Abstract
In order to reason about effects, we can define quantitative formulas to describe behavioural aspects of effectful programs. These formulas can for example express probabilities that (or sets of correct starting states for which) a program satisfies a property. Fundamental to this approach is the notion of quantitative modality, which is used to lift a property on values to a property on computations. Taking all formulas together, we say that two terms are equivalent if they satisfy all formulas to the same quantitative degree. Under sufficient conditions on the quantitative modalities, this equivalence is equal to a notion of Abramsky's applicative bisimilarity, and is moreover a congruence. We investigate these results in the context of Levy's call-by-push-value with general recursion and algebraic effects. For example, the results apply to (combinations of) nondeterministic choice, probabilistic choice, global store, and error.
Highlights
There are many notions of program equivalence for languages with effects
We explore the notion of behavioural equivalence, which states that programs may be considered behaviourally equivalent if they satisfy the same behavioural properties
This can be made rigorous by defining a logic, where each formula φ denotes a certain behavioural property
Summary
There are many notions of program equivalence for languages with effects. In this paper, we explore the notion of behavioural equivalence, which states that programs may be considered behaviourally equivalent if they satisfy the same behavioural properties. This paper provides a general framework for quantitative logics for expressing behavioural properties of programs with effects, generalising the Boolean-valued framework from [28]. As in [28], we are able to establish that the behavioural equivalence defined as above is a congruence, as long as suitable properties on the quantitative modalities are satisfied. These properties require notions of monotonicity, continuity, and a notion of preservation over sequencing called decomposability. The other main contributions are the examples illustrating the quantitative approach Some examples such as the combination of nondeterminism with probabilistic choice, or with global store, do not fit into the Boolean-valued framework of [28], but do work here.
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