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Ugo Dal Lago

Publications and source records attributed to Ugo Dal Lago.

At least 55 records · Page 3Linked to original sources
arXiv2021

Resource Transition Systems and Full Abstraction for Linear Higher-Order Effectful Systems

We investigate program equivalence for linear higher-order(sequential) languages endowed with primitives for computational effects. More specifically, we study operationally-based notions of program equivalence for a linear $λ$-calculus with explicit copying and algebraic effects \emph{à la} Plotkin and Power. Such a calculus makes explicit the interaction between copying and linearity, which are intensional aspects of computation, with effects, which are, instead, \emph{extensional}. We review some of the notions of equivalences for linear calculi proposed in the literature and show their limitations when applied to effectful calculi where copying is a first-class citizen. We then introduce resource transition systems, namely transition systems whose states are built over tuples of programs representing the available resources, as an operational semantics accounting for both intensional and extensional interactive behaviors of programs. Our main result is a sound and complete characterization of contextual equivalence as trace equivalence defined on top of resource transition systems.

cs.PL↗
arXiv2021

On Counting Propositional Logic

We study counting propositional logic as an extension of propositional logic with counting quantifiers. We prove that the complexity of the underlying decision problem perfectly matches the appropriate level of Wagner's counting hierarchy, but also that the resulting logic admits a satisfactory proof-theoretical treatment. From the latter, a type system for a probabilistic lambda-calculus is derived in the spirit of the Curry-Howard correspondence, showing the potential of counting propositional logic as a useful tool in several fields of theoretical computer science.

cs.LO↗
arXiv2021

The Space of Interaction (long version)

The space complexity of functional programs is not well understood. In particular, traditional implementation techniques are tailored to time efficiency, and space efficiency induces time inefficiencies, as it prefers re-computing to saving. Girard's geometry of interaction underlies an alternative approach based on the interaction abstract machine (IAM), claimed as space efficient in the literature. It has also been conjectured to provide a reasonable notion of space for the lambda-calculus, but such an important result seems to be elusive. In this paper we introduce a new intersection type system precisely measuring the space consumption of the IAM on the typed term. Intersection types have been repeatedly used to measure time, which they achieve by dropping idempotency, turning intersections into multisets. Here we show that the space consumption of the IAM is connected to a further structural modification, turning multisets into trees. Tree intersection types lead to a finer understanding of some space complexity results from the literature. They also shed new light on the conjecture about reasonable space: we show that the usual way of encoding Turing machines into the lambda calculus cannot be used to prove that the space of the IAM is a reasonable cost model.

cs.LO↗
arXiv2021

On Measure Quantifiers in First-Order Arithmetic (Long Version)

We study the logic obtained by endowing the language of first-order arithmetic with second-order measure quantifiers. This new kind of quantification allows us to express that the argument formula is true in a certain portion of all possible interpretations of the quantified variable. We show that first-order arithmetic with measure quantifiers is capable of formalizing simple results from probability theory and, most importantly, of representing every recursive random function. Moreover, we introduce a realizability interpretation of this logic in which programs have access to an oracle from the Cantor space.

cs.LO↗
arXiv2021

Modal Reasoning = Metric Reasoning, via Lawvere

Graded modal types systems and coeffects are becoming a standard formalism to deal with context-dependent computations where code usage plays a central role. The theory of program equivalence for modal and coeffectful languages, however, is considerably underdeveloped if compared to the denotational and operational semantics of such languages. This raises the question of how much of the theory of ordinary program equivalence can be given in a modal scenario. In this work, we show that coinductive equivalences can be extended to a modal setting, and we do so by generalising Abramsky's applicative bisimilarity to coeffectful behaviours. To achieve this goal, we develop a general theory of ternary program relations based on the novel notion of a comonadic lax extension, on top of which we define a modal extension of Abramsky's applicative bisimilarity (which we dub modal applicative bisimilarity). We prove such a relation to be a congruence, this way obtaining a compositional technique for reasoning about modal and coeffectful behaviours. But this is not the end of the story: we also establish a correspondence between modal program relations and program distances. This correspondence shows that modal applicative bisimilarity and (a properly extended) applicative bisimilarity distance coincide, this way revealing that modal program equivalences and program distances are just two sides of the same coin.

