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ABSTRACT A new concept for the implementation of abstract data types is proposed: Given algebraic specifications SPECO and SPEC1 of abstract data types ADTO and ADT1 an implementation of ADTO by ADT1 is defined separately on the... more
ABSTRACT A new concept for the implementation of abstract data types is proposed: Given algebraic specifications SPECO and SPEC1 of abstract data types ADTO and ADT1 an implementation of ADTO by ADT1 is defined separately on the syntactical level of specifications and on the semantical level of algebras. This concept is shown to satisfy a number of conceptual requirements for the implementation of abstract data types. Several correctness criteria are given and illustrating examples are provided.
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ABSTRACT If an algebraic specification is designed in a structured way, a small specification is stepwise enriched by more complex operations and their defining equations. Based on normalization properties of term reductions we present... more
ABSTRACT If an algebraic specification is designed in a structured way, a small specification is stepwise enriched by more complex operations and their defining equations. Based on normalization properties of term reductions we present sufficient local conditions for the completeness and consistency of enrichment steps, which can be efficiently verified in many cases where other attempts to prove the enrichment property syntactically have failed so far.
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This chapter deals with several theories derived from a Horn clause specification. Each theory is complete with respect to a subclass of Mod(SIG,AX) (cf. Section 2.3). Different theories represent different concepts of semantical... more
This chapter deals with several theories derived from a Horn clause specification. Each theory is complete with respect to a subclass of Mod(SIG,AX) (cf. Section 2.3). Different theories represent different concepts of semantical abstraction. Some of them correspond to the theory of a single SIG-structure, say B. Vice versa, if we start out from a SIG-structure A as the formalization of a data type, an axiom set AX is called correct w.r.t. A if A coincides with B.
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Mathematics and Axiom
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... DEDUCTION AND DECLARATIVE PROGRAMMING PETER PADAWITZ Department of Computer Science University 0_f Dortmund P] CAMBRIDGE UNIVERSITY ... reductive validity Sample confluence proofs On rewriting logics $|$|$c':\°| 8BB353l§|$|5|... more
... DEDUCTION AND DECLARATIVE PROGRAMMING PETER PADAWITZ Department of Computer Science University 0_f Dortmund P] CAMBRIDGE UNIVERSITY ... reductive validity Sample confluence proofs On rewriting logics $|$|$c':\°| 8BB353l§|$|5| $|G|3B|~oloo| o\|u1ml ...
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We summarize a number of new results concerning inductive-theorem proving in the area of design specifications using Horn logic with equality. Induction is explicit here because induction orderings must be integrated into the... more
We summarize a number of new results concerning inductive-theorem proving in the area of design specifications using Horn logic with equality. Induction is explicit here because induction orderings must be integrated into the specification. However, the proofs need less guidance if the specification is ground confluent and strongly terminating. Calculi for verifying these conditions are presented along with a list of useful applications.
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Google, Inc. (search). ...
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States build up hidden sorts. Consequently, labelled transition systems can be specified as ternary predicates within many-sorted specifications with visible (static, constructor-based) as well as hidden (dynamic, state-based) components.... more
States build up hidden sorts. Consequently, labelled transition systems can be specified as ternary predicates within many-sorted specifications with visible (static, constructor-based) as well as hidden (dynamic, state-based) components. In modal logic, transition systems determine the equality of states, usually called bisimilarity. In many sorted type logic, the equality of hidden data usually comes as contextual or behavioural equivalence. We integrate both concepts and specify transition systems in terms of functional, relational and/or transitional actions.
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Expander2 is a flexible multi-purpose workbench for inter- active rewriting, verification, constraint solving, flow graph analysis and other procedures that build up proofs or computation sequences. More- over, tailor-made interpreters... more
Expander2 is a flexible multi-purpose workbench for inter- active rewriting, verification, constraint solving, flow graph analysis and other procedures that build up proofs or computation sequences. More- over, tailor-made interpreters display terms as two-dimensional struc- tures ranging from trees and rooted graphs to a variety of pictorial rep- resentations that include tables, matrices, alignments, piles, partitions, fractals and turtle systems. Proofs and computations performed with Expander2 follow the rules and the semantics of swinging types. Swinging types are based on many- sorted predicate logic and combine visible constructor-based types with hidden state-based types. The former come as initial term models, the lat- ter as final models consisting of context interpretations. Relation symbols are interpreted as least or greatest solutions of their respective axioms. This paper presents an overview of Expander2 with particular emphasis on the system's prover capabilities.
