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Finite algebraic specifications of semicomputable data types [chapter]

G. Marongiu, S. Tulipani
<span title="">1987</span> <i title="Springer Berlin Heidelberg"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/2w3awgokqne6te4nvlofavy5a4" style="color: black;">Lecture Notes in Computer Science</a> </i> &nbsp;
This is a revised version of previous (unpublished) paper which circulated under the title: Finite Specification of Data Types with extra Operations.  ...  Therefore, the significant models of a data type specification E in a signature Z are given by the class Algm(Z,E) whose members are the models of E which are finitely generated by elements named as const  ...  The specification method (for abstract data types) by means of a finite conditional equation consistent extension with no additional sorts, is complete for the class of semicomputable data types. m 1)  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/3-540-17660-8_51">doi:10.1007/3-540-17660-8_51</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/wh75z63w25ga5askxd4rryagfe">fatcat:wh75z63w25ga5askxd4rryagfe</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20180727053545/https://link.springer.com/content/pdf/10.1007%2F3-540-17660-8_51.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/e9/15/e915c2e5047ce782b133d7c213fbe7044cb07766.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/3-540-17660-8_51"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> springer.com </button> </a>

On a conjecture of Bergstra and Tucker

G. Marongiu, S. Tulipani
<span title="">1989</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/elaf5sq7lfdxfdejhkqbtz6qoq" style="color: black;">Theoretical Computer Science</a> </i> &nbsp;
Bergstra and Tucker (1983) conjectured that a semicomputable (abstract) data type has a finite hidden enrichment specification under its initial algebra semantics.  ...  Here, following the line and the proof techniques of the previous paper, we examine a nontrivial case in which the conjecture has a positive answer. * Work performed under the auspices of the Italian "  ...  Finite (conditional) equation specifications define, with respect to the initial algebra semantics, abstract data types which are isomorphism classes of semicomputa le data structures. common now to think  ... 
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Page 3434 of Mathematical Reviews Vol. , Issue 89F [page]

<span title="">1989</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
V. (4-LEED-C) Algebraic specifications of computable and semicomputable data types. Theoret. Comput. Sci. 50 (1987), no. 2, 137-181.  ...  The abstract data type specified by an algebraic specification is the isomorphism class of its initial algebras.  ... 
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Algebraic specifications of computable and semicomputable data types

J.A. Bergstra, J.V. Tucker
<span title="">1987</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/elaf5sq7lfdxfdejhkqbtz6qoq" style="color: black;">Theoretical Computer Science</a> </i> &nbsp;
An extensive survey is given of the properties of various specification mechanisms based on initial algebra semantics.  ...  Rodenburg, and an anonymous referee for useful criticism on earlier versions of this paper. We thank Ms. Judith Thursby for typing the manuscript.  ...  We are left with two basic questions: Are any finite algebraic specifications either complete for the semicomputable data types, or adequate for the computable data types?  ... 
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Generalizing Computability Theory to Abstract Algebras [chapter]

J. V. Tucker, J. I. Zucker
<span title="">2015</span> <i title="Springer International Publishing"> Turing&#39;s Revolution </i> &nbsp;
We are grateful to Mark Armstrong, Sol Feferman, Diogo Poças and an anonymous referee for very helpful comments on earlier drafts of this chapter.  ...  We also thank the editors, Giovanni Sommaruga and Thomas Strahm, for the opportunity to participate in this volume, and for their helpfulness during the preparation of this chapter.  ...  Consider an adequate specification language S over an abstract data type dt.  ... 
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Terminal algebra semantics and retractions for abstract data types [chapter]

Günter Hornung, Peter Raulefs
<span title="">1980</span> <i title="Springer Berlin Heidelberg"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/2w3awgokqne6te4nvlofavy5a4" style="color: black;">Lecture Notes in Computer Science</a> </i> &nbsp;
Montanari, Observability concepts in abstrac t data types specification, Proc . of 5th MFCS symposium, LNCS 4 5 (576-587) 1976 .  ...  . -30 -On the other hand, given GI/0 (A) and a finite set of (conditional) equation s E the initial algebra of GI/0 U E is semicomputable, as are all initia l algebras of decidable sets of axioms .  ... 
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Initial and Final Algebra Semantics for Data Type Specifications: Two Characterization Theorems

J. A. Bergstra, J. V. Tucker
<span title="">1983</span> <i title="Society for Industrial &amp; Applied Mathematics (SIAM)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/7dys7zoberdktmxyjciuy5bnse" style="color: black;">SIAM journal on computing (Print)</a> </i> &nbsp;
And we characterize the computable data types as those data types which may be specified by conditional equation specifications using both initial algebra semantics and final algebra semantics.  ...  We prove that those data types which may be defined by conditional equation specifications and final algebra semantics are exactly the cosemicomputable data typesthose data types which are effectively  ...  Then the following are equivalent: FINITE CONDITIONAL SPECIFICATIONS + INITIAL SEMANTICS SEMICOMPUTABLE DATA TYPES, FINITE CONDITIONAL SPECIFICATIONS +FINAL SEMANTICS COSEMICOMPUTABLE DATA TYPES,  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1137/0212024">doi:10.1137/0212024</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/kmous3qll5epblieg57c4savga">fatcat:kmous3qll5epblieg57c4savga</a> </span>
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Equational specifications, complete term rewriting systems, and computable and semicomputable algebras