cs.LO↗
arXiv2020

Intersection Types and (Positive) Almost-Sure Termination

Randomized higher-order computation can be seen as being captured by a lambda calculus endowed with a single algebraic operation, namely a construct for binary probabilistic choice. What matters about such computations is the probability of obtaining any given result, rather than the possibility or the necessity of obtaining it, like in (non)deterministic computation. Termination, arguably the simplest kind of reachability problem, can be spelled out in at least two ways, depending on whether it talks about the probability of convergence or about the expected evaluation time, the second one providing a stronger guarantee. In this paper, we show that intersection types are capable of precisely characterizing both notions of termination inside a single system of types: the probability of convergence of any lambda-term can be underapproximated by its type, while the underlying derivation's weight gives a lower bound to the term's expected number of steps to normal form. Noticeably, both approximations are tight -- not only soundness but also completeness holds. The crucial ingredient is non-idempotency, without which it would be impossible to reason on the expected number of reduction steps which are necessary to completely evaluate any term. Besides, the kind of approximation we obtain is proved to be optimal recursion theoretically: no recursively enumerable formal system can do better than that.

cs.LO↗
arXiv2020

The (In)Efficiency of Interaction

Evaluating higher-order functional programs through abstract machines inspired by the geometry of the interaction is known to induce $\textit{space}$ efficiencies, the price being $\textit{time}$ performances often poorer than those obtainable with traditional, environment-based, abstract machines. Although families of lambda-terms for which the former is exponentially less efficient than the latter do exist, it is currently unknown how \emph{general} this phenomenon is, and how far the inefficiencies can go, in the worst case. We answer these questions formulating four different well-known abstract machines inside a common definitional framework, this way being able to give sharp results about the relative time efficiencies. We also prove that non-idempotent intersection type theories are able to precisely reflect the time performances of the interactive abstract machine, this way showing that its time-inefficiency ultimately descends from the presence of higher-order types.

cs.PL↗
arXiv2020

The Abstract Machinery of Interaction (Long Version)

This paper revisits the Interaction Abstract Machine (IAM), a machine based on Girard's Geometry of Interaction, introduced by Mackie and Danos & Regnier. It is an unusual machine, not relying on environments, presented on linear logic proof nets, and whose soundness proof is convoluted and passes through various other formalisms. Here we provide a new direct proof of its correctness, based on a variant of Sands's improvements, a natural notion of bisimulation. Moreover, our proof is carried out on a new presentation of the IAM, defined as a machine acting directly on $λ$-terms, rather than on linear logic proof nets.

cs.LO↗
arXiv2020

Decomposing Probabilistic Lambda-calculi

A notion of probabilistic lambda-calculus usually comes with a prescribed reduction strategy, typically call-by-name or call-by-value, as the calculus is non-confluent and these strategies yield different results. This is a break with one of the main advantages of lambda-calculus: confluence, which means results are independent from the choice of strategy. We present a probabilistic lambda-calculus where the probabilistic operator is decomposed into two syntactic constructs: a generator, which represents a probabilistic event; and a consumer, which acts on the term depending on a given event. The resulting calculus, the Probabilistic Event Lambda-Calculus, is confluent, and interprets the call-by-name and call-by-value strategies through different interpretations of the probabilistic operator into our generator and consumer constructs. We present two notions of reduction, one via fine-grained local rewrite steps, and one by generation and consumption of probabilistic events. Simple types for the calculus are essentially standard, and they convey strong normalization. We demonstrate how we can encode call-by-name and call-by-value probabilistic evaluation.

cs.LO↗
arXiv2020

On the Versatility of Open Logical Relations: Continuity, Automatic Differentiation, and a Containment Theorem

Logical relations are one of the most powerful techniques in the theory of programming languages, and have been used extensively for proving properties of a variety of higher-order calculi. However, there are properties that cannot be immediately proved by means of logical relations, for instance program continuity and differentiability in higher-order languages extended with real-valued functions. Informally, the problem stems from the fact that these properties are naturally expressed on terms of non-ground type (or, equivalently, on open terms of base type), and there is no apparent good definition for a base case (i.e. for closed terms of ground types). To overcome this issue, we study a generalization of the concept of a logical relation, called \emph{open logical relation}, and prove that it can be fruitfully applied in several contexts in which the property of interest is about expressions of first-order type. Our setting is a simply-typed $λ$-calculus enriched with real numbers and real-valued first-order functions from a given set, such as the one of continuous or differentiable functions. We first prove a containment theorem stating that for any such a collection of functions including projection functions and closed under function composition, any well-typed term of first-order type denotes a function belonging to that collection. Then, we show by way of open logical relations the correctness of the core of a recently published algorithm for forward automatic differentiation. Finally, we define a refinement-based type system for local continuity in an extension of our calculus with conditionals, and prove the soundness of the type system using open logical relations.

