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Towards a typed geometry of interaction
, 2005
"... We introduce a typed version of Girard’s Geometry of Interaction, called Multiobject GoI (MGoI) semantics. We give an MGoI interpretation for multiplicative linear logic (MLL) without units which applies to new kinds of models, including finite dimensional vector spaces. For MGoI (i) we develop a v ..."
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Cited by 7 (2 self)
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We introduce a typed version of Girard’s Geometry of Interaction, called Multiobject GoI (MGoI) semantics. We give an MGoI interpretation for multiplicative linear logic (MLL) without units which applies to new kinds of models, including finite dimensional vector spaces. For MGoI (i) we develop a version of partial traces and trace ideals (related to previous work of Abramsky, Blute, and Panangaden); (ii) we do not require the existence of a reflexive object for our interpretation (the original GoI 1 and 2 were untyped and hence involved a bureaucracy of domain equation isomorphisms); (iii) we introduce an abstract notion of orthogonality (related to work of Hyland and Schalk) and use this to develop a version of Girard’s theory of types, datum and algorithms in our setting, (iv) we prove appropriate Soundness and Completeness Theorems for our interpretations in partially traced categories with orthogonality; (v) we end with an application to completeness of (the original) untyped GoI in a unique decomposition category.
Čirovič, On some equivalence notions of synchronous systems
 Proceedings, 11th International Conference on Automata and Formal Languages, Dogogókő, Hungary
, 2005
"... An important optimization tool in the design of synchronous systems is retiming, which in many cases allows a significant reduction in the length of the systems ’ clock period. Even though the internal structure of systems changes upon retiming, their inputoutput behavior remains essentially the sa ..."
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An important optimization tool in the design of synchronous systems is retiming, which in many cases allows a significant reduction in the length of the systems ’ clock period. Even though the internal structure of systems changes upon retiming, their inputoutput behavior remains essentially the same. The original system and the one after the retiming can simulate each other in a suitable way. The equivalence notion arising from this kind of mutual simulation is called simulation equivalence, and the aim of this paper is to characterize simulation equivalence in an algebraic setting. It is shown that simulation equivalence is a congruence relation of the algebra of synchronous schemes, and that this congruence is the smallest one containing retiming equivalence and finitary strong equivalence. An axiomatization of these equivalences is presented in the general framework of strictly monoidal categories with feedback. 1
The parallel composition of processes
, 904
"... Abstract. We suggest that the canonical parallel operation of processes is composition in a wellsupported compact closed category of spans of reflexive graphs. We present the parallel operations of classical process algebras as derived operations arising from monoid objects in such a ..."
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Abstract. We suggest that the canonical parallel operation of processes is composition in a wellsupported compact closed category of spans of reflexive graphs. We present the parallel operations of classical process algebras as derived operations arising from monoid objects in such a
TANGLED CIRCUITS
"... Abstract. We consider commutative Frobenius algebras in braided strict monoidal categories in the study of the circuits and communicating systems which occur in Computer Science, including circuits in which the wires are tangled. We indicate also some possible novel geometric interest in such algebr ..."
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Abstract. We consider commutative Frobenius algebras in braided strict monoidal categories in the study of the circuits and communicating systems which occur in Computer Science, including circuits in which the wires are tangled. We indicate also some possible novel geometric interest in such algebras. For example, we show how Armstrong’s description ([1, 2]) of knot colourings and knot groups fit into this context. 1.
unknown title
, 2009
"... Cospans and spans of graphs: a categorical algebra for the sequential and parallel composition of discrete systems ..."
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Cospans and spans of graphs: a categorical algebra for the sequential and parallel composition of discrete systems
Simulation equivalence of automata and circuits
"... Automata over a symmetric monoidal category M are introduced, and a multistep simulation is defined among such automata. The collection of Mautomata is given the structure of a 2category on the same objects as M, in which the vertical structure is determined by groups of indistinguishable simula ..."
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Automata over a symmetric monoidal category M are introduced, and a multistep simulation is defined among such automata. The collection of Mautomata is given the structure of a 2category on the same objects as M, in which the vertical structure is determined by groups of indistinguishable simulations. TwoMautomata are called simulation equivalent if they are connected by an isomorphism of 2cells in this 2category. It is shown that the category of simulation equivalent Mautomata is monoidal, and it satisfies all the axioms of traced monoidal categories, except the one that explicitly kills the delay. 1
Blockedbraid Groups
, 2014
"... We introduce and study a family of groupsBBn, called the blockedbraid groups, which are quotients of Artin’s braid groups Bn, and have the corresponding symmetric groups Σn as quotients. They are defined by adding a certain class of geometrical modifications to braids. They arise in the study of c ..."
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We introduce and study a family of groupsBBn, called the blockedbraid groups, which are quotients of Artin’s braid groups Bn, and have the corresponding symmetric groups Σn as quotients. They are defined by adding a certain class of geometrical modifications to braids. They arise in the study of commutative Frobenius algebras and tangle algebras in braided strict monoidal categories. A fundamental equation true in BBn is Dirac’s Belt Trick; that torsion through 4pi is equal to the identity. We show that BBn is finite for n = 1, 2 and 3 but infinite for n> 3. 1 ar X iv