Metric structures and probabilistic computation

Journal of Computer and System Sciences, 1984

A logic, PrDL, is presented, which enables formal reasoning about probabilistic programs or, alternatively, reasoning probabilistically about conventional programs. The syntax of PrDL derives from Pratt's first-order dynamic logic and the semantics extends Kozen's semantics of probabilistic programs. An axiom system for PrDL is presented and shown to be complete relative to an extension of first-order analysis. For discrete probabilities it is shown that first-order analysis actually suffkes. Examples are presented, both of the expressive power of PrDL, and of a proof in the axiom system.

2012

A nonconstructive proof can be used to prove the existence of an object with some properties without providing an explicit example of such an object. A special case is a probabilistic proof where we show that an object with required properties appears with some positive probability in some random process. Can we use such arguments to prove the existence of a computable infinite object? Sometimes yes: following [8], we show how the notion of a layerwise computable mapping can be used to prove a computable version of Lovász local lemma.

We begin with a rather easy random model which illustrates many of the concepts we shall deal with. We call it the simple unary predicate with parameters n, p and denote it by SU(n, p). The model is over a universe of size n, a positive integer. We imagine each x ∈ flipping a coin to decide if U(x) holds, and the coin comes up heads with probability p. Here we have p real, 0 ≤ p ≤ 1. Formally we have a probability space on the possible U over defined by the properties Pr( U(x)) = p for all x ∈ and the events U(x) being mutually independent. We consider sentences in the first order language. In this language we have only equality (we shall always assume we have equality) and the unary predicate U. (The cognescenti should note that has no further structure and in particular is not considere d an ordered set.)

Lecture Notes in Computer Science

We show the equivalence of several di erent axiomatizations of the notion of (abstract) probabilistic domain in the category of dcpo's and continuous functions. The axiomatization with the richest set of operations provides probabilistic selection among a nite number of possibilities with arbitrary probabilities, whereas the poorest one has binary choice with equal probabilities as the only operation. The remaining theories lie in between; one of them is the theory of binary choice by Graham 1].

We study three aspects of the power of space-bounded probabilistic Turing machines. First, we give a simple alternative proof of Simon's result that space-bounded probabilistic complexity classes are closed under complement. Second, we demonstrate that any language recognizable by an alternating Turing machine in log n space with a constant number of alternations (the log n space "alternation hierarchy") also can be recognized by a log n spacebounded probabilistic Turing machine with small error probability; this is a generalization of Gill's result that any language in NSPACE (log n) can be recognized by such a machine. Third, we give a new definition of space-bounded oracle machines, and use it to define a space-bounded "oracle hierarchy" analogous to the original definition of the polynomial time hierarchy. Unlike its polynomial time analogue, the entire log n space "alternation hierarchy" is contained in the second level of the log n space "oracle hierarchy." However, the entire log n space "oracle hierarchy" is still contained in bounded-error probabilistic space log n.

2002

Probabilistic computation has proven to be a challenging and interesting area of research, both from the theoretical perspective of denotational semantics and the practical perspective of reasoning about probabilistic algorithms. On the theoretical side, the probabilistic powerdomain of Jones and Plotkin represents a significant advance. Further work, especially by Alvarez-Manilla, has greatly improved our understanding of the probabilistic powerdomain, and has helped clarify its relation to classical measure and integration theory. On the practical side, many researchers such as Kozen, Segala, Desharnais, and Kwiatkowska, among others, study problems of verification for probabilistic computation by defining various suitable logics for the classes of processes under study. The work reported here begins to bridge the gap between the domain theoretic and verification (model checking) perspectives on probabilistic computation by exhibiting sound and complete logics for probabilistic powerdomains that arise directly from given logics for the underlying domains. The category in which the construction is carried out generalizes Scott's Information Systems by taking account of full classical sequents. Via Stone duality, following Abramsky's Domain Theory in Logical Form, all known interesting categories of domains are embedded as subcategories. So the results reported here properly generalize similar constructions on specific categories of domains. The category offers a promising universe of semantic domains characterized by a very rich structure and good preservation properties of standard constructions. Furthermore, because the logical constructions make use of full classical sequents, the morphisms have a natural non-deterministic interpretation. Thus the category is a natural one in which to investigate the relationship between probabilistic and non-deterministic computation. We discuss the problem of integrating probabilistic and non-deterministic computation after presenting the construction of logics for probabilistic powerdomains.

Computers & Mathematics with Applications, 1993

This paper explores some topological features in order to analyse the consistent region in Probabilistic Logic. Using the L1 norm enables us to reduce and stabilize the consistent area associated with the probability of a predicate in a set of beliefs. The concept of facts and rules is approached as a particular problem. We present the program of the method used and propose an application to predicates in first-order logic. A study of the accuracy and the program complexity is made and compared to other methods.

Kolmogorov’s Heritage in Mathematics

Within classical propositional logic, assigning probabilities to formulas is shown to be equivalent to assigning probabilities to valuations. A novel notion of probabilistic entailment enjoying desirable properties of logical consequence is proposed and shown to collapse into the classical entailment when the language is left unchanged. Motivated by this result, a decidable conservative enrichment of propositional logic is proposed by giving the appropriate semantics to a new language construct that allows the constraining of the probability of a formula. A sound and weakly complete axiomatization is provided using the de-cidability of the theory of real closed ordered fields.

Motivation for this rather abstract work comes from the study of Genetics and Systems Biology. The data sets in these fields are usually small, extremely high dimensional, noisy and with complex interactions, which makes inferring causal interactions extremely difficult and unreliable. In this work, we approach non- parametric inference through Gromov’s Metric Measure Space Theory. We define the space of computable probability measures from this point of view and pro- vide an invariant representation based on distance matrices, through which one can construct inference algorithms for any arbitrary data set, even with a mix of different kinds of data types. Using the theory of Ergodic Automorphisms over the space of metrics, we give an estimate for the metric. Finally, through con- centration of measure phenomena, we are able to prove dimension independent convergence rates for our invariant representation. As a corollary we show that the approach in Kernel Embedding of Distributions in RKHS can be seen as a special case.