arXiv subjects

Ugo Dal Lago

Publications and source records attributed to Ugo Dal Lago.

At least 73 records · Page 4Linked to original sources

arXiv2017

The Geometry of Parallelism. Classical, Probabilistic, and Quantum Effects

We introduce a Geometry of Interaction model for higher-order quantum computation, and prove its adequacy for a full quantum programming language in which entanglement, duplication, and recursion are all available. Our model comes with a multi-token machine, a proof net system, and a PCF-style language. The approach we develop is not specific to quantum computation, and our model is an instance of a new framework whose main feature is the ability to model commutative effects in a parallel setting. Being based on a multi-token machine equipped with a memory, it has a concrete nature which makes it well suited for building low-level operational descriptions of higher-order languages.

cs.LO↗

arXiv2017

Automating Sized Type Inference for Complexity Analysis (Technical Report)

This paper introduces a new methodology for the complexity analysis of higher-order functional programs, which is based on three ingredients: a powerful type system for size analysis and a sound type inference procedure for it, a ticking monadic transformation, and constraint solving. Noticeably, the presented methodology can be fully automated, and is able to analyse a series of examples which cannot be handled by most competitor methodologies. This is possible due to the choice of adopting an abstract index language and index polymorphism at higher ranks. A prototype implementation is available.

cs.LO↗

arXiv2017

Automated Sized-Type Inference and Complexity Analysis

This paper introduces a new methodology for the complexity analysis of higher-order functional programs, which is based on three components: a powerful type system for size analysis and a sound type inference procedure for it, a ticking monadic transformation and a concrete tool for constraint solving. Noticeably, the presented methodology can be fully automated, and is able to analyse a series of examples which cannot be handled by most competitor methodologies. This is possible due to various key ingredients, and in particular an abstract index language and index polymorphism at higher ranks. A prototype implementation is available.

cs.LO↗

arXiv2017

The Geometry of Concurrent Interaction: Handling Multiple Ports by Way of Multiple Tokens (Long Version)

We introduce a geometry of interaction model for Mazza's multiport interaction combinators, a graph-theoretic formalism which is able to faithfully capture concurrent computation as embodied by process algebras like the $π$-calculus. The introduced model is based on token machines in which not one but multiple tokens are allowed to traverse the underlying net at the same time. We prove soundness and adequacy of the introduced model. The former is proved as a simulation result between the token machines one obtains along any reduction sequence. The latter is obtained by a fine analysis of convergence, both in nets and in token machines.

cs.LO↗

arXiv2017

Effectful Applicative Bisimilarity: Monads, Relators, and Howe's Method (Long Version)

We study Abramsky's applicative bisimilarity abstractly, in the context of call-by-value $λ$-calculi with algebraic effects. We first of all endow a computational $λ$-calculus with a monadic operational semantics. We then show how the theory of relators provides precisely what is needed to generalise applicative bisimilarity to such a calculus, and to single out those monads and relators for which applicative bisimilarity is a congruence, thus a sound methodology for program equivalence. This is done by studying Howe's method in the abstract.

cs.LO↗

arXiv2017

A Lambda-Calculus Foundation for Universal Probabilistic Programming

We develop the operational semantics of an untyped probabilistic lambda-calculus with continuous distributions, as a foundation for universal probabilistic programming languages such as Church, Anglican, and Venture. Our first contribution is to adapt the classic operational semantics of lambda-calculus to a continuous setting via creating a measure space on terms and defining step-indexed approximations. We prove equivalence of big-step and small-step formulations of this distribution-based semantics. To move closer to inference techniques, we also define the sampling-based semantics of a term as a function from a trace of random samples to a value. We show that the distribution induced by integrating over all traces equals the distribution-based semantics. Our second contribution is to formalize the implementation technique of trace Markov chain Monte Carlo (MCMC) for our calculus and to show its correctness. A key step is defining sufficient conditions for the distribution induced by trace MCMC to converge to the distribution-based semantics. To the best of our knowledge, this is the first rigorous correctness proof for trace MCMC for a higher-order functional language.

cs.PL↗

arXiv2017

Metric Reasoning About $λ$-Terms: The General Case (Long Version)

