Eilenberg–Moore algebras for stochastic relations

IFIP Advances in Information and Communication Technology

2010

This paper describes some basic relationships between mathematical structures that are relevant in quantum logic and probability, namely convex sets, effect algebras, and a new class of functors that we call 'convex functors'; they include what are usually called probability distribution functors. These relationships take the form of three adjunctions. Two of these three are 'dual' adjunctions for convex sets, one time with the Boolean truth values {0, 1} as dualising object, and one time with the probablity values [0,1]. The third adjunction is between effect algebras and convex functors.

“…It goes back to [28] and also applies to compact Hausdorff spaces [20] or Polish spaces [6]. For convenience, a proof sketch is included.…”

Section: Convex Setsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Convexity, Duality and Effects

IFIP Advances in Information and Communication Technology

2010

“…FINAL REMARKS Somewhat remarkably, the proof of the adjunction EMod op EM(G) in Theorem 19 does not require a precise characterisation of the category of algebras EM(G) of the Giry monad. One such characterisation is elaborated in [24]. A closer connection to the category of convex compact Hausdorff spaces, used in [23], see als [25], will be elaborated in an extended version of the current paper.…”

Section: Predicates and Statesmentioning

confidence: 99%

Measurable Spaces and Their Effect Logic

2013 28th Annual ACM/IEEE Symposium on Logic in Computer Science

2013

Abstract-So-called effect algebras and modules are basic mathematical structures that were first identified in mathematical physics, for the study of quantum logic and quantum probability. They incorporate a double negation law p ⊥⊥ = p. Since then it has been realised that these effect structures form a useful abstraction that covers not only quantum logic, but also Boolean logic and probabilistic logic. Moreover, the duality between effect and convex structures lies at the heart of the duality between predicates and states. These insights are leading to a uniform framework for the semantics of computation and logic. This framework has been elaborated elsewhere for settheoretic, discrete probabilistic, and quantum computation. Here the missing case of continuous probability is shown to fit in the same uniform framework. On a technical level, this involves an investigation of the logical aspects of the Giry monad on measurable spaces and of Lebesgue integration.

“…is the category PCA of positive convex algebras and convex maps [11]. We instantiate the functor M to B = 1 and the subprobability distribution monad D, and get…”

Section: Generative Probabilistic Systemsmentioning

confidence: 99%

Trace Semantics via Determinization

Coalgebraic Methods in Computer Science

Silva

Sokolova

2012

Abstract. This paper takes a fresh look at the topic of trace semantics in the theory of coalgebras. The first development of coalgebraic trace semantics used final coalgebras in Kleisli categories, stemming from an initial algebra in the underlying category. This approach requires some non-trivial assumptions, like dcpo enrichment, which do not always hold, even in cases where one can reasonably speak of traces (like for weighted automata). More recently, it has been noticed that trace semantics can also arise by first performing a determinization construction. In this paper, we develop a systematic approach, in which the two approaches correspond to different orders of composing a functor and a monad, and accordingly, to different distributive laws. The relevant final coalgebra that gives rise to trace semantics does not live in a Kleisli category, but more generally, in a category of Eilenberg-Moore algebras. In order to exploit its finality, we identify an extension operation, that changes the state space of a coalgebra into a free algebra, which abstractly captures determinization of automata. Notably, we show that the two different views on trace semantics are equivalent, in the examples where both approaches are applicable.