A Sound and Complete Logic for Algebraic Effects

Matache, Cristina; Staton, Sam

doi:10.1007/978-3-030-17127-8_22

Cited by 10 publications

(6 citation statements)

References 37 publications

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“…There is a very large body of methods for proving program equivalence, and in particular contextual equivalence, in higher-order languages with state, probabilities, and effects; see e.g. [Benton et al 2014;Bizjak and Birkedal 2015;Crubillé and Lago 2015;Jung et al 2015;Matache and Staton 2019;Pitts and Stark 1998]. Many of these methods have been applied to reason about security and privacy, using the natural view of adversaries as contexts.…”

Section: Related Workmentioning

confidence: 99%

Higher-order probabilistic adversarial computations: categorical semantics and program logics

Aguirre

Barthe

Gaboardi

et al. 2021

Proc. ACM Program. Lang.

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Adversarial computations are a widely studied class of computations where resource-bounded probabilistic adversaries have access to oracles, i.e., probabilistic procedures with private state. These computations arise routinely in several domains, including security, privacy and machine learning. In this paper, we develop program logics for reasoning about adversarial computations in a higher-order setting. Our logics are built on top of a simply typed λ-calculus extended with a graded monad for probabilities and state. The grading is used to model and restrict the memory footprint and the cost (in terms of oracle calls) of computations. Under this view, an adversary is a higher-order expression that expects as arguments the code of its oracles. We develop unary program logics for reasoning about error probabilities and expected values, and a relational logic for reasoning about coupling-based properties. All logics feature rules for adversarial computations, and yield guarantees that are valid for all adversaries that satisfy a fixed resource policy. We prove the soundness of the logics in the category of quasi-Borel spaces, using a general notion of graded predicate liftings, and we use logical relations over graded predicate liftings to establish the soundness of proof rules for adversaries. We illustrate the working of our logics with simple but illustrative examples.

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Section: Related Workmentioning

confidence: 99%

Higher-order probabilistic adversarial computations: categorical semantics and program logics

Aguirre

Barthe

Gaboardi

et al. 2021

Proc. ACM Program. Lang.

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“…This is a desirable property, since any two programs related by a congruence are contextually equivalent. Moreover, in [17,18] it is shown that for a continuation-passing style language, such a program equivalence coincides with contextual equivalence.…”

Section: Simulations and Equalitymentioning

confidence: 99%

“…We capture such observations with a collection of predicate liftings [27,18]. These consist of a set of tokens denoting each possible observation, and for each token a device that lifts a predicate on return values to a predicate on coinductive trees over these values.…”

Section: Introductionmentioning

confidence: 99%

Inductive and Coinductive Predicate Liftings for Effectful Programs

Veltri

Voorneveld

2021

Electron. Proc. Theor. Comput. Sci.

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We formulate a framework for describing behaviour of effectful higher-order recursive programs. Examples of effects are implemented using effect operations, and include: execution cost, nondeterminism, global store and interaction with a user. The denotational semantics of a program is given by a coinductive tree in a monad, which combines potential return values of the program in terms of effect operations.Using a simple test logic, we construct two sorts of predicate liftings, which lift predicates on a result type to predicates on computations that produce results of that type, each capturing a facet of program behaviour. Firstly, we study inductive predicate liftings which can be used to describe effectful notions of total correctness. Secondly, we study coinductive predicate liftings, which describe effectful notions of partial correctness. The two constructions are dual in the sense that one can be used to disprove the other.The predicate liftings are used as a basis for an endogenous logic of behavioural properties for higher-order programs. The program logic has a derivable notion of negation, arising from the duality of the two sorts of predicate liftings, and it generates a program equivalence which subsumes a notion of bisimilarity. Appropriate definitions of inductive and coinductive predicate liftings are given for a multitude of effect examples.The whole development has been fully formalized in the Agda proof assistant.

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“…Plotkin and Pretnar [2008] propose a generic logic for algebraic effects that encompasses Moggi's computational λ-calculus, evaluation logic, and Hennesy-Milner logic, but does not extend to Hoare-style reasoning for state. Simpson and Voorneveld [2018] and Matache and Staton [2019] explore logics for algebraic effects by specifying the effectful behaviour of algebraic operations using a collection of effectspecific modalities instead of equations. Their modalities are closely related to how we derive effect observations θ : M → W Pure and thus program specifications from M-algebras on P in §4.4, as intuitively the conditions they impose on their modalities ensure that these can be collectively treated as an M-algebra on P. In recent work concurrent to ours, Voorneveld [2019] studies a logic based on quantitative modalities by considering truth objects richer than P, including S → P for stateful and [0, 1] for probabilistic computation.…”

Section: Related Workmentioning

confidence: 99%

Dijkstra monads for all

Maillard

Ahman

Atkey

et al. 2019

Proc. ACM Program. Lang.

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This paper proposes a general semantic framework for verifying programs with arbitrary monadic side-effects using Dijkstra monads, which we define as monad-like structures indexed by a specification monad. We prove that any monad morphism between a computational monad and a specification monad gives rise to a Dijkstra monad, which provides great flexibility for obtaining Dijkstra monads tailored to the verification task at hand. We moreover show that a large variety of specification monads can be obtained by applying monad transformers to various base specification monads, including predicate transformers and Hoare-style preand postconditions. For defining correct monad transformers, we propose a language inspired by Moggi's monadic metalanguage that is parameterized by a dependent type theory. We also develop a notion of algebraic operations for Dijkstra monads, and start to investigate two ways of also accommodating effect handlers. We implement our framework in both Coq and F ⋆ , and illustrate that it supports a wide variety of verification styles for effects such as exceptions, nondeterminism, state, input-output, and general recursion.

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A Sound and Complete Logic for Algebraic Effects

Cited by 10 publications

References 37 publications

Higher-order probabilistic adversarial computations: categorical semantics and program logics

Higher-order probabilistic adversarial computations: categorical semantics and program logics

Inductive and Coinductive Predicate Liftings for Effectful Programs

Dijkstra monads for all

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