709 research outputs found
A Recipe for State-and-Effect Triangles
In the semantics of programming languages one can view programs as state
transformers, or as predicate transformers. Recently the author has introduced
state-and-effect triangles which capture this situation categorically,
involving an adjunction between state- and predicate-transformers. The current
paper exploits a classical result in category theory, part of Jon Beck's
monadicity theorem, to systematically construct such a state-and-effect
triangle from an adjunction. The power of this construction is illustrated in
many examples, covering many monads occurring in program semantics, including
(probabilistic) power domains
Healthiness Conditions for Predicate Transformers
AbstractThe behavior of a program can be modeled by describing how it transforms input states to output states, the state transformer semantics. Alternatively, for verification purposes one is interested in a 'predicate transformer semantics' which, for every condition on the output, yields the weakest precondition on the input that guarantees the desired property for the output.In the presence of computational effects like nondeterministic or probabilistic choice, a computation will be modeled by a map t:X→TY, where T is an appropriate computational monad. The corresponding predicate transformer assigns predicates on Y to predicates on X. One looks for necessary and, if possible, sufficient conditions (healthiness conditions) on predicate transformers that correspond to state transformers t:X→TY.In this paper we propose a framework for establishing healthiness conditions for predicate transformers. As far as the author knows, it fits to almost all situations in which healthiness conditions for predicate transformers have been worked out. It may serve as a guideline for finding new results; but it also shows quite narrow limitations
An expectation transformer approach to predicate abstraction and data independence for probabilistic programs
In this paper we revisit the well-known technique of predicate abstraction to
characterise performance attributes of system models incorporating probability.
We recast the theory using expectation transformers, and identify transformer
properties which correspond to abstractions that yield nevertheless exact bound
on the performance of infinite state probabilistic systems. In addition, we
extend the developed technique to the special case of "data independent"
programs incorporating probability. Finally, we demonstrate the subtleness of
the extended technique by using the PRISM model checking tool to analyse an
infinite state protocol, obtaining exact bounds on its performance
A New Proof Rule for Almost-Sure Termination
An important question for a probabilistic program is whether the probability
mass of all its diverging runs is zero, that is that it terminates "almost
surely". Proving that can be hard, and this paper presents a new method for
doing so; it is expressed in a program logic, and so applies directly to source
code. The programs may contain both probabilistic- and demonic choice, and the
probabilistic choices may depend on the current state.
As do other researchers, we use variant functions (a.k.a.
"super-martingales") that are real-valued and probabilistically might decrease
on each loop iteration; but our key innovation is that the amount as well as
the probability of the decrease are parametric.
We prove the soundness of the new rule, indicate where its applicability goes
beyond existing rules, and explain its connection to classical results on
denumerable (non-demonic) Markov chains.Comment: V1 to appear in PoPL18. This version collects some existing text into
new example subsection 5.5 and adds a new example 5.6 and makes further
remarks about uncountable branching. The new example 5.6 relates to work on
lexicographic termination methods, also to appear in PoPL18 [Agrawal et al,
2018
Healthiness from Duality
Healthiness is a good old question in program logics that dates back to
Dijkstra. It asks for an intrinsic characterization of those predicate
transformers which arise as the (backward) interpretation of a certain class of
programs. There are several results known for healthiness conditions: for
deterministic programs, nondeterministic ones, probabilistic ones, etc.
Building upon our previous works on so-called state-and-effect triangles, we
contribute a unified categorical framework for investigating healthiness
conditions. We find the framework to be centered around a dual adjunction
induced by a dualizing object, together with our notion of relative
Eilenberg-Moore algebra playing fundamental roles too. The latter notion seems
interesting in its own right in the context of monads, Lawvere theories and
enriched categories.Comment: 13 pages, Extended version with appendices of a paper accepted to
LICS 201
Quantitative program logic and expected time bounds in probabilistic distributed algorithms
AbstractIn this paper we show how quantitative program logic (Morgan et al., ACM Trans. Programming Languages Systems 18 (1996) 325) provides a formal framework in which to promote standard techniques of program analysis to a context where probability and nondeterminism interact, a situation common to probabilistic distributed algorithms. We show that overall expected time can be formulated directly in the logic and that it can be derived from local properties of components. We illustrate the methods with an analysis of expected running time of the probabilistic dining philosophers (Lehmann and Ravin, Proc 8th Annu. ACM. Symp. on principles of Programming Languages, ACM, New York, 1981, p. 133)
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