this paper was published as Walicki, M.A. and Meldal, S., 1995, Nondeterministic Operators in Algebraic Frameworks, Tehnical Report No. CSL--TR--95--664, Stanford University

Abstract. Following earlier models, we lift standard deterministic and nondeterministic semantics of imperative programs to probabilistic semantics. This semantics allows for random external inputs of known or unknown probability and random number generators. We then propose a method of analysis of programs according to this semantics, in the general framework of abstract interpretation. This method lifts an “ordinary ” abstract lattice, for non-probabilistic programs, to one suitable for probabilistic programs. Our construction is highly generic. We discuss the influence of certain parameters on the precision of the analysis, basing ourselves on experimental results. 1

We develop a notion of predicate transformer and, in particular, the weakest precondition, appropriate for quantum computation. We show that there is a Stone-type duality between the usual state-transformer semantics and the weakest precondition semantics. Rather than trying to reduce quantum computation to probabilistic programming we develop a notion that is directly taken from concepts used in quantum computation. The proof that weakest preconditions exist for completely positive maps follows immediately from the Kraus representation theorem. As an example we give the semantics of Selinger’s language in terms of our weakest preconditions. We also cover some specific situations and exhibit an interesting link with stabilizers.

We introduce a characterisation of probabilistic transition systems (PTS) in terms of linear operators on some suitably defined vector space representing the set of states. Various notions of process equivalences can then be re-formulated as abstract linear operators related to the concrete PTS semantics via a probabilistic abstract interpretation. These process equivalences can be turned into corresponding approximate notions by identifying processes whose abstract operators "differ" by a given quantity, which can be calculated as the norm of the difference operator. We argue that this number can be given a statistical interpretation in terms of the tests needed to distinguish two behaviours.

As probabilistic computations play an increasing role in solving various problems, researchers have designed probabilistic languages which treat probability distributions as primitive datatypes. Most probabilistic languages, however, focus only on discrete distributions and have limited expressive power. This paper presents a probabilistic language, called λ○, whose expressive power is beyond discrete distributions. Rich expressiveness of λ ○ is due to its use of sampling functions, i.e., mappings from the unit interval (0.0, 1.0] to probability domains, in specifying probability distributions. As such, λ ○ enables programmers to formally express and reason about sampling methods developed in simulation theory. The use of λ ○ is demonstrated with three applications in robotics: robot localization, people tracking, and robotic mapping. All experiments have been carried out with real robots.

We present a method for approximating the semantics of probabilistic programs to the purpose of constructing semantics-based analyses of such programs. The method resembles the one based on Galois connection as developed in the Cousot framework for abstract interpretation. The main difference between our approach and the standard theory of abstract interpretation is the choice of linear space structures instead of order-theoretic ones as semantical (concrete and abstract) domains. We show that our method generates "best approximations" according to an appropriate notion of precision defined in terms of a norm. Moreover, if re-casted in a order-theoretic setting these approximations are correct in the sense of classical abstract interpretation theory. We use Concurrent ...

Lambda--Upsilon--Omega ( \Upsilon\Omega ) is a research tool designed to assist the average case analysis of some well defined classes of algorithms and data structures. This cookbook consists of an informal introduction to the system together with eighteen examples of programmes that are automatically analyzed. Amongst the applications treated here, we find: addition chains, quantitative concurrency analysis of simple systems, symbolic manipulation algorithms such as formal differentiation, simplification and rewriting systems, as well as combinatorial models including various tree and permutation statistics and functional graphs with applications to integer factorisation.

We interpret the modal µ-calculus over a new model [10], to give a temporal logic suitable for systems exhibiting both probabilistic and demonic nondeterminism. The logical formulae are real-valued, and the statements are not limited to properties that hold with probability 1. In achieving that conceptual step, our technical contribution is to determine the correct quantitative generalisation of the Boolean operators: one that allows many of the standard Boolean-based temporal laws to carry over the reals with little or no structural alteration, even for properties that hold with probability strictly between 0 and 1. The generalisation is not obvious, but is dictated by our discovery elsewhere of the algebraic property that characterises the next-time operator over the new model: it is arithmetic ‘sublinearity ’ [20, Fig.4 p.342], which replaces the Boolean conjunctivity that characterises next-time in a modal algebra. We confirm by example that the new modal laws can be used for quantitative reasoning about probabilistic/demonic behaviour. The random walk is treated using only those laws and real-number arithmetic: arguing from precise numeric premises, more specific than simply ‘with some non-zero probability’, we reach numeric conclusions that are not simply ‘with probability 1’.

ation is to use digital signatures. Here you would verify a digital signature that is computed over the program using TrustMe's private key. But this is not much help in the scenario above. It merely provides you with confirmation that the program came from TrustMe so that they can be held accountable if some day you discover that the program did misbehave. By that time there is no telling how many "data warehouses" [13] already store the information. To appear in SIGACT News, 1998 But suppose we have a formal system, or logic, in which to reason about a program's ability to preserve privacy. Then our trust in a program could be based on the program itself, not on some digital signature for it. Further, depending on the logic, we might even have an algorithm for deciding whether programs have "privacy proofs" in the logic. And this in turn could lead to an efficient static program analyzer. All this req

Probabilistic predicate transformers extend standard predicate transformers by adding probabilistic choice to (transformers for) sequential programs. Demonic nondeterminism is retained. For finite state spaces, the basic theory is set out elsewhere [15], together with a statement of the probabilistic `healthiness conditions' that generalise the `positive conjunctivity' of ordinary predicate transformers. Here we extend the earlier results to infinite state spaces, and investigate the structure of the transformer space generally: as Back and von Wright [1] did for `standard' (non-probabilistic) transformers, we nest deterministic, demonic and demonic/angelic transformers, showing how each can be constructed from the one before. In the end we thus find healthiness conditions for a system in which deterministic, demonic, probabilistic and angelic choices all coexist.