which lead us to discover exciting new data types. There have been several approaches to systematically organize data types: algebraic specification methods using algebras, domain theory using complete partially ordered sets and type theory using the connection between logics and data types. Here, we use category theory. Category theory has proved to be remarkably good at revealing the nature of mathematical objects, and we use it to understand the true nature of data types in programming.
On Observational Equivalence and Algebraic Specification
, 1987
"... The properties of a simple and natural notion of observational equivalence of algebras and the corresponding specificationbuilding operation are studied. We begin with a defmition of observational equivalence which is adequate to handle reachable algebras only, and show how to extend it to cope wit ..."
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Cited by 66 (17 self)
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The properties of a simple and natural notion of observational equivalence of algebras and the corresponding specificationbuilding operation are studied. We begin with a defmition 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.
Inductive Families
 Formal Aspects of Computing
, 1997
"... A general formulation of inductive and recursive definitions in MartinLof's type theory is presented. It extends Backhouse's `DoItYourself Type Theory' to include inductive definitions of families of sets and definitions of functions by recursion on the way elements of such sets are generated. Th ..."
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Cited by 66 (13 self)
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A general formulation of inductive and recursive definitions in MartinLof's type theory is presented. It extends Backhouse's `DoItYourself Type Theory' to include inductive definitions of families of sets and definitions of functions by recursion on the way elements of such sets are generated. The formulation is in natural deduction and is intended to be a natural generalization to type theory of MartinLof's theory of iterated inductive definitions in predicate logic. Formal criteria are given for correct formation and introduction rules of a new set former capturing definition by strictly positive, iterated, generalized induction. Moreover, there is an inversion principle for deriving elimination and equality rules from the formation and introduction rules. Finally, there is an alternative schematic presentation of definition by recursion. The resulting theory is a flexible and powerful language for programming and constructive mathematics. We hint at the wealth of possible applic...
Completeness of manysorted equational logic
 Houston Journal of Mathematics
, 1985
"... ABSTRACT. Assuming that manysorted oquationallogic "goes just as for the onesorted case " has led to incorrect statements of results in manysorted universal algebra; in fact, the onesorted rules are not sound for manysortededuction. This paper gives sound and complete rules, and characterizes w ..."
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Cited by 62 (6 self)
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ABSTRACT. Assuming that manysorted oquationallogic "goes just as for the onesorted case " has led to incorrect statements of results in manysorted universal algebra; in fact, the onesorted rules are not sound for manysortededuction. This paper gives sound and complete rules, and characterizes when the onesorted rules can still be used safely; it also characterizes the related question of when manysorted algebras can be represented as onesorted algebras. The paper contains a detailed introduction to Hall's theory of clones (later developed into "algebraic theories " by Lawvere and Benabou); this allows a full algebraization of manysorted equational deduction that is not possible with the usual fully invariant congruences on the free algebra on countably many generators. 1. Introduction. The
Automatically Checking an Implementation against Its Formal Specification
 IEEE Transactions on Software Engineering
, 2000
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