In any setting in which observable properties have a quantitative flavour, it is natural to compare computational objects by way of \emph{metrics} rather than equivalences or partial orders. This holds, in particular, for probabilistic higher-order programs. A natural notion of comparison, then, becomes context distance, the metric analogue of Morris' context equivalence. In this paper, we analyze the main properties of the context distance in fully-fledged probabilistic $λ$-calculi, this way going beyond the state of the art, in which only affine calculi were considered. We first of all study to which extent the context distance trivializes, giving a sufficient condition for trivialization. We then characterize context distance by way of a coinductively defined, tuple-based notion of distance in one of those calculi, called $Λ^\oplus_!$. We finally derive pseudometrics for call-by-name and call-by-value probabilistic $λ$-calculi, and prove them fully-abstract.

cs.LO↗

arXiv2017

Probabilistic Termination by Monadic Affine Sized Typing (Long Version)

We introduce a system of monadic affine sized types, which substantially generalise usual sized types, and allows this way to capture probabilistic higher-order programs which terminate almost surely. Going beyond plain, strong normalisation without losing soundness turns out to be a hard task, which cannot be accomplished without a richer, quantitative notion of types, but also without imposing some affinity constraints. The proposed type system is powerful enough to type classic examples of probabilistically terminating programs such as random walks. The way typable programs are proved to be almost surely terminating is based on reducibility, but requires a substantial adaptation of the technique.

cs.PL↗

arXiv2016

Infinitary $λ$-Calculi from a Linear Perspective (Long Version)

We introduce a linear infinitary $λ$-calculus, called $\ellΛ_{\infty}$, in which two exponential modalities are available, the first one being the usual, finitary one, the other being the only construct interpreted coinductively. The obtained calculus embeds the infinitary applicative $λ$-calculus and is universal for computations over infinite strings. What is particularly interesting about $\ellΛ_{\infty}$, is that the refinement induced by linear logic allows to restrict both modalities so as to get calculi which are terminating inductively and productive coinductively. We exemplify this idea by analysing a fragment of $\ellΛ$ built around the principles of $\mathsf{SLL}$ and $\mathsf{4LL}$. Interestingly, it enjoys confluence, contrarily to what happens in ordinary infinitary $λ$-calculi.

cs.LO↗

arXiv2016

(Leftmost-Outermost) Beta Reduction is Invariant, Indeed

Slot and van Emde Boas' weak invariance thesis states that reasonable machines can simulate each other within a polynomially overhead in time. Is lambda-calculus a reasonable machine? Is there a way to measure the computational complexity of a lambda-term? This paper presents the first complete positive answer to this long-standing problem. Moreover, our answer is completely machine-independent and based over a standard notion in the theory of lambda-calculus: the length of a leftmost-outermost derivation to normal form is an invariant cost model. Such a theorem cannot be proved by directly relating lambda-calculus with Turing machines or random access machines, because of the size explosion problem: there are terms that in a linear number of steps produce an exponentially long output. The first step towards the solution is to shift to a notion of evaluation for which the length and the size of the output are linearly related. This is done by adopting the linear substitution calculus (LSC), a calculus of explicit substitutions modeled after linear logic proof nets and admitting a decomposition of leftmost-outermost derivations with the desired property. Thus, the LSC is invariant with respect to, say, random access machines. The second step is to show that LSC is invariant with respect to the lambda-calculus. The size explosion problem seems to imply that this is not possible: having the same notions of normal form, evaluation in the LSC is exponentially longer than in the lambda-calculus. We solve such an impasse by introducing a new form of shared normal form and shared reduction, deemed useful. Useful evaluation avoids those steps that only unshare the output without contributing to beta-redexes, i.e. the steps that cause the blow-up in size. The main technical contribution of the paper is indeed the definition of useful reductions and the thorough analysis of their properties.

cs.PL↗

arXiv2015

Parallelism and Synchronization in an Infinitary Context (Long Version)

We study multitoken interaction machines in the context of a very expressive logical system with exponentials, fixpoints and synchronization. The advantage of such machines is to provide models in the style of the Geometry of Interaction, i.e., an interactive semantics which is close to low-level implementation. On the one hand, we prove that despite the inherent complexity of the framework, interaction is guaranteed to be deadlock free. On the other hand, the resulting logical system is powerful enough to embed PCF and to adequately model its behaviour, both when call-by-name and when call-by-value evaluation are considered. This is not the case for single-token stateless interactive machines.

cs.LO↗

arXiv2015

Applicative Bisimulation and Quantum $λ$-Calculi (Long Version)

