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Geometry of Interaction and Linear Combinatory Algebras
, 2000
"... this paper was quite di#erent, stemming from the axiomatics of categories of tangles (although the authors were aware of possible connections to iteration theories. In fact, similar axiomatics in the symmetric case, motivated by flowcharts and "flownomials" had been developed some years ea ..."
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this paper was quite di#erent, stemming from the axiomatics of categories of tangles (although the authors were aware of possible connections to iteration theories. In fact, similar axiomatics in the symmetric case, motivated by flowcharts and "flownomials" had been developed some years earlier by Stefanescu (Stefanescu 2000).) However, the first author realized, following a stimulating discussion with Gordon Plotkin, that traced monoidal categories provided a common denominator for the axiomatics of both the Girardstyle and AbramskyJagadeesanstyle versions of the Geometry of Interaction, at the basic level of the multiplicatives. This insight was presented in (Abramsky 1996), in which Girardstyle GoI was dubbed "particlestyle", since it concerns information particles or tokens flowing around a network, while the AbramskyJagadeesan style GoI was dubbed "wavestyle", since it concerns the evolution of a global information state or "wave". Formally, this distinction is based on whether the tensor product (i.e. the symmetric monoidal structure) in the underlying category is interpreted as a coproduct (particle style) or as a product (wave style). This computational distinction between coproduct and product interpretations of the same underlying network geometry turned out to have been partially anticipated, in a rather di#erent context, in a pioneering paper by E. S. Bainbridge (Bainbridge 1976), as observed by Dusko Pavlovic. These two forms of interpretation, and ways of combining them, have also been studied recently in (Stefanescu 2000). He uses the terminology "additive" for coproductbased (i.e. our "particlestyle") and "multiplicative" for productbased (i.e. our "wavestyle"); this is not suitable for our purposes, because of the clash with Linear Logic term...
ON THE FUNCTOR ℓ 2
"... and both categories are enriched over algebraic domains. The functor preserves daggers, monoidal structures, enrichment, and various (co)limits, but has no adjoints. Up to unitaries, its direct image consists precisely of the partial isometries, but its essential image consists of all continuous lin ..."
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and both categories are enriched over algebraic domains. The functor preserves daggers, monoidal structures, enrichment, and various (co)limits, but has no adjoints. Up to unitaries, its direct image consists precisely of the partial isometries, but its essential image consists of all continuous linear maps between Hilbert spaces. 1.
Symmetries and transitions of bounded Turing machines
, 2008
"... We consider the structures given by repeatedly generalising the definition of finite state automata by symmetry considerations, and constructing analogues of the transition monoid at each step. This approach first gives us nondeterministic automata, then (nondeterministic) twoway automata and bou ..."
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We consider the structures given by repeatedly generalising the definition of finite state automata by symmetry considerations, and constructing analogues of the transition monoid at each step. This approach first gives us nondeterministic automata, then (nondeterministic) twoway automata and bounded Turing machines — that is, Turing machines where the read / write head is unable to move past the end of the input word. In the case of twoway automata, the transition monoids generalise to endomorphism monoids in compact closed categories. These use Girard’s resolution formula (from the Geometry of Interaction representation of linear logic) to construct the images of singleton words. In the case of bounded Turing machines, the transition homomorphism generalises to a monoid homomorphism from N to a monoid constructed from the union of endomorphism monoids of a compact closed category, together with an appropriate composition. These use Girard’s execution formula (also from the Geometry of Interaction representation of linear logic) to construct images of singletons. 1
The Categorical Theory Of SelfSimilarity
, 1999
"... . We demonstrate how the identity N\Omega N = N in a monoidal category allows us to construct a functor from the full subcategory generated by N and\Omega to the endomorphism monoid of the object N . This provides a categorical foundation for oneobject analogues of the symmetric monoidal c ..."
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. We demonstrate how the identity N\Omega N = N in a monoidal category allows us to construct a functor from the full subcategory generated by N and\Omega to the endomorphism monoid of the object N . This provides a categorical foundation for oneobject analogues of the symmetric monoidal categories used by J.Y. Girard in his Geometry of Interaction series of papers, and explicitly described in terms of inverse semigroup theory in [6, 11]. This functor also allows the construction of oneobject analogues of other categorical structures. We give the example of oneobject analogues of the categorical trace, and compact closedness. Finally, we demonstrate how the categorical theory of selfsimilarity can be related to the algebraic theory (as presented in [11]), and Girard's dynamical algebra, by considering oneobject analogues of projections and inclusions. 1. Introduction It is wellknown, [12], that any oneobject monoidal category is an abelian monoid with respect ...
A class of subgroups of Thompson's group V
, 2007
"... We prove that the subgroups of the Thompson group V which are closed under a binary operation introduced by K. S. Brown are in bijective correspondence with a class of varieties of algebras with a single binary operation. In addition, we prove that there are exactly three such proper subgroups which ..."
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We prove that the subgroups of the Thompson group V which are closed under a binary operation introduced by K. S. Brown are in bijective correspondence with a class of varieties of algebras with a single binary operation. In addition, we prove that there are exactly three such proper subgroups which contain the Thompson group F.