The Power of Convex Algebras

Probabilistic automata (PA) combine probability and nondeterminism. They can be given different semantics, like strong bisimilarity, convex bisimilarity, or (more recently) distribution bisimilarity. The latter is based on the view of PA as transformers of probability distributions, also called belief states, and promotes distributions to first-class citizens. We give a coalgebraic account of the latter semantics, and explain the genesis of the belief-state transformer from a PA. To do so, we make explicit the convex algebraic structure present in PA and identify belief-state transformers as transition systems with state space that carries a convex algebra. As a consequence of our abstract approach, we can give a sound proof technique which we call bisimulation up-to convex hull.Comment: Full (extended) version of a CONCUR 2017 paper, to be submitted to LMC

Generic Trace Semantics via Coinduction

Trace semantics has been defined for various kinds of state-based systems, notably with different forms of branching such as non-determinism vs. probability. In this paper we claim to identify one underlying mathematical structure behind these "trace semantics," namely coinduction in a Kleisli category. This claim is based on our technical result that, under a suitably order-enriched setting, a final coalgebra in a Kleisli category is given by an initial algebra in the category Sets. Formerly the theory of coalgebras has been employed mostly in Sets where coinduction yields a finer process semantics of bisimilarity. Therefore this paper extends the application field of coalgebras, providing a new instance of the principle "process semantics via coinduction."Comment: To appear in Logical Methods in Computer Science. 36 page

Free Steiner loops

A Steiner loop, or a sloop, is a grupoid (L; · ,1), where · is a binary operation and 1 is a constant, satisfying the identities 1 · x = x, x · y = y · x, x · (x · y) = y. There is a one-to-one correspondence between Steiner triple systems and finite sloops. Two constructions of free objects in the variety of sloops are presented in this paper. They both allow recursive construction of a free sloop with a free base X, provided that X is recursively defined set. The main results besides the constructions are: Each subsloop of a free sloop is free two. A free sloop S with a free finite bases X, |X| ≥ 3, has a free subsloop with a free base of any finite cardinality and a free subsloop with a free base of cardinality ω as well; also S has a (non free) base of any finite cardinality k ≥ |X|. We also show that the word problem for the variety of sloops is solvable, due to embedding property

Free Steiner loops

A Steiner loop, or a sloop, is a grupoid (L; · ,1), where · is a binary operation and 1 is a constant, satisfying the identities 1 · x = x, x · y = y · x, x · (x · y) = y. There is a one-to-one correspondence between Steiner triple systems and finite sloops. Two constructions of free objects in the variety of sloops are presented in this paper. They both allow recursive construction of a free sloop with a free base X, provided that X is recursively defined set. The main results besides the constructions are: Each subsloop of a free sloop is free two. A free sloop S with a free finite bases X, |X| ≥ 3, has a free subsloop with a free base of any finite cardinality and a free subsloop with a free base of cardinality ω as well; also S has a (non free) base of any finite cardinality k ≥ |X|. We also show that the word problem for the variety of sloops is solvable, due to embedding property

Termination in Convex Sets of Distributions

Convex algebras, also called (semi)convex sets, are at the heart of modelling probabilistic systems including probabilistic automata. Abstractly, they are the Eilenberg-Moore algebras of the finitely supported distribution monad. Concretely, they have been studied for decades within algebra and convex geometry. In this paper we study the problem of extending a convex algebra by a single point. Such extensions enable the modelling of termination in probabilistic systems. We provide a full description of all possible extensions for a particular class of convex algebras: For a fixed convex subset D of a vector space satisfying additional technical condition, we consider the algebra of convex subsets of D. This class contains the convex algebras of convex subsets of distributions, modelling (nondeterministic) probabilistic automata. We also provide a full description of all possible extensions for the class of free convex algebras, modelling fully probabilistic systems. Finally, we show that there is a unique functorial extension, the so-called black-hole extension