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A Functorial Semantics for MultiAlgebras and Partial Algebras, With Applications to Syntax
, 2000
"... Multialgebras allow for the modeling of nondeterminism in an algebraic framework by interpreting operators as functions from individual arguments to sets of possible results. We propose a functorial presentation of various categories of multialgebras and partial algebras, analogous to the classica ..."
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Cited by 14 (7 self)
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Multialgebras allow for the modeling of nondeterminism in an algebraic framework by interpreting operators as functions from individual arguments to sets of possible results. We propose a functorial presentation of various categories of multialgebras and partial algebras, analogous to the classical presentation of algebras over a signature as cartesian functors from the algebraic theory over to Set. We introduce two dierent notions of theory over a signature, both having a structure weaker than cartesian, and we consider functors from them to Rel or Pfn, the categories of sets and relations or partial functions, respectively. Next we discuss how the functorial presentation provides guidelines for the choice of syntactical notions for a class of algebras, and as an application we argue that the natural generalization of usual terms are \conditioned terms" for partial algebras, and \term graphs" for multialgebras. Contents 1 Introduction 2 2 A short recap on multialgebras 4 3...
Allegories of Circuits
 Proc. Logical Foundations of Computer Science
, 1994
"... This paper presents three paradigms for circuit design, and investigates the relationships between them. These paradigms are syntactic (based on Freyd and Scedrov's unitary pretabular allegories (upas)), pictorial (based on the net list model of circuit connectivity), and relational (based on Sheer ..."
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Cited by 12 (0 self)
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This paper presents three paradigms for circuit design, and investigates the relationships between them. These paradigms are syntactic (based on Freyd and Scedrov's unitary pretabular allegories (upas)), pictorial (based on the net list model of circuit connectivity), and relational (based on Sheeran's relational model of circuit design Ruby). We show that net lists over a given signature \Sigma constitute the free upa on \Sigma. Our proof demonstrates that nets and upas are equally expressive, and that nets provide a normal form for both upas and pictures. We use Freyd and Scedrov's representation theorem for upas to show that our relational interpretations constitute a sound and complete class of models for the upa axioms. Thus we can reason about circuits using either the upa axioms, pictures or relations. By considering garbage collection, we show that there is no faithful representation of nets in Rel: we conjecture that a semantics for nets which takes garbage collection into ac...
A Relational Approach To Optimization Problems
, 1996
"... The main contribution of this thesis is a study of the dynamic programming and greedy strategies for solving combinatorial optimization problems. The study is carried out in the context of a calculus of relations, and generalises previous work by using a loop operator in the imperative programming s ..."
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Cited by 6 (0 self)
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The main contribution of this thesis is a study of the dynamic programming and greedy strategies for solving combinatorial optimization problems. The study is carried out in the context of a calculus of relations, and generalises previous work by using a loop operator in the imperative programming style for generating feasible solutions, rather than the fold and unfold operators of the functional programming style. The relationship between fold operators and loop operators is explored, and it is shown how to convert from the former to the latter. This fresh approach provides additional insights into the relationship between dynamic programming and greedy algorithms, and helps to unify previously distinct approaches to solving combinatorial optimization problems. Some of the solutions discovered are new and solve problems which had previously proved difficult. The material is illustrated with a selection of problems and solutions that is a mixture of old and new. Another contribution is the invention of a new calculus, called the graph calculus, which is a useful tool for reasoning in the relational calculus and other nonrelational calculi. The graph
A Graphical Calculus
 Mathematics of Program Construction. SpringerVerlag LNCS 947
, 1995
"... . We present a graphical calculus, which allows mathematical formulae to be represented and reasoned about using a visual representation. We define how a formula may be represented by a graph, and present a number of laws for transforming graphs, and describe the effects these transformations have o ..."
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Cited by 6 (1 self)
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. We present a graphical calculus, which allows mathematical formulae to be represented and reasoned about using a visual representation. We define how a formula may be represented by a graph, and present a number of laws for transforming graphs, and describe the effects these transformations have on the corresponding formulae. We then use these transformation laws to perform proofs. We illustrate the graphical calculus by applying it to the relational and sequential calculi. The graphical calculus makes formulae easier to understand, and so often makes the next step in a proof more obvious. Furthermore, it is more expressive, and so allows a number of proofs that cannot otherwise be undertaken in a pointfree way. 1 Introduction Traditionally, mathematical formulae are written down on a single line. For example, in the relational calculus [9], given four relations P , Q, R and S, we can write P ;Q " R;S to represent the relation that relates two elements x and y iff there exist u and...
