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A Relational Model of NonDeterministic Dataflow
 In CONCUR'98, volume 1466 of LNCS
, 1998
"... . We recast dataflow in a modern categorical light using profunctors as a generalisation of relations. The well known causal anomalies associated with relational semantics of indeterminate dataflow are avoided, but still we preserve much of the intuitions of a relational model. The development fits ..."
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Cited by 27 (13 self)
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. We recast dataflow in a modern categorical light using profunctors as a generalisation of relations. The well known causal anomalies associated with relational semantics of indeterminate dataflow are avoided, but still we preserve much of the intuitions of a relational model. The development fits with the view of categories of models for concurrency and the general treatment of bisimulation they provide. In particular it fits with the recent categorical formulation of feedback using traced monoidal categories. The payoffs are: (1) explicit relations to existing models and semantics, especially the usual axioms of monotone IO automata are read off from the definition of profunctors, (2) a new definition of bisimulation for dataflow, the proof of the congruence of which benefits from the preservation properties associated with open maps and (3) a treatment of higherorder dataflow as a biproduct, essentially by following the geometry of interaction programme. 1 Introduction A fundament...
Shadows and traces in bicategories
 J. Homotopy Relat. Struct
"... Abstract. Traces in symmetric monoidal categories are wellknown and have many applications; for instance, their functoriality directly implies the Lefschetz fixed point theorem. However, for some applications, such as generalizations of the Lefschetz theorem, one needs “noncommutative ” traces, s ..."
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Abstract. Traces in symmetric monoidal categories are wellknown and have many applications; for instance, their functoriality directly implies the Lefschetz fixed point theorem. However, for some applications, such as generalizations of the Lefschetz theorem, one needs “noncommutative ” traces, such as the HattoriStallings trace for modules over noncommutative rings. In this paper we study a generalization of the symmetric monoidal trace which applies to noncommutative situations; its context is a bicategory equipped with an extra structure called a “shadow. ” In particular, we prove its functoriality and 2functoriality, which are essential to its applications in fixedpoint theory. Throughout we make use of an appropriate “cylindrical ” type of string
From coalgebraic to monoidal traces
 Coalgebraic Methods in Computer Science (CMCS 2010), volume 264 of Elect. Notes in Theor. Comp. Sci
, 2010
"... The main result of this paper shows how coalgebraic traces, in suitable Kleisli categories, give rise to traced monoidal structure in those Kleisli categories, with finite coproducts as monoidal structure. At the heart of the matter lie partially additive monads inducing partially additive structure ..."
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Cited by 8 (2 self)
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The main result of this paper shows how coalgebraic traces, in suitable Kleisli categories, give rise to traced monoidal structure in those Kleisli categories, with finite coproducts as monoidal structure. At the heart of the matter lie partially additive monads inducing partially additive structure in their Kleisli categories. By applying the standard “Int ” construction one obtains compact closed categories for “bidirectional monadic computation”. 1
Traced Premonoidal Categories
, 1999
"... Motivated by some examples from functional programming, we propose a generalization of the notion of trace to symmetric premonoidal categories and of Conway operators to Freyd categories. We show that in a Freyd category, these notions are equivalent, generalizing a wellknown theorem relating trace ..."
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Cited by 7 (0 self)
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Motivated by some examples from functional programming, we propose a generalization of the notion of trace to symmetric premonoidal categories and of Conway operators to Freyd categories. We show that in a Freyd category, these notions are equivalent, generalizing a wellknown theorem relating traces and Conway operators in cartesian categories.
Memoryful Geometry of Interaction From Coalgebraic Components to Algebraic Effects
"... Girard’s Geometry of Interaction (GoI) is interaction based semantics of linear logic proofs and, via suitable translations, of functional programs in general. Its mathematical cleanness identifies essential structures in computation; moreover its use as a compilation technique from programs to s ..."
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Cited by 4 (3 self)
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Girard’s Geometry of Interaction (GoI) is interaction based semantics of linear logic proofs and, via suitable translations, of functional programs in general. Its mathematical cleanness identifies essential structures in computation; moreover its use as a compilation technique from programs to state machines—“GoI implementation, ” so to speak—has been worked out by Mackie, Ghica and others. In this paper, we develop Abramsky’s idea of resumption based GoI systematically into a generic framework that accounts for computational effects (nondeterminism, probability, exception, global states, interactive I/O, etc.). The framework is categorical: Plotkin & Power’s algebraic operations provide an interface to computational effects; the framework is built on the categorical axiomatization of GoI by Abramsky, Haghverdi and Scott; and, by use of the coalgebraic formalization of component calculus, we describe explicit construction of state machines as interpretations of functional programs. The resulting interpretation is shown to be sound with respect to equations between algebraic operations, as well as to Moggi’s equations for the computational lambda calculus. We illustrate the construction by concrete examples.
