A Probabilistic Kleene Theorem

International audienceWe provide a Kleene Theorem for (Rabin) probabilistic automata over finite words. Probabilistic automata generalize deterministic finite automata and assign to a word an acceptance probability. We provide probabilistic expressions with probabilistic choice, guarded choice, concatenation, and a star operator. We prove that probabilistic expressions and probabilistic automata are expressively equivalent. Our result actually extends to two-way probabilistic automata with pebbles and corresponding expressions

Probabilistic Logic, Probabilistic Regular Expressions, and Constraint Temporal Logic

The classic theorems of Büchi and Kleene state the expressive equivalence of finite automata to monadic second order logic and regular expressions, respectively. These fundamental results enjoy applications in nearly every field of theoretical computer science. Around the same time as Büchi and Kleene, Rabin investigated probabilistic finite automata. This equally well established model has applications ranging from natural language processing to probabilistic model checking. Here, we give probabilistic extensions Büchi\\\''s theorem and Kleene\\\''s theorem to the probabilistic setting. We obtain a probabilistic MSO logic by adding an expected second order quantifier. In the scope of this quantifier, membership is determined by a Bernoulli process. This approach turns out to be universal and is applicable for finite and infinite words as well as for finite trees. In order to prove the expressive equivalence of this probabilistic MSO logic to probabilistic automata, we show a Nivat-theorem, which decomposes a recognisable function into a regular language, homomorphisms, and a probability measure. For regular expressions, we build upon existing work to obtain probabilistic regular expressions on finite and infinite words. We show the expressive equivalence between these expressions and probabilistic Muller-automata. To handle Muller-acceptance conditions, we give a new construction from probabilistic regular expressions to Muller-automata. Concerning finite trees, we define probabilistic regular tree expressions using a new iteration operator, called infinity-iteration. Again, we show that these expressions are expressively equivalent to probabilistic tree automata. On a second track of our research we investigate Constraint LTL over multidimensional data words with data values from the infinite tree. Such LTL formulas are evaluated over infinite words, where every position possesses several data values from the infinite tree. Within Constraint LTL on can compare these values from different positions. We show that the model checking problem for this logic is PSPACE-complete via investigating the emptiness problem of Constraint Büchi automata

Compositional bisimulation metric reasoning with Probabilistic Process Calculi

We study which standard operators of probabilistic process calculi allow for compositional reasoning with respect to bisimulation metric semantics. We argue that uniform continuity (generalizing the earlier proposed property of non-expansiveness) captures the essential nature of compositional reasoning and allows now also to reason compositionally about recursive processes. We characterize the distance between probabilistic processes composed by standard process algebra operators. Combining these results, we demonstrate how compositional reasoning about systems specified by continuous process algebra operators allows for metric assume-guarantee like performance validation

Weak bisimulation for coalgebras over order enriched monads

The paper introduces the notion of a weak bisimulation for coalgebras whose type is a monad satisfying some extra properties. In the first part of the paper we argue that systems with silent moves should be modelled coalgebraically as coalgebras whose type is a monad. We show that the visible and invisible part of the functor can be handled internally inside a monadic structure. In the second part we introduce the notion of an ordered saturation monad, study its properties, and show that it allows us to present two approaches towards defining weak bisimulation for coalgebras and compare them. We support the framework presented in this paper by two main examples of models: labelled transition systems and simple Segala systems.Comment: 44 page

Guarded Kleene Algebra with Tests: Verification of Uninterpreted Programs in Nearly Linear Time (Invited Talk)

Guarded Kleene Algebra with Tests (GKAT) is a variation on Kleene Algebra with Tests (KAT) that arises by restricting the union (+) and iteration (*) operations from KAT to predicate-guarded versions. We develop the (co)algebraic theory of GKAT and show how it can be efficiently used to reason about imperative programs. In contrast to KAT, whose equational theory is PSPACE-complete, we show that the equational theory of GKAT is (almost) linear time. We also provide a full Kleene theorem and prove completeness for an analogue of Salomaa\u27s axiomatization of Kleene Algebra. We will also discuss how this result has practical implications in the verification of programs, with examples from network and probabilistic programming. This is joint work with Nate Foster, Justin Hsu, Tobias Kappe, Dexter Kozen, and Steffen Smolka

Probabilistic Rely-guarantee Calculus

Jones' rely-guarantee calculus for shared variable concurrency is extended to include probabilistic behaviours. We use an algebraic approach which combines and adapts probabilistic Kleene algebras with concurrent Kleene algebra. Soundness of the algebra is shown relative to a general probabilistic event structure semantics. The main contribution of this paper is a collection of rely-guarantee rules built on top of that semantics. In particular, we show how to obtain bounds on probabilities by deriving rely-guarantee rules within the true-concurrent denotational semantics. The use of these rules is illustrated by a detailed verification of a simple probabilistic concurrent program: a faulty Eratosthenes sieve.Comment: Preprint submitted to TCS-QAP

Computing the Least Fixed Point of Positive Polynomial Systems

We consider equation systems of the form X_1 = f_1(X_1, ..., X_n), ..., X_n = f_n(X_1, ..., X_n) where f_1, ..., f_n are polynomials with positive real coefficients. In vector form we denote such an equation system by X = f(X) and call f a system of positive polynomials, short SPP. Equation systems of this kind appear naturally in the analysis of stochastic models like stochastic context-free grammars (with numerous applications to natural language processing and computational biology), probabilistic programs with procedures, web-surfing models with back buttons, and branching processes. The least nonnegative solution mu f of an SPP equation X = f(X) is of central interest for these models. Etessami and Yannakakis have suggested a particular version of Newton's method to approximate mu f. We extend a result of Etessami and Yannakakis and show that Newton's method starting at 0 always converges to mu f. We obtain lower bounds on the convergence speed of the method. For so-called strongly connected SPPs we prove the existence of a threshold k_f such that for every i >= 0 the (k_f+i)-th iteration of Newton's method has at least i valid bits of mu f. The proof yields an explicit bound for k_f depending only on syntactic parameters of f. We further show that for arbitrary SPP equations Newton's method still converges linearly: there are k_f>=0 and alpha_f>0 such that for every i>=0 the (k_f+alpha_f i)-th iteration of Newton's method has at least i valid bits of mu f. The proof yields an explicit bound for alpha_f; the bound is exponential in the number of equations, but we also show that it is essentially optimal. Constructing a bound for k_f is still an open problem. Finally, we also provide a geometric interpretation of Newton's method for SPPs.Comment: This is a technical report that goes along with an article to appear in SIAM Journal on Computing

Computing Least Fixed Points of Probabilistic Systems of Polynomials

We study systems of equations of the form X1 = f1(X1, ..., Xn), ..., Xn = fn(X1, ..., Xn), where each fi is a polynomial with nonnegative coefficients that add up to 1. The least nonnegative solution, say mu, of such equation systems is central to problems from various areas, like physics, biology, computational linguistics and probabilistic program verification. We give a simple and strongly polynomial algorithm to decide whether mu=(1, ..., 1) holds. Furthermore, we present an algorithm that computes reliable sequences of lower and upper bounds on mu, converging linearly to mu. Our algorithm has these features despite using inexact arithmetic for efficiency. We report on experiments that show the performance of our algorithms.Comment: Published in the Proceedings of the 27th International Symposium on Theoretical Aspects of Computer Science (STACS). Technical Report is also available via arxiv.or