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15
On the Search for a Finitizable Algebraization of First Order Logic
, 2000
"... We give an algebraic version of first order logic without equality in which the class of representable algebras forms a nitely based equational class. Further, the representables are dened in terms of set algebras, and all operations of the latter are permutation invariant. The algebraic form of thi ..."
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We give an algebraic version of first order logic without equality in which the class of representable algebras forms a nitely based equational class. Further, the representables are dened in terms of set algebras, and all operations of the latter are permutation invariant. The algebraic form of this result is Theorem 1.1 (a concrete version of which is given by Theorems 2.8 and 4.2), while its logical form is Corollary 5.2. For first order logic with equality we give a result weaker than the one for rst order logic without equality. Namely, in this case  instead of finitely axiomatizing the corresponding class of all representable algebras  we finitely axiomatize only the equational theory of that class. See Subsection 6.1, especially Remark 6.6 there. The proof of Theorem 1.1 is elaborated in Sections 3 and 4. These sections contain theorems which are interesting of their own rights, too, e.g. Theorem 4.2 is a purely semigroup theoretic result. Cf. also "Further main results" in the
Between Functions and Relations in Calculating Programs
, 1992
"... This thesis is about the calculational approach to programming, in which one derives programs from specifications. One such calculational paradigm is Ruby, the relational calculus developed by Jones and Sheeran for describing and designing circuits. We identify two shortcomings with derivations made ..."
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Cited by 15 (4 self)
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This thesis is about the calculational approach to programming, in which one derives programs from specifications. One such calculational paradigm is Ruby, the relational calculus developed by Jones and Sheeran for describing and designing circuits. We identify two shortcomings with derivations made using Ruby. The first is that the notion of a program being an implementation of a specification has never been made precise. The second is to do with types. Fundamental to the use of type information in deriving programs is the idea of having types as special kinds of programs. In Ruby, types are partial equivalence relations (pers). Unfortunately, manipulating some formulae involving types has proved difficult within Ruby. In particular, the preconditions of the `induction' laws that are much used within program derivation often work out to be assertions about types; such assertions have typically been verified either by informal arguments or by using predicate calculus, rather than by ap...
Prototyping Relational Specifications Using HigherOrder Objects
, 1994
"... An approach is described for the generation of certain mathematical objects (like sets, correspondences, mappings) in terms of relations using relationalgebraic descriptions of higherorder objects. From nonconstructivecharacterizations executable relational specifications are obtained. We als ..."
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Cited by 8 (4 self)
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An approach is described for the generation of certain mathematical objects (like sets, correspondences, mappings) in terms of relations using relationalgebraic descriptions of higherorder objects. From nonconstructivecharacterizations executable relational specifications are obtained. We also showhowtodevelop more efficient algorithms from the frequently inefficient specifications within the calculus of binary relations.
Pointfree foundations for (generic) lossless decomposition
, 2007
"... This report presents a typed, “pointfree ” generalization of relational data dependency theory expressed not in the standard settheoretic way, “à la Codd”, but in the calculus of binary relations which, initiated by De Morgan in the 1860s, is the core of modern algebra of programming. Contrary to t ..."
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Cited by 8 (5 self)
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This report presents a typed, “pointfree ” generalization of relational data dependency theory expressed not in the standard settheoretic way, “à la Codd”, but in the calculus of binary relations which, initiated by De Morgan in the 1860s, is the core of modern algebra of programming. Contrary to the intuition that a binary relation is just a particular case of an nary relation, this report shows the effectiveness of the former in “explaining ” and reasoning about the latter. Data dependency theory, which is central to relational database design, is addressed in pointfree calculational style instead of reasoning about (sets of) tuples in conventional “implicationfirst ” logic style. It turns out that the theory becomes more general, more structured and simpler. Elegant expressions replace lengthy formulæ and easytofollow calculations replace pointwise proofs with lots of “··· ” notation, case analysis and natural language explanations for “obvious ” steps. In particular, attributes are generalized to arbitrary (observation) functions and the principle of lossless decomposition is established for arbitrary such functions. The report concludes by showing how the proposed generalization of data dependency theory paves the way to interesting synergies with other branches of computer science, namely formal modeling and transition systems theory. A number of open topics for research in the field are proposed as future work. Pointfree Foundations for (Generic) Lossless
Fork algebras: past, present and future
 Journal of Relational Methods in Computer Science
, 2004
"... Abstract. Fork algebras have interesting connections with Computing, Algebra and Logic. This paper presents a survey of ideas and results about fork algebras, with special emphasis on current developments and promising research lines. 1 ..."
