A unification of probabilistic choice within a design-based model of reversible computation

Abstract

We see reversible computing as a generalisation of sequential computation obtained by revoking the law of the excluded miracle. Our execution language includes naked guarded commands and non-deterministic choice. Choices which lead to miraculous continuations invoke reverse computation, and non-deterministic choice plays the rôle of provisional choice within a backtracking context. We require probabilistic choice for symmetry breaking and sampling large search spaces, but must formulate it differently from previous approaches to obtain the required interactions between probabilistic choice and non-deterministic choice and between probabilistic choice and feasibility. Our formulation allows us to derive the post-distributions which characterise a program, and we use these to construct a relational model. We consider refinement as containment of convex closures within distribution space, qualified with additional conditions to avoid over-refinement. We link the non-probabilistic and probabilistic versions of the model with a Galois connection and show that classical designs are a retract of our probabilistic designs. We consider the interaction between probabilistic and non-deterministic choice and find the same initially counter-intuitive results that have been noted by other investigators. We provide an alternative formulation, within the same model, of oblivious non-determinism, which allows all non-deterministic choices to be moved to the start of a computation. We consider the interaction between probabilistic choice and feasibility that is required to match an operational interpretation in which infeasible commands provoke reverse execution, and we present a small case study to show how the interaction between probabilistic choice and feasibility can be exploited in a practical program. All programming structures described here are supported by our implementation platform, the Reversible Virtual Machine, whose development has accompanied our theoretical investigations.This technical report is a working document. A peer reviewed version of this work has been accepted for publication in the Formal Aspects of Computing Special Issue on Unifying Theories of Programming.

Probabilistic Choice, Reversibility, Loops, and Miracles

“…In [17,18] we add probabilistic choice to our programming language. In common with pGSL, pGCL and other formalisms we use S p ⊕ T for the operation that will choose S with probability p and T with probability 1−p.…”

“…The theory of probabilistic programming given here is more fully described in our paper "A Unification of Probabilistic Choice within a Design-based Model of Reversible Computation" [17] and in an associated technical report [18]. The original contributions of the current paper are the re-expression of our theory in GSL, the establishment of a semantic foundation for probabilistic iterations in our formalism based on fixed point theory, the practical proof treatment of loops with probabilistic loop bodies, and consideration of probabilistic loop termination within a formalism which has a strict approach to probabilistic termination, and some reflections on the total correctness abstraction in the context of backtracking and probabilistic choice.…”

“…The original contributions of the current paper are the re-expression of our theory in GSL, the establishment of a semantic foundation for probabilistic iterations in our formalism based on fixed point theory, the practical proof treatment of loops with probabilistic loop bodies, and consideration of probabilistic loop termination within a formalism which has a strict approach to probabilistic termination, and some reflections on the total correctness abstraction in the context of backtracking and probabilistic choice. Aspects of our approach which are covered in our previous report [17] include: the ability to derive a relational model from a probabilistic program text, the expression of blind nondeterminism within the same model as demonic nondeterminism, the characterisation of probabilistic refinement as containment of convex closures in distribution space, and the linking of probabilistic and non-probabilistic models via a Galois connection.…”

Preference and Non-deterministic Choice

“…Furthermore, probabilistic choice on the RVM (and in our formalisms) is subject to revision by backtracking, and thus presents itself as a candidate for expressing preference. Our expectation calculus for probabilistic choice (including its combination with non-deterministic choice) is given in [9]; here we resume the details required for the current investigation.…”

“…However, we cannot just set p to 1 and obtain a preferential-choice operator, since in this case probabilistic choice collapses and we are left with just S [9]. However, it seems we could use the same kind of guarded-bunch formalism in expressing preference as in expressing probabilistic choice, namely by introducing a preferential choice [>, which prefers its first operand, with the property…”