A semantical approach to equilibria and rationality
, 905
"... ”An equilibrium does not appear because agents are rational, but rather agents appear to be rational because an equilibrium has been reached.[...] The task for game theory is to formulate a notion of rationality.” Larry Samuelson [20, p. 3] Abstract. Game theoretic equilibria are mathematical expres ..."
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”An equilibrium does not appear because agents are rational, but rather agents appear to be rational because an equilibrium has been reached.[...] The task for game theory is to formulate a notion of rationality.” Larry Samuelson [20, p. 3] Abstract. Game theoretic equilibria are mathematical expressions of rationality. Rational agents are used to model not only humans and their software representatives, but also organisms, populations, species and genes, interacting with each other and with the environment. Rational behaviors are achieved not only through conscious reasoning, but also through spontaneous stabilization at equilibrium points. Formal theories of rationality are usually guided by informal intuitions, which are acquired by observing some concrete economic, biological, or network processes. Treating such processes as instances of computation, we reconstruct and refine some basic notions of equilibrium and rationality from the some basic structures of computation. It is, of course, well known that equilibria arise as fixed points; the point is that semantics of computation of fixed points seems to be providing novel methods, algebraic and coalgebraic, for reasoning about them. 1
Monoidal indeterminates and categories of possible worlds
 In Proc. of MFPS XXV
, 2009
"... Given any symmetric monoidal category C, a small symmetric monoidal category Σ and a strong monoidal functor j:Σ C, we construct C[x: jΣ], the polynomial category with a system of (freely adjoined) monoidal indeterminates x: I j(w), natural in w ∈ Σ. As a special case, we construct the free coaffin ..."
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Given any symmetric monoidal category C, a small symmetric monoidal category Σ and a strong monoidal functor j:Σ C, we construct C[x: jΣ], the polynomial category with a system of (freely adjoined) monoidal indeterminates x: I j(w), natural in w ∈ Σ. As a special case, we construct the free coaffine category (symmetric monoidal category with initial unit) on a given small symmetric monoidal category. We then exhibit all the known categories of “possible worlds ” used to treat languages that allow for dynamic creation of “new ” variables, locations, or names as instances of this construction and explicate their associated universality properties. As an application of the resulting characterisation of O(W), Oles’s category of possible worlds, we present an O(W)indexed Lawvere theory of manysorted storage, generalizing the singlesorted one introduced by J. Power, and we describe explicitly an associated
and uniform parameterized fixpoint operators
"... A note on strong dinaturality, initial algebras ..."
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Language, Theory
"... We introduce a lambda calculus TFG for transformations of finite graphs by generalizing and extending an existing calculus UnCAL. Whereas UnCAL can treat only unordered graphs, TFG can treat a variety of graph models: directed edgelabeled graphs whose branch styles are represented by monads T. For ..."
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We introduce a lambda calculus TFG for transformations of finite graphs by generalizing and extending an existing calculus UnCAL. Whereas UnCAL can treat only unordered graphs, TFG can treat a variety of graph models: directed edgelabeled graphs whose branch styles are represented by monads T. For example, TFG can treat unordered graphs, ordered graphs, weighted graphs, probability graphs, and so on, by using the powerset monad, list monad, multiset monad, probability monad, respectively. In TFG, graphs are considered as extension of tree data structures, i.e. as infinite (regular) trees, so the semantics is given with bisimilarity. A remarkable feature of UnCAL and TFG is structural recursion for graphs, which gives a systematic programming basis like that for trees. Despite the nonwellfoundedness of graphs, by suitably restricting the structural recursion, UnCAL and TFG ensures that there is a termination property and that all transformations preserve the finiteness of the graphs. The structural recursion is defined in a "divideandaggregate " way; "aggregation " is done by connecting graphs with "edges, which are similar to the "transitions of automata. We give a suitable general definition of bisimilarity, taking account of "edges; then we show that the structural recursion is well defined with respect to the bisimilarity.
Categories of Possible Worlds
, 2007
"... freely adjoining to C a system of monoidal indeterminates for every element of W. It is then shown that all of the categories of “possible worlds ” used to treat languages that allow for dynamic creation of “new ” variables, locations, or names are instances of this construction and hence have appro ..."
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freely adjoining to C a system of monoidal indeterminates for every element of W. It is then shown that all of the categories of “possible worlds ” used to treat languages that allow for dynamic creation of “new ” variables, locations, or names are instances of this construction and hence have appropriate universality properties. 1.