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Cited by 5 (3 self)
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Abstract. Fork algebras have interesting connections with Computing, Algebra and Logic. This paper presents a survey of ideas and results about fork algebras, with special emphasis on current developments and promising research lines. 1
A Relational Derivation of a Functional Program
 In Proc. STOP Summer School on Constructive Algorithmics, Ameland, The
, 1992
"... This article is an introduction to the use of relational calculi in deriving programs. We present a derivation in a relational language of a functional program that adds one bit to a binary number. The resulting program is unsurprising, being the standard `column of halfadders', but the deriv ..."
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This article is an introduction to the use of relational calculi in deriving programs. We present a derivation in a relational language of a functional program that adds one bit to a binary number. The resulting program is unsurprising, being the standard `column of halfadders', but the derivation illustrates a number of points about working with relations rather than functions. 1 Ruby Our derivation is made within the relational calculi developed by Jones and Sheeran [14, 15]. Their language, called Ruby , is designed specifically for the derivation of `hardwarelike' programs that denote finite networks of simple primitives. Ruby has been used to derive a number of different kinds of hardwarelike programs [13, 22, 23, 16]. Programs in Ruby are built piecewise from smaller programs using a simple set of combining forms. Ruby is not meant as a programming language in its own right, but as a tool for developing and explaining algorithms. Fundamental to Ruby is the use of terse not...
Representability of Pairing Relation Algebras Depends on your Ontology
 Fundamenta Informaticae
, 1997
"... We consider classes of relation algebras expanded with new operations based on the formation of ordered pairs. Examples for such algebras are pairing (or projection) algebras of algebraic logic and fork algebras of computer science. It is proved in SainN'emeti [36] that there is no `strong ..."
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Cited by 3 (1 self)
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We consider classes of relation algebras expanded with new operations based on the formation of ordered pairs. Examples for such algebras are pairing (or projection) algebras of algebraic logic and fork algebras of computer science. It is proved in SainN'emeti [36] that there is no `strong' representation theorem for all abstract pairing algebras in most set theories including ZFC as well as most nonwellfounded set theories. Such a `strong' representation theorem would state that every abstract pairing algebra is isomorphic to a set relation algebra having projection elements which are defined with the help of the real (set theoretic) pairing function. Here we show that, by choosing an appropriate (nonwellfounded) set theory as our metatheory, pairing algebras and fork algebras admit such `strong' representation theorems. 1 Introduction This paper is about representation (or axiomatization) problems of algebraic logic. In particular, we discuss the representation proble...
Algebraic Graph Derivations for Graphical Calculi
 Graph Theoretic Concepts in Computer Science (WG'96), volume 1197 of LNCS
, 1997
"... this paper, but only refer to it for comparison with one of the main streams of related work in the literature. In [BH94], an approach to transformations of expressions in UPAs via transformations of graphs has been presented and proven correct. The approach has been developed with a bias towards VL ..."
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this paper, but only refer to it for comparison with one of the main streams of related work in the literature. In [BH94], an approach to transformations of expressions in UPAs via transformations of graphs has been presented and proven correct. The approach has been developed with a bias towards VLSI circuit development and the formalisation and drawings reflect this. More or less building on the approach of [BH94], another approach to graphical calculi has been presented in [CL95], where a gentler introduction is given and an attempt is made to somewhat generalise beyond UPAs. Both approaches, however, present the transformation rules as lowlevel graph manipulation rules and do not resort to any established graph transformation mechanism. As a result, there is only a fixed set of transformation rules that correspond to the basic axioms of the calculus, but no general mechanism to formulate new rules corresponding to proven theorems or special definitions. In this paper we start from a slightly more general definition of diagram as basic data structure for our graphical calculus, and we proceed to give algebraic definitions of rule application and derivation. We cleanly separate the syntax and the semantics of our diagrams and we define correctness of rules on a high level. For reasons of space we do not present any proofs, but concentrate on giving ample motivation and at least a few examples. I gratefully acknowledge the comments of an anonymous referee. 2 Type and Relation Terms
Representability and Program Construction within Fork Algebras
 LOGIC JOURNAL OF THE IGPL
, 1998
"... The representation theorem for fork algebras was always misunderstood regarding its applications in program construction. Its application was always described as "the portability of properties of the problem domain into the abstract calculus of fork algebras". In this paper we show that th ..."
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The representation theorem for fork algebras was always misunderstood regarding its applications in program construction. Its application was always described as "the portability of properties of the problem domain into the abstract calculus of fork algebras". In this paper we show that the results provided by the representation theorem are by far more important. We show that not only the heuristic power coming from concrete binary relations is captured inside the abstract calculus, but also design strategies for program development can be successfully expressed. This result makes fork algebras a programming calculus by far more powerful than it was previously thought. Keywords : Fork algebras, program construction, algorithm design strategies.
Logic Journal of the IGPL, Volume 8, No. 4
, 2000
"... Algebraic Logic. In preparation. Manuscript. ..."