Applicative bisimulation is a coinductive technique to check program equivalence in higher-order functional languages. It is known to be sound, and sometimes complete, with respect to context equivalence. In this paper we show that applicative bisimulation also works when the underlying language of programs takes the form of a linear $λ$-calculus extended with features such as probabilistic binary choice, but also quantum data, the latter being a setting in which linearity plays a role. The main results are proofs of soundness for the obtained notions of bisimilarity.

cs.LO↗

arXiv2015

Analysing the Complexity of Functional Programs: Higher-Order Meets First-Order (Long Version)

We show how the complexity of higher-order functional programs can be analysed automatically by applying program transformations to a defunctionalized versions of them, and feeding the result to existing tools for the complexity analysis of first-order term rewrite systems. This is done while carefully analysing complexity preservation and reflection of the employed transformations such that the complexity of the obtained term rewrite system reflects on the complexity of the initial program. Further, we describe suitable strategies for the application of the studied transformations and provide ample experimental data for assessing the viability of our method.

cs.LO↗

arXiv2015

On Equivalences, Metrics, and Polynomial Time (Long Version)

Interactive behaviors are ubiquitous in modern cryptography, but are also present in $λ$-calculi, in the form of higher-order constructions. Traditionally, however, typed $λ$-calculi simply do not fit well into cryptography, being both deterministic and too powerful as for the complexity of functions they can express. We study interaction in a $λ$-calculus for probabilistic polynomial time computable functions. In particular, we show how notions of context equivalence and context metric can both be characterized by way of traces when defined on linear contexts. We then give evidence on how this can be turned into a proof methodology for computational indistinguishability, a key notion in modern cryptography. We also hint at what happens if a more general notion of a context is used.

cs.LO↗

arXiv2015

Metric Reasoning about $λ$-Terms: the Affine Case (Long Version)

Terms of Church's $λ$-calculus can be considered equivalent along many different definitions, but context equivalence is certainly the most direct and universally accepted one. If the underlying calculus becomes probabilistic, however, equivalence is too discriminating: terms which have totally unrelated behaviours are treated the same as terms which behave very similarly. We study the problem of evaluating the distance between affine $λ$-terms. The most natural definition for it, namely a natural generalisation of context equivalence, is shown to be characterised by a notion of trace distance, and to be bounded from above by a coinductively defined distance based on the Kantorovich metric on distributions. A different, again fully-abstract, tuple-based notion of trace distance is shown to be able to handle nontrivial examples.

cs.LO↗

arXiv2015

Proceedings Tenth International Workshop on Developments in Computational Models

This volume contains the papers presented at the Tenth International Workshop on Developments in Computational Models (DCM) held in Vienna, Austria on 13th July 2014, as part of the Vienna Summer of Logic. Several new models of computation have emerged in the last years, and many developments of traditional computational models have been proposed with the aim of taking into account the new demands of computer systems users and the new capabilities of computation engines. A new computational model, or a new feature in a traditional one, usually is reflected in a new family of programming languages, and new paradigms of software development. The aim of this workshop is to bring together researchers who are currently developing new computational models or new features for traditional computational models, in order to foster their interaction, to provide a forum for presenting new ideas and work in progress, and to enable newcomers to learn about current activities in this area. Topics of interest include all abstract models of computation and their applications to the development of programming languages and systems.

cs.LO↗

arXiv2015

Wave-Style Token Machines and Quantum Lambda Calculi

Particle-style token machines are a way to interpret proofs and programs, when the latter are written following the principles of linear logic. In this paper, we show that token machines also make sense when the programs at hand are those of a simple quantum lambda-calculus with implicit qubits. This, however, requires generalising the concept of a token machine to one in which more than one particle travel around the term at the same time. The presence of multiple tokens is intimately related to entanglement and allows us to give a simple operational semantics to the calculus, coherently with the principles of quantum computation.

cs.LO↗

arXiv2015

On Sharing, Memoization, and Polynomial Time (Long Version)

We study how the adoption of an evaluation mechanism with sharing and memoization impacts the class of functions which can be computed in polynomial time. We first show how a natural cost model in which lookup for an already computed value has no cost is indeed invariant. As a corollary, we then prove that the most general notion of ramified recurrence is sound for polynomial time, this way settling an open problem in implicit computational complexity.

cs.CC↗