J. A. Bergstra, J. V. Tucker
<span title="1995-11-01">1995</span> <i title="Association for Computing Machinery (ACM)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/r72grtxx6jad5hjzl7fxcunwhi" style="color: black;">Journal of the ACM</a> </i> &nbsp;
We classify the computable and semicomputable algebras in terms of finite equational initial algebra specifications and their properties as term term rewriting systems, such as completeness.  ...  Equational specifications, complete term rewriting systems, and computable and semicomputable algebras Bergstra, J.A.; Tucker, J.V. Abstract.  ...  We examine the term rewriting properties of finite equational hidden function specifications of data types and we give an algebraic characterization of the computable data types in terms of complete term  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1145/227683.227687">doi:10.1145/227683.227687</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/7tk5kkawmnagdecsvdxmg62hju">fatcat:7tk5kkawmnagdecsvdxmg62hju</a> </span>
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Deterministic and nondeterministic computation, and horn programs, on abstract data types

J.V. Tucker, J.I. Zucker
<span title="">1992</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/7qdabej4lbaczotmeg3ldn4ioy" style="color: black;">The Journal of Logic Programming</a> </i> &nbsp;
We investigate the notion of "semicomputability," intended to generalize the notion of recursive enumerability of relations to abstract structures.  ...  Two characterizations are considered and shown to be equivalent: one in terms of " partial computable functions' ' (for a suitable notion of computability over abstract structures) and one in terms of  ...  Lassez for inviting us to give a detailed exposition of these results.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/0743-1066(92)90020-4">doi:10.1016/0743-1066(92)90020-4</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/ghcujd36yzaebmrd33x6yfy6by">fatcat:ghcujd36yzaebmrd33x6yfy6by</a> </span>
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Notions of semicomputability in topological algebras over the reals

Mark Armstrong, Jeffery Zucker
<span title="2018-02-19">2018</span> <i title="IOS Press"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/rtyr26r4evfy3hccqovstp6k5m" style="color: black;">Computability - The Journal of the Assosiation</a> </i> &nbsp;
We consider several results from the classical theory in the context of topological algebra of the reals: closure of semicomputable sets under finite union (Chapter 4 Theorem 2, p.33), the equivalence  ...  of semicomputable and projectively (semi)computable sets (Chapter 5 Theorem 7, p.45), and Post's Theorem (i.e. a set is computable iff both it and its complement are semicomputable) (Appendix B Theorem  ...  does not hold for its While semicomputable subsets.Define a signature Σ D :signature Σ D sorts data, bool functions 0 : → data F, G : data data eq : data 2 → bool and algebra D: The  ... 
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The data type variety of stack algebras

J.A Bergstra, J.V Tucker
<span title="">1995</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/bnojym2hjzcgnpa4wixi2axhnq" style="color: black;">Annals of Pure and Applied Logic</a> </i> &nbsp;
There is a finite equational specification (Cast, E",) with a hidden sort and two hidden operations that specifies the class Alg,."(&, E") of all stack algebras and its initial algebra T (&" E") .  ...  represent the consensus properties of the behaviour of stacks with the capacity to store k data.  ...  In algebraic data type theory, data types are specified by means of equational or conditional equational axioms, most commonly up to isomorphism using initial algebra semantics for the specification.  ... 
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Computation by 'While' programs on topological partial algebras

J.V. Tucker, J.I. Zucker
<span title="">1999</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/elaf5sq7lfdxfdejhkqbtz6qoq" style="color: black;">Theoretical Computer Science</a> </i> &nbsp;
In the theory of data, continuous data types are modelled by topological many-sorted algebras.  ...  The language of while programs is a fundamental model for imperative programming on any data type.  ...  Introduction The theory of computable functions and sets on many-sorted algebras is a mathematical theory for the analysis of finite deterministic computation on any kind of data.  ... 
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Page 5134 of Mathematical Reviews Vol. , Issue 92i [page]

<span title="">1992</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
As a consequence every semicomputable or cosemicomputable equality- test data type is computable.  ...  A data type is an isomorphism class of a data structure.  ... 
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Which data types have ω-complete initial algebra specifications?

J.A. Bergstra, J. Heering
<span title="">1994</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/elaf5sq7lfdxfdejhkqbtz6qoq" style="color: black;">Theoretical Computer Science</a> </i> &nbsp;
We also show that in the case of finite data types the hidden sorts can be dispensed with.  ...  theory of the initial model is identical to the equational theory of the specification, As the latter is recursively enumerable, the initial model of an w-complete algebraic specification is a data type  ...  So initial algebra specifications give rise to semicomputable data types.  ... 
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Page 1592 of Mathematical Reviews Vol. , Issue 93c [page]

<span title="">1993</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
J. (4-EDIN-C); Wells, Charles [Wells, Charles F.] (1-CWR) A formalism for the specification of essentially-algebraic structures in 2-categories. Math. Structures Comput. Sci. 2 (1992), no. 1, 1-28.  ...  weak observational equiva- lence of finite transition systems, and language containment for finite w-automata.  ... 
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