Probabilistic Semantics: Metric and Logical Character\ua8ations for Nondeterministic Probabilistic Processes

In this thesis we focus on processes with nondeterminism and probability in the PTS model, and we propose novel techniques to study their semantics, in terms of both classic behavioral relations and the more recent behavioral metrics. Firstly, we propose a method for decomposing modal formulae in a probabilistic extension of the Hennessy-Milner logic. This decomposition method allows us to derive the compositional properties of probabilistic (bi)simulations. Then, we propose original notions of metrics measuring the disparities in the behavior of processes with respect to (decorated) trace and testing semantics. To capture the differences in the expressive power of the metrics we order them by the relation `makes processes further than'. Thus, we obtain the first spectrum of behavioral metrics on the PTS model. From this spectrum we derive an analogous one for the kernels of the metrics, ordered by the relation `makes strictly less identification than'. Finally, we introduce a novel technique for the logical characterization of both behavioral metrics and their kernels, based on the notions of mimicking formula and distance on formulae. This kind of characterization allows us to obtain the first example of a spectrum of distances on processes obtained directly from logics. Moreover, we show that the kernels of the metrics can be characterized by simply comparing the mimicking formulae of processes

Calibrating Generative Models: The Probabilistic Chomsky-Schützenberger Hierarchy

A probabilistic Chomsky–Schützenberger hierarchy of grammars is introduced and studied, with the aim of understanding the expressive power of generative models. We offer characterizations of the distributions definable at each level of the hierarchy, including probabilistic regular, context-free, (linear) indexed, context-sensitive, and unrestricted grammars, each corresponding to familiar probabilistic machine classes. Special attention is given to distributions on (unary notations for) positive integers. Unlike in the classical case where the "semi-linear" languages all collapse into the regular languages, using analytic tools adapted from the classical setting we show there is no collapse in the probabilistic hierarchy: more distributions become definable at each level. We also address related issues such as closure under probabilistic conditioning

Probabilistic Semantics: Metric and Logical Character¨ations for Nondeterministic Probabilistic Processes

In this thesis we focus on processes with nondeterminism and probability in the PTS model, and we propose novel techniques to study their semantics, in terms of both classic behavioral relations and the more recent behavioral metrics. Firstly, we propose a method for decomposing modal formulae in a probabilistic extension of the Hennessy-Milner logic. This decomposition method allows us to derive the compositional properties of probabilistic (bi)simulations. Then, we propose original notions of metrics measuring the disparities in the behavior of processes with respect to (decorated) trace and testing semantics. To capture the differences in the expressive power of the metrics we order them by the relation `makes processes further than'. Thus, we obtain the first spectrum of behavioral metrics on the PTS model. From this spectrum we derive an analogous one for the kernels of the metrics, ordered by the relation `makes strictly less identification than'. Finally, we introduce a novel technique for the logical characterization of both behavioral metrics and their kernels, based on the notions of mimicking formula and distance on formulae. This kind of characterization allows us to obtain the first example of a spectrum of distances on processes obtained directly from logics. Moreover, we show that the kernels of the metrics can be characterized by simply comparing the mimicking formulae of processes

kk-Schur functions and affine Schubert calculus

This book is an exposition of the current state of research of affine Schubert calculus and $k$-Schur functions. This text is based on a series of lectures given at a workshop titled "Affine Schubert Calculus" that took place in July 2010 at the Fields Institute in Toronto, Ontario. The story of this research is told in three parts: 1. Primer on $k$-Schur Functions 2. Stanley symmetric functions and Peterson algebras 3. Affine Schubert calculusComment: 213 pages; conference website: http://www.fields.utoronto.ca/programs/scientific/10-11/schubert/, updates and corrections since v1. This material is based upon work supported by the National Science Foundation under Grant No. DMS-065264