cs.PL↗
arXiv2020

On Higher-Order Cryptography (Long Version)

Type-two constructions abound in cryptography: adversaries for encryption and authentication schemes, if active, are modeled as algorithms having access to oracles, i.e. as second-order algorithms. But how about making cryptographic schemes themselves higher-order? This paper gives an answer to this question, by first describing why higher-order cryptography is interesting as an object of study, then showing how the concept of probabilistic polynomial time algorithm can be generalized so as to encompass algorithms of order strictly higher than two, and finally proving some positive and negative results about the existence of higher-order cryptographic primitives, namely authentication schemes and pseudorandom functions.

cs.LO↗
arXiv2020

A Diagrammatic Calculus for Algebraic Effects

We introduce a new diagrammatic notation for representing the result of (algebraic) effectful computations. Our notation explicitly separates the effects produced during a computation from the possible values returned, this way simplifying the extension of definitions and results on pure computations to an effectful setting. Additionally, we show a number of algebraic and order-theoretic laws on diagrams, this way laying the foundations for a diagrammatic calculus of algebraic effects. We give a formal foundation for such a calculus in terms of Lawvere theories and generic effects.

cs.PL↗
arXiv2019

On randomised strategies in the $λ$-calculus (long version)

In this work we study randomised reduction strategies,a notion already known in the context of abstract reduction systems, for the $λ$-calculus. We develop a simple framework that allows us to prove a randomised strategy to be positive almost-surely normalising. Then we propose a simple example of randomised strategy for the $λ$-calculus that has such a property and we show why it is non-trivial with respect to classical deterministic strategies such as leftmost-outermost or rightmost-innermost. We conclude studying this strategy for two sub-$λ$-calculi, namely those where duplication and erasure are syntactically forbidden, showing some non-trivial properties.

cs.LO↗
arXiv2019

Differential Logical Relations, Part I: The Simply-Typed Case (Long Version)

We introduce a new form of logical relation which, in the spirit of metric relations, allows us to assign each pair of programs a quantity measuring their distance, rather than a boolean value standing for their being equivalent. The novelty of differential logical relations consists in measuring the distance between terms not (necessarily) by a numerical value, but by a mathematical object which somehow reflects the interactive complexity, i.e. the type, of the compared terms. We exemplify this concept in the simply-typed lambda-calculus, and show a form of soundness theorem. We also see how ordinary logical relations and metric relations can be seen as instances of differential logical relations. Finally, we show that differential logical relations can be organised in a cartesian closed category, contrarily to metric relations, which are well-known not to have such a structure, but only that of a monoidal closed category.

cs.LO↗
arXiv2019

On the Taylor Expansion of Probabilistic $λ$-Terms (Long Version)

We generalise Ehrhard and Regnier's Taylor expansion from pure to probabilistic $λ$-terms through notions of probabilistic resource terms and explicit Taylor expansion. We prove that the Taylor expansion is adequate when seen as a way to give semantics to probabilistic $λ$-terms, and that there is a precise correspondence with probabilistic Böhm trees, as introduced by the second author.

cs.LO↗
arXiv2019

The Geometry of Bayesian Programming

We give a geometry of interaction model for a typed lambda-calculus endowed with operators for sampling from a continuous uniform distribution and soft conditioning, namely a paradigmatic calculus for higher-order Bayesian programming. The model is based on the category of measurable spaces and partial measurable functions, and is proved adequate with respect to both a distribution-based and a sampling based operational semantics.

cs.PL↗
arXiv2018

On Probabilistic Term Rewriting

We study the termination problem for probabilistic term rewrite systems. We prove that the interpretation method is sound and complete for a strengthening of positive almost sure termination, when abstract reduction systems and term rewrite systems are considered. Two instances of the interpretation method - polynomial and matrix interpretations - are analyzed and shown to capture interesting and nontrivial examples when automated. We capture probabilistic computation in a novel way by way of multidistribution reduction sequences, this way accounting for both the nondeterminism in the choice of the redex and the probabilism intrinsic in firing each rule.

cs.SC↗