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Expander2 is a flexible multi-purpose workbench for interactive term rewriting, graph transformation, theorem proving, constraint solving, flow graph analysis and other procedures that build up proofs or other rewrite sequences. Moreover,... more
Expander2 is a flexible multi-purpose workbench for interactive term rewriting, graph transformation, theorem proving, constraint solving, flow graph analysis and other procedures that build up proofs or other rewrite sequences. Moreover, tailor-made interpreters display terms as two-dimensional structures ranging from trees and rooted graphs to a variety of pictorial representations that include tables, matrices, alignments, partitions, fractals and various tree-like or rectangular graph layouts.
Refinements of paramodulation (cf. Section 5.3) such as goal reduction (cf. Chapter 7) and narrowing (cf. Chapter 8) are complete only if certain requirements on are fulfilled, which, in turn, depend on the division of into a base... more
Refinements of paramodulation (cf. Section 5.3) such as goal reduction (cf. Chapter 7) and narrowing (cf. Chapter 8) are complete only if certain requirements on are fulfilled, which, in turn, depend on the division of into a base specification and the rest of . The base signature BSIG contains all sorts, predicates and sort-building or constructor functions of SIG. Hence the remaining part of SIG consists of operation symbols called non-constructor or, with the name indicating their meaning,destructor, inquiry, state-transition or value-returning functions. Accordingly, BAX specifies the predicates, in particular the equality predicates, with the help of constructor functions. Non-base axioms must be conditional equations and should only be used to specify non-constructor functions.
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We saw in Chapters 7 and B that Church-Rosser properties are crucial for the completeness of reduction and narrowing. Our general Church-Rosser criterion (Theorem 7.B.2) presumes confluence and BAX- (or ~BAX-) compatibility of NAX (or... more
We saw in Chapters 7 and B that Church-Rosser properties are crucial for the completeness of reduction and narrowing. Our general Church-Rosser criterion (Theorem 7.B.2) presumes confluence and BAX- (or ~BAX-) compatibility of NAX (or ~NAX, respectively). This chapter is mainly devoted to confluence and compatibility criteria based upon the convergence of critical pairs.
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Design problems are inherently difficult to solve. Re-itman (Reitman, 1964) and Simon (Simon, 1973) them ill-structured and ill-defined, Rittel and Webber (Rittel and Webber, 1973) wicked problems. Due to the difficulties in tackling... more
Design problems are inherently difficult to solve. Re-itman (Reitman, 1964) and Simon (Simon, 1973) them ill-structured and ill-defined, Rittel and Webber (Rittel and Webber, 1973) wicked problems. Due to the difficulties in tackling complex design problems, design automation has been limited to routine and detail design. Routine or detail design means to follow a specified schema with expected results. However, recently interest has shifted toward the automation of complex design tasks, in particular tasks which require creativity. By creative design we understand the development of new, unexpected features or solutions. Design is considered to be a search process with vast and complex search space. Creativity is needed if the search space is not well-defined and only a few heuristics are known for guiding the search. Unfortunately, very little is known about the human creative process. However, certain creative strategies can be distinguished. Moreover, two approaches, those drawi...
Dieser Bericht ist im Wortlaut identisch mit: Peter Padawitz, Church-Rosser-Eigenschaften von Graph-Grammatiken und Anwendungen auf die Semantik von LISP, Diplomarbeit 1978.
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Expander2 is a flexible multi-purpose workbench for interactive rewriting, verification, constraint solving, flow graph analysis and other procedures that build up proofs or computation sequences. Moreover, tailormade interpreters display... more
Expander2 is a flexible multi-purpose workbench for interactive rewriting, verification, constraint solving, flow graph analysis and other procedures that build up proofs or computation sequences. Moreover, tailormade interpreters display terms as two-dimensional structures ranging from trees and rooted graphs to a variety of pictorial representations that include tables, matrices, alignments, partitions, fractals and turtle systems. Proofs and computations performed with Expander2 follow the rules and the semantics of swinging types. Swinging types are based on manysorted predicate logic and combine constructor-based types with destructorbased (e.g. state-based) ones. The former come as initial term models, the latter as final models consisting of context interpretations. Relation symbols are interpreted as least or greatest solutions of their respective axioms. This paper presents an overview of Expander2 with particular emphasis on the system’s prover and rewriter capabilities. 1
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Swinging types (STs) provide an axiomatic specification formalism for designing and verifying software in terms of many-sorted logic and canonical models. STs are one-tiered insofar as static and dynamic, structural and behavioral aspects... more
Swinging types (STs) provide an axiomatic specification formalism for designing and verifying software in terms of many-sorted logic and canonical models. STs are one-tiered insofar as static and dynamic, structural and behavioral aspects of a system are treated on the same syntactic and semantic level. Canonical models interpret relations as least or greatest fixpoints. All reasoning about a particular ST can be reduced to deductive processes, from built-in simplifications via resolution upon relations, narrowing upon functions, up to interactive proofs employing induction and coinduction rules. In this paper, the different possibilities of building up an ST are clearly separated from each other. The designer of an ST may choose among six specification patterns when extending a given ST by new components. Semantically, this leads to stratified models, similar to those known from the semantics of stratified logic programs. Predicates (relations interpreted as least fixpoints) and fu...
