Equational Axioms for Probabilistic Bisimilarity (Preliminary Report)

This paper gives an equational axiomatization of probabilistic bisimulation equivalence for a class of finite-state agents previously studied by Stark and Smolka ((2000) Proof, Language, and Interaction: Essays in Honour of Robin Milner, pp. 571-595). The axiomatization is obtained by extending the general axioms of iteration theories (or iteration algebras), which characterize the equational properties of the fixed point operator on (omega-)continuous or monotonic functions, with three axiom schemas that express laws that are specific to probabilistic bisimilarity. Hence probabilistic bisimilarity (over finite-state agents) has an equational axiomatization relative to iteration algebras

Continuous Additive Algebras and Injective Simulations of Synchronization Trees

The (in)equational properties of the least fixed point operation on(omega-)continuous functions on (omega-)complete partially ordered sets arecaptured by the axioms of (ordered) iteration algebras, or iterationtheories. We show that the inequational laws of the sum operation inconjunction with the least fixed point operation in continuous additivealgebras have a finite axiomatization over the inequations of orderediteration algebras. As a byproduct of this relative axiomatizability result, we obtain complete infinite inequational and finite implicationalaxiomatizations. Along the way of proving these results, we give a concrete description of the free algebras in the corresponding variety ofordered iteration algebras. This description uses injective simulations of regular synchronization trees. Thus, our axioms are also sound andcomplete for the injective simulation (resource bounded simulation) of(regular) processes.Keywords: equational logic, fixed points, synchronization trees, simulation

Iteration Algebras for UnQL Graphs and Completeness for Bisimulation

This paper shows an application of Bloom and Esik's iteration algebras to model graph data in a graph database query language. About twenty years ago, Buneman et al. developed a graph database query language UnQL on the top of a functional meta-language UnCAL for describing and manipulating graphs. Recently, the functional programming community has shown renewed interest in UnCAL, because it provides an efficient graph transformation language which is useful for various applications, such as bidirectional computation. However, no mathematical semantics of UnQL/UnCAL graphs has been developed. In this paper, we give an equational axiomatisation and algebraic semantics of UnCAL graphs. The main result of this paper is to prove that completeness of our equational axioms for UnCAL for the original bisimulation of UnCAL graphs via iteration algebras. Another benefit of algebraic semantics is a clean characterisation of structural recursion on graphs using free iteration algebra.Comment: In Proceedings FICS 2015, arXiv:1509.0282

An Inverse Method for Policy-Iteration Based Algorithms

We present an extension of two policy-iteration based algorithms on weighted graphs (viz., Markov Decision Problems and Max-Plus Algebras). This extension allows us to solve the following inverse problem: considering the weights of the graph to be unknown constants or parameters, we suppose that a reference instantiation of those weights is given, and we aim at computing a constraint on the parameters under which an optimal policy for the reference instantiation is still optimal. The original algorithm is thus guaranteed to behave well around the reference instantiation, which provides us with some criteria of robustness. We present an application of both methods to simple examples. A prototype implementation has been done

Automated verification of refinement laws

Demonic refinement algebras are variants of Kleene algebras. Introduced by von Wright as a light-weight variant of the refinement calculus, their intended semantics are positively disjunctive predicate transformers, and their calculus is entirely within first-order equational logic. So, for the first time, off-the-shelf automated theorem proving (ATP) becomes available for refinement proofs. We used ATP to verify a toolkit of basic refinement laws. Based on this toolkit, we then verified two classical complex refinement laws for action systems by ATP: a data refinement law and Back's atomicity refinement law. We also present a refinement law for infinite loops that has been discovered through automated analysis. Our proof experiments not only demonstrate that refinement can effectively be automated, they also compare eleven different ATP systems and suggest that program verification with variants of Kleene algebras yields interesting theorem proving benchmarks. Finally, we apply hypothesis learning techniques that seem indispensable for automating more complex proofs

Corecursive Algebras, Corecursive Monads and Bloom Monads

An algebra is called corecursive if from every coalgebra a unique coalgebra-to-algebra homomorphism exists into it. We prove that free corecursive algebras are obtained as coproducts of the terminal coalgebra (considered as an algebra) and free algebras. The monad of free corecursive algebras is proved to be the free corecursive monad, where the concept of corecursive monad is a generalization of Elgot's iterative monads, analogous to corecursive algebras generalizing completely iterative algebras. We also characterize the Eilenberg-Moore algebras for the free corecursive monad and call them Bloom algebras

Clifford Algebraic Remark on the Mandelbrot Set of Two--Component Number Systems

We investigate with the help of Clifford algebraic methods the Mandelbrot set over arbitrary two-component number systems. The complex numbers are regarded as operator spinors in D\times spin(2) resp. spin(2). The thereby induced (pseudo) normforms and traces are not the usual ones. A multi quadratic set is obtained in the hyperbolic case contrary to [1]. In the hyperbolic case a breakdown of this simple dynamics takes place.Comment: LaTeX, 27 pages, 6 fig. with psfig include