Carolyn Brown and Alan Jeffrey. Allegories of circuits. In Proc. LFCS, 1994.  Home/Search   Document Details and Download   Summary   Related Articles   Check

.... [1] and [2] on game semantics and Interaction Categories; 11] on cyclic structures and recursive computation; 14] on premonoidal categories, and mixed data and control ow graphs; 3] on nuclear and trace ideals, and probabilistic relations ; 4] on linearly distributive categories; 15] and [5] on relational models of circuit design ; and [27] on the semantics of asynchronous communication in networks of parallel processes. This paper is part of a project ( 24] 9] 23] 31] 20] 21] 22] 18] 19] that aims to develop compositional models of distributed systems using ....

....can become unmanageable. Furthermore, there is no description of the internal state of the circuit: for example, there are only two such relations which model circuits with no input and no output. For more on how the category of relations can be used to model circuits, the reader is referred to [5] and [15] 3 Circuits In this section, a very general (and abstract) notion of a circuit will be introduced, as well as operations on these circuits. These circuits do not form a monoidal category, but, rather, a monoidal bicategory. The structure of this bicategory will not be described, as ....

Brown C, Jerey A, Allegories of circuits, in: Proc. Logical Foundations of Computer Science, St. Petersburg, 1994.

.... allows us to infer our properties from analogous results already presented in e.g. the theory of relational algebras [2] In fact, even if the overall justi cation for the study of the categories of spans have been their connections with relations [6,5] and it was already clear since [3] the one to one correspondence between unitary pretabular allegories [12] and cartesian bicategories) a careful analysis of these connections may result in a few surprises. In Section 2 we make the correspondence precise exploiting the notion of multirelation. For our purposes, the ....

C. Brown and A. Jerey. Allegories of circuits. In A. Nerode and Y. Matiyasevich, editors, Logic Foundations of Computer Science, volume 813 of Lect. Notes in Comp. Science, pages 56-68. Springer Verlag, 1994.

....sequences (r i ) i2N satisfying, for all i 2 N, the recurrence relation r i 2 = r i 1 r i . In Section 2.3 we will see that i o defines a homomorphism of bicategories. Relations of the form X N Y N provide a useful calculus for specifying (input output behaviours of) circuits. In fact, in [7] and [19] relations were used in this way to model circuits. Note, however, that the input output behaviours of circuits with input and output I (such as the example above involving the nor function) are trivial there are only two relations from I N to I N . More will be said on behaviours ....

Brown C, Jeffrey A, Allegories of circuits, in: Proc. Logical Foundations of Computer Science, St. Petersburg, 1994.

....syntax for diagrammatic presentations of, e.g. nets and circuits. The initial works can be considered those introducing the ownomial calculus [13,14] proposed as a calculus for owgraphs. Similar considerations arose independently also in di erent contexts, such as the allegories of circuits [3], based on the relational language ruby for circuit design [29] or the languages for (term) graph rewriting [10,11,20] and contextual net processes [22] or the calculi for concurrent and distributed processes [21,24,28] mostly arising from the work on pre monoidal categories [36] On the other ....

....supported by CNR Project Metodi per la Veri ca di Sistemi Eterogenei ; by MURST Project TOSCA; by Esprit Working Groups CONFER2 and COORDINA. by the need of modeling a particular graphical formalism, are semantic in nature. In general terms, they are tightly related, since it was already clear in [3] that there is a one to one correspondence between certain kind of allegories, called unitary pretabular allegories [18] and cartesian bicategories. It would then be important to identify a natural categorical structure that could be used as a semantic domain, still retaining suitable ....

C. Brown and A. Jerey. Allegories of circuits. In Logic in Computer Science. IEEE Computer Society Press, 1994.

....et cetera are modelled by Elgot automata processes in a category with sums. The algebra of such processes is that of monoidal bicategories equipped with an operation called feedback. Compact closed categories and Cartesian bicategories ( 6] have been used as a model for circuit design ( 11] [4]) as well as a paradigm for the semantics of computation ( 1] Due to the symmetry of the structures there investigated, these models, unlike the one here presented, are unable to treat the asymmetric nature of the roles of input and output in processes. After presenting some basic definitions in ....

....the circuits with input X and output Y that have the property: if (x i ) i2N R(y j ) j2N then there exists a behaviour of the circuit with corresponding input and output behaviours (x i ) i2N and (y j ) j2N . The category Rel has been used to model real circuits in both these ways in [11] and [4]. In this sense, the theory of circuits here developed is more concrete than theories using only locally ordered bicategories such as Rel. We wish to point out that though Rel models neither the internal state nor the dynamics of processes, it is still of interest since calculations there are ....

Brown C. and Jeffrey A., Allegories of circuits, in: Proc. Logical Foundations of Computer Science (St. Petersburg, 1994).