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Expander2 is a flexible multi-purpose workbench for interactive term rewriting, graph transformation, theorem proving, constraint solving, flow graph analysis and other procedures that build up proofs or other rewrite sequences. Moreover,... more
Expander2 is a flexible multi-purpose workbench for interactive term rewriting, graph transformation, theorem proving, constraint solving, flow graph analysis and other procedures that build up proofs or other rewrite sequences. Moreover, tailor-made interpreters display terms as two-dimensional structures ranging from trees and rooted graphs to a variety of pictorial representations that include tables, matrices, alignments, partitions, fractals and various tree-like or rectangular graph layouts. An Expander specification consists of a signature with functions, predicates, axioms, theorems and conjectures (terms to be rewritten or formulas to be solved or proved). It describes a set of algebraic (constructor-based) and/or coalgebraic (destructor-based) types (formerly called swinging types). Syntactically, it follows Haskell (for presenting functions) and usual mathematical notations (for presenting relations and propositional, predicate-logic, modal and temporal operators). Predic...
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Swinging types 18] provide an integrated framework for specifying software on the basis of many-sorted logic in terms of \static" functions and relations as well as \dynamic" transition systems. Swinging types combine... more
Swinging types 18] provide an integrated framework for specifying software on the basis of many-sorted logic in terms of \static" functions and relations as well as \dynamic" transition systems. Swinging types combine equational, Horn and modal logic for the purpose of using evaluation and proof rules from all three logics for rapid prototyp-ing and veriication. A swinging speciication separates from each other visible sorts that denote domains of data identiied by their structure; hidden sorts that denote domains of data identiied by their behavior in response to observers;-predicates (least relations) that represent in-ductive(ly provable) properties; and-predicates (greatest relations) that represent complementary \coinductive" properties. The paper at hand deals with structured speciications with swinging components. Vertical structuring is supported by a deduction-oriented reenement criterion that admits, for instance, to implement visible sorts by hidden sorts a...
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Prinzipiell lässt sich jedes Modell eines Softwaresystems einer von zwei Klassen zuordnen: den konstruktorbasierten white-box-Modellen oder den destruktorbasierten black-box-Modellen. Ein gröÿeres System setzt sich konstruktorbzw.... more
Prinzipiell lässt sich jedes Modell eines Softwaresystems einer von zwei Klassen zuordnen: den konstruktorbasierten white-box-Modellen oder den destruktorbasierten black-box-Modellen. Ein gröÿeres System setzt sich konstruktorbzw. destruktorbasierter Teilen zusammen. Endliche, durch kontextfreie Grammatiken beschriebene Datenstrukturen gehören in die erste Klasse, unendliche Datenstrukturen (z.B. Ströme) sowie Automaten, Transitionssysteme und alle Arten von zustandsund objektorientierten Modellen in die zweite Klasse. Konstruktorwie destruktorbasierte Modelle verwenden Funktionsterme. Im ersten Fall liefern diese die Bausteine des Modells selbst, im zweiten Fall beschreiben sie Versuchsaufbauten , mit deren Hilfe das Modell und sein Verhalten beobachtet wird und somit seine Bausteine identi ziert werden.
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ABSTRACT A precise mathematical approach to stepwise refinement of software systems is given within the framework of algebraic specifications. Since our new concept for the implementation of abstract data types — recently introduced in... more
ABSTRACT A precise mathematical approach to stepwise refinement of software systems is given within the framework of algebraic specifications. Since our new concept for the implementation of abstract data types — recently introduced in another paper — corresponds to a single refinement step, the composition problem for algebraic implementations is studied in this paper. It is shown that in general algebraic implementations are not closed under composition unless we have the special case of persistent implementations. For other types of implementations sufficient consistency conditions are given to achieve closure under composition. These results can be extended to compound algebraic implementations which are syntactically defined to be sequences of (weak) implementations according to the idea of stepwise refinement of software systems.
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ABSTRACT [For the entire collection see Zbl 0534.00019.] It is shown that certain parameterized data types have a ”typical” initial algebra which captures the equational theory of the data type. Hence the characteristic of initial... more
ABSTRACT [For the entire collection see Zbl 0534.00019.] It is shown that certain parameterized data types have a ”typical” initial algebra which captures the equational theory of the data type. Hence the characteristic of initial algebras, namely that term induction and rewriting provide a proof method for equational theorems, also applied to parameterized data types.