Term Graph Syntax for MultiAlgebras
, 2000
"... Multialgebras allow to model nondeterminism in an algebraic framework by interpreting operators as functions from individual arguments to sets of possible results. Starting from a functorial presentation of multialgebras based on gsmonoidal theories, we argue that speci cations for multialgebras ..."
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Cited by 5 (4 self)
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Multialgebras allow to model nondeterminism in an algebraic framework by interpreting operators as functions from individual arguments to sets of possible results. Starting from a functorial presentation of multialgebras based on gsmonoidal theories, we argue that speci cations for multialgebras should be based on the notion of term graphs instead of on standard terms. We consider the simplest case of (term graph) equational specification, showing that it enjoys an unrestricted form of substitutivity. We discuss the expressive power of equational specification for multialgebras, and we sketch possible extensions of the calculus.
Producing Design Diagrams From Declarative Descriptions
 in Proc. Fourth Int. Conf. on CAD/CG
, 1995
"... The declarative language Ruby provides a coherent framework for representing and developing designs. Sketching diagrams for Ruby programs by hand is, however, timeconsuming and errorprone. This paper describes a design sketcher which automates the production of a diagram from a Ruby description. 1 ..."
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Cited by 4 (2 self)
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The declarative language Ruby provides a coherent framework for representing and developing designs. Sketching diagrams for Ruby programs by hand is, however, timeconsuming and errorprone. This paper describes a design sketcher which automates the production of a diagram from a Ruby description. 1 INTRODUCTION Textbased languages, such as VHDL, 3 are becoming increasingly popular for developing designs. Their popularity is mainly due to their facilities for parametrising designs, and it is a great bonus if both behaviour and structure can be expressed in a single notation. Moreover, pictorial representations such as circuit schematics can be tedious to create and to modify. Providing visual aid in hardware design is, nevertheless, important. Circuit diagrams, when appropriately drawn, make explicit the basic structure and size of components, allowing designers to obtain rapidly an overview of a design and to locate specific parts on which they can focus. There have been attempts ...
Some Notes on Logic Programming with a Relational Machine (Extended Abstract)
 Relational Methods in Computer Science, Technical Report Nr. 199803
, 1998
"... James Lipton Dept. of Mathematics Wesleyan University Emily Chapman Dept. of Mathematics Wesleyan University Abstract We study the use of relation calculi for compilation and execution of Horn Clause programs with an extended notion of input and output. We consider various other extensions to the Pr ..."
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Cited by 3 (0 self)
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James Lipton Dept. of Mathematics Wesleyan University Emily Chapman Dept. of Mathematics Wesleyan University Abstract We study the use of relation calculi for compilation and execution of Horn Clause programs with an extended notion of input and output. We consider various other extensions to the Prolog core.
Proofs with Graphs
 Science of Computer Programming
, 1995
"... We present a graphical calculus, which allows mathematical formulae to be represented and reasoned about using a visual representation. We define how a formula may be represented by a graph, and present a number of laws for transforming graphs, and describe the effects these transformations have on ..."
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Cited by 3 (1 self)
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We present a graphical calculus, which allows mathematical formulae to be represented and reasoned about using a visual representation. We define how a formula may be represented by a graph, and present a number of laws for transforming graphs, and describe the effects these transformations have on the corresponding formulae. We then use these transformation laws to perform proofs. We illustrate the graphical calculus by applying it to the relational and sequential calculi. The graphical calculus makes formulae easier to understand, and so often makes the next step in a proof more obvious. Furthermore, it is more expressive, and so allows a number of proofs that cannot otherwise be undertaken in a pointfree way. 1 Introduction Traditionally, mathematical formulae are written down on a single line. For example, in the relational calculus [12], given four relations P , Q, R and S, we can write P ;Q " R;S to represent the relation that relates two elements x and y iff there exist u and...
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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Cited by 2 (1 self)
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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