....[17] have developed a calculus of pictures, oriented towards circuit design. Their pictures are built up from basic cells and wires using sequential composition, intersection and reciprocation. They give a semantics to pictures in terms of relations, in a manner very similar to our approach. In [17, 18] it is shown that their calculus is complete in that two pictures are equivalent with respect to their transformation rules if and only if they represent the same relation for all interpretations of the basic cells; this proof proceeds by viewing pictures as arrows in a unitary pretabular allegory ....

Carolyn Brown and Alan Jeffrey. Allegories of circuits. In Third International Symposium, Logical Foundations of Computer Science, volume 813 of Lecture Notes in Computer Science, pages 56--68. Springer-Verlag, 1994.

....Fibonacci circuit described above then i o(F ) I R N is the subset of sequences (r i ) i2N satisfying, for all i 2 N, the recurrence relation r i 2 = r i 1 r i . Relations of the form X N Y N provide a useful calculus for specifying (input output behaviours of) circuits. In fact, in [7] and [18] relations were used in this way to model circuits. Note, however, that the i o behaviour of circuits with input and output I (such as the example above involving the nor function) are trivial as there are only two relations from I N to I N . More will be said on behaviours in the ....

Brown C, Jeffrey A, Allegories of circuits, in: Proc. Logical Foundations of Computer Science, St. Petersburg, 1994.

....[2] have developed a calculus of pictures, oriented towards circuit design. Their pictures are built up from basic cells and wires using sequential composition, intersection and reciprocation. They give a semantics to pictures in terms of relations, in a manner very similar to our approach. In [2, 3] it is shown that their calculus is complete in that two pictures are equivalent with respect to their transformation rules if and only if they represent the same relation for all interpretations of the basic cells; this proof proceeds by viewing pictures as arrows in a unitary pretabular allegory ....

Carolyn Brown and Alan Jeffrey. Allegories of circuits. In Third International Symposium, Logical Foundations of Computer Science, volume 813 of Lecture Notes in Computer Science, pages 56--68. Springer-Verlag, 1994.

....[2] have developed a calculus of pictures, oriented towards circuit design. Their pictures are built up from basic cells and wires using sequential composition, intersection and reciprocation. They give a semantics to pictures in terms of relations, in a manner very similar to our approach. In [2,3] it is shown that their calculus is complete in that two pictures are equivalent with respect to their transformation rules if and only if they rep17 resent the same relation for all interpretations of the basic cells; this proof proceeds by viewing pictures as arrows in a unitary pretabular ....

Carolyn Brown and Alan Jeffrey. Allegories of circuits. In Third International Symposium, Logical Foundations of Computer Science, volume 813 of Lecture Notes in Computer Science, pages 56--68. Springer-Verlag, 1994.

....wires, and an interface connect, which connects the process behaviours to the internal wires. connect t a r s r c . P 1 P 2 P n . Figure 3. 10: Model of a Process This graphical presentation of processes has been influenced by a graphical model for electronic circuits [BJ94]. We use ideas from the categorical presentation of this model of circuits to formalize our model of processes categorically. 3.3.2 The Model We will now formally define a category Proc of processes, as process behaviour plus interface information, as outlined in the previous section. The model ....

Carolyn Brown and Alan Jeffrey. Allegories of circuits. In Proc. LFCS, 1994.

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Carolyn Brown and Alan Jeffrey. Allegories of circuits. In Proc. LFCS 94, 1994.

....behaviours to the internal wires. Chapter 3. A Categorical Model for Typed Concurrent Programming Languages 40 connect t a r s r c . P 1 P 2 P n . Figure 3. 10: Model of a Process This graphical presentation of processes has been influenced by a graphical model for electronic circuits [BJ94]. We use ideas from the categorical presentation of this model of circuits to formalize our model of processes categorically. 3.3.2 The Model We will now formally define a category Proc of processes, as process behaviour plus interface information, as outlined in the previous section. The model ....

Carolyn Brown and Alan Jeffrey. Allegories of circuits. In Proc. LFCS 94, 1994.

....new wires and components) Two pictures are provably equivalent if and only if they are mutually homomorphic, which is if and only if they denote the same relation for any interpretation of their basic components. These results lead us to a simple decision procedure for equivalence of circuits [3], which has been implemented [9] Our results encourage the use of pictures in deriving circuits, by providing a formal foundation for that use. Pictures are easier and quicker to understand than syntactic terms, and so their use speeds up the process of circuit design. This paper illustrates ....

....two pictures P and P 0 denote the same relation whenever there are homomorphisms from P to P 0 and from P 0 to P; which is precisely when the pictures are provably equivalent using the axioms of a upa. Since any upa has a faithful representation in a power of Rel, the allegory of sets [3, 5], we obtain a completeness theorem stating that two pictures are provably equivalent if and only if, for every interpretation of their basic components, they denote the same relation. Soundness and completeness show that we can rely on pictures when deriving circuits. In the appendix, we picture ....

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Carolyn Brown and Alan Jeffrey, Allegories of Circuits, to appear in Proc. Symposium on Logical Foundations of Computer Science, St Petersburg, 1994.

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