Finite-State Abstractions for Probabilistic Computation Tree Logic

Probabilistic Computation Tree Logic (PCTL) is the established temporal logic for probabilistic verification of discrete-time Markov chains. Probabilistic model checking is a technique that verifies or refutes whether a property specified in this logic holds in a Markov chain. But Markov chains are often infinite or too large for this technique to apply. A standard solution to this problem is to convert the Markov chain to an abstract model and to model check that abstract model. The problem this thesis therefore studies is whether or when such finite abstractions of Markov chains for model checking PCTL exist. This thesis makes the following contributions. We identify a sizeable fragment of PCTL for which 3-valued Markov chains can serve as finite abstractions; this fragment is maximal for those abstractions and subsumes many practically relevant specifications including, e.g., reachability. We also develop game-theoretic foundations for the semantics of PCTL over Markov chains by capturing the standard PCTL semantics via a two-player games. These games, finally, inspire a notion of p-automata, which accept entire Markov chains. We show that p-automata subsume PCTL and Markov chains; that their languages of Markov chains have pleasant closure properties; and that the complexity of deciding acceptance matches that of probabilistic model checking for p-automata representing PCTL formulae. In addition, we offer a simulation between p-automata that under-approximates language containment. These results then allow us to show that p-automata comprise a solution to the problem studied in this thesis

From computability to executability : a process-theoretic view on automata theory

The theory of automata and formal language was devised in the 1930s to provide models for and to reason about computation. Here we mean by computation a procedure that transforms input into output, which was the sole mode of operation of computers at the time. Nowadays, computers are systems that interact with us and also each other; they are non-deterministic, reactive systems. Concurrency theory, split off from classical automata theory a few decades ago, provides a model of computation similar to the model given by the theory of automata and formal language, but focuses on concurrent, reactive and interactive systems. This thesis investigates the integration of the two theories, exposing the differences and similarities between them. Where automata and formal language theory focuses on computations and languages, concurrency theory focuses on behaviour. To achieve integration, we look for process-theoretic analogies of classic results from automata theory. The most prominent difference is that we use an interpretation of automata as labelled transition systems modulo (divergence-preserving) branching bisimilarity instead of treating automata as language acceptors. We also consider similarities such as grammars as recursive specifications and finite automata as labelled finite transition systems. We investigate whether the classical results still hold and, if not, what extra conditions are sufficient to make them hold. We especially look into three levels of Chomsky's hierarchy: we study the notions of finite-state systems, pushdown systems, and computable systems. Additionally we investigate the notion of parallel pushdown systems. For each class we define the central notion of automaton and its behaviour by associating a transition system with it. Then we introduce a suitable specification language and investigate the correspondence with the respective automaton (via its associated transition system). Because we not only want to study interaction with the environment, but also the interaction within the automaton, we make it explicit by means of communicating parallel components: one component representing the finite control of the automaton and one component representing the memory. First, we study finite-state systems by reinvestigating the relation between finite-state automata, left- and right-linear grammars, and regular expressions, but now up to (divergence-preserving) branching bisimilarity. For pushdown systems we augment the finite-state systems with stack memory to obtain the pushdown automata and consider different termination styles: termination on empty stack, on final state, and on final state and empty stack. Unlike for language equivalence, up to (divergence-preserving) branching bisimilarity the associated transition systems for the different termination styles fall into different classes. We obtain (under some restrictions) the correspondence between context-free grammars and pushdown automata for termination on final state and empty stack. We show how for contrasimulation, a weaker equivalence than branching bisimilarity, we can obtain the correspondence result without some of the restrictions. Finally, we make the interaction within a pushdown automaton explicit, but in a different way depending on the termination style. By analogy of pushdown systems we investigate the parallel pushdown systems, obtained by augmenting finite-state systems with bag memory, and consider analogous termination styles. We investigate the correspondence between context-free grammars that use parallel composition instead of sequential composition and parallel pushdown automata. While the correspondence itself is rather tight, it unfortunately only covers a small subset of the parallel pushdown automata, i.e. the single-state parallel pushdown automata. When making the interaction within parallel pushdown automata explicit, we obtain a rather uniform result for all termination styles. Finally, we study computable systems and the relation with exective and computable transition systems and Turing machines. For this we present the reactive Turing machine, a classical Turing machine augmented with capabilities for interaction. Again, we make the interaction in the reactive Turing machine between its finite control and the tape memory explicit