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Michele Pagani

Publications and source records attributed to Michele Pagani.

13 recordsLinked to original sources
arXiv2025

JAX Autodiff from a Linear Logic Perspective (Extended Version)

Autodiff refers to the core of the automatic differentiation systems developed in projects like JAX and Dex. Autodiff has recently been formalised in a linear typed calculus by Radul et al in arXiv:2204.10923. Although this formalisation suffices to express the main program transformations of Autodiff, the calculus is very specific to this task, and it is not clear whether the type system yields a substructural logic that has interest on its own. We propose an encoding of Autodiff into a linear $λ$-calculus that enjoys a Curry-Howard correspondence with Girard's linear logic. We prove that the encoding is sound both qualitatively (the encoded terms are extensionally equivalent to the original ones) and quantitatively (the encoding preserves the original work cost as described in arXiv:2204.10923). As a byproduct, we show that unzipping, one of the transformations used to implement backpropagation in Autodiff, is, in fact, optional.

cs.PL↗
arXiv2025

Variable Elimination as Rewriting in a Linear Lambda Calculus

Variable Elimination (VE) is a classical exact inference algorithm for probabilistic graphical models such as Bayesian Networks, computing the marginal distribution of a subset of the random variables in the model. Our goal is to understand Variable Elimination as an algorithm acting on programs, here expressed in an idealized probabilistic functional language -- a linear simply-typed $λ$-calculus suffices for our purpose. Precisely, we express VE as a term rewriting process, which transforms a global definition of a variable into a local definition, by swapping and nesting let-in expressions. We exploit in an essential way linear types.

cs.LO↗
arXiv2021

Automatic Differentiation in PCF

We study the correctness of automatic differentiation (AD) in the context of a higher-order, Turing-complete language (PCF with real numbers), both in forward and reverse mode. Our main result is that, under mild hypotheses on the primitive functions included in the language, AD is almost everywhere correct, that is, it computes the derivative or gradient of the program under consideration except for a set of Lebesgue measure zero. Stated otherwise, there are inputs on which AD is incorrect, but the probability of randomly choosing one such input is zero. Our result is in fact more precise, in that the set of failure points admits a more explicit description: for example, in case the primitive functions are just constants, addition and multiplication, the set of points where AD fails is contained in a countable union of zero sets of non-identically-zero polynomials.

cs.LO↗
arXiv2020

Revisiting Call-by-value Böhm trees in light of their Taylor expansion

The call-by-value lambda calculus can be endowed with permutation rules, arising from linear logic proof-nets, having the advantage of unblocking some redexes that otherwise get stuck during the reduction. We show that such an extension allows to define a satisfying notion of Böhm(-like) tree and a theory of program approximation in the call-by-value setting. We prove that all lambda terms having the same Böhm tree are observationally equivalent, and characterize those Böhm-like trees arising as actual Böhm trees of lambda terms. We also compare this approach with Ehrhard's theory of program approximation based on the Taylor expansion of lambda terms, translating each lambda term into a possibly infinite set of so-called resource terms. We provide sufficient and necessary conditions for a set of resource terms in order to be the Taylor expansion of a lambda term. Finally, we show that the normal form of the Taylor expansion of a lambda term can be computed by performing a normalized Taylor expansion of its Böhm tree. From this it follows that two lambda terms have the same Böhm tree if and only if the normal forms of their Taylor expansions coincide.

cs.LO↗
arXiv2020

The Benefit of Being Non-Lazy in Probabilistic λ-calculus

We consider the probabilistic applicative bisimilarity (PAB), a coinductive relation comparing the applicative behaviour of probabilistic untyped lambda terms according to a specific operational semantics. This notion has been studied with respect to the two standard parameter passing policies, call-by-value (cbv) and call-by-name (cbn), using a lazy reduction strategy not reducing within the body of a function. In particular, PAB has been proven to be fully abstract with respect to the contextual equivalence in cbv but not in lazy cbn. We overcome this issue of cbn by relaxing the laziness constraint: we prove that PAB is fully abstract with respect to the standard head reduction contextual equivalence. Our proof is based on the Leventis Separation Theorem, using probabilistic Nakajima trees as a tree-like representation of the contextual equivalence classes. Finally, we prove also that the inequality full abstraction fails, showing that the probabilistic applicative similarity is strictly contained in the contextual preorder.

cs.LO↗
arXiv2019

Backpropagation in the Simply Typed Lambda-calculus with Linear Negation

Backpropagation is a classic automatic differentiation algorithm computing the gradient of functions specified by a certain class of simple, first-order programs, called computational graphs. It is a fundamental tool in several fields, most notably machine learning, where it is the key for efficiently training (deep) neural networks. Recent years have witnessed the quick growth of a research field called differentiable programming, the aim of which is to express computational graphs more synthetically and modularly by resorting to actual programming languages endowed with control flow operators and higher-order combinators, such as map and fold. In this paper, we extend the backpropagation algorithm to a paradigmatic example of such a programming language: we define a compositional program transformation from the simply-typed lambda-calculus to itself augmented with a notion of linear negation, and prove that this computes the gradient of the source program with the same efficiency as first-order backpropagation. The transformation is completely effect-free and thus provides a purely logical understanding of the dynamics of backpropagation.

cs.LO↗
arXiv2017

Measurable Cones and Stable, Measurable Functions

We define a notion of stable and measurable map between cones endowed with measurability tests and show that it forms a cpo-enriched cartesian closed category. This category gives a denotational model of an extension of PCF supporting the main primitives of probabilistic functional programming, like continuous and discrete probabilistic distributions, sampling, conditioning and full recursion. We prove the soundness and adequacy of this model with respect to a call-by-name operational semantics and give some examples of its denotations.

cs.LO↗
arXiv2016

Strong Normalizability as a Finiteness Structure via the Taylor Expansion of λ-terms

In the folklore of linear logic, a common intuition is that the structure of finiteness spaces, introduced by Ehrhard, semantically reflects the strong normalization property of cut-elimination. We make this intuition formal in the context of the non-deterministic λ-calculus by introducing a finiteness structure on resource terms, which is such that a λ-term is strongly normalizing iff the support of its Taylor expansion is finitary. An application of our result is the existence of a normal form for the Taylor expansion of any strongly normalizable non-deterministic λ-term.

cs.LO↗
arXiv2015

Full abstraction for probabilistic PCF

We present a probabilistic version of PCF, a well-known simply typed universal functional language. The type hierarchy is based on a single ground type of natural numbers. Even if the language is globally call-by-name, we allow a call-by-value evaluation for ground type arguments in order to provide the language with a suitable algorithmic expressiveness. We describe a denotational semantics based on probabilistic coherence spaces, a model of classical Linear Logic developed in previous works. We prove an adequacy and an equational full abstraction theorem showing that equality in the model coincides with a natural notion of observational equivalence.

cs.LO↗
arXiv2013

Call-by-value non-determinism in a linear logic type discipline

We consider the call-by-value lambda-calculus extended with a may-convergent non-deterministic choice and a must-convergent parallel composition. Inspired by recent works on the relational semantics of linear logic and non-idempotent intersection types, we endow this calculus with a type system based on the so-called Girard's second translation of intuitionistic logic into linear logic. We prove that a term is typable if and only if it is converging, and that its typing tree carries enough information to give a bound on the length of its lazy call-by-value reduction. Moreover, when the typing tree is minimal, such a bound becomes the exact length of the reduction.

cs.LO↗
arXiv2013

Applying quantitative semantics to higher-order quantum computing

Finding a denotational semantics for higher order quantum computation is a long-standing problem in the semantics of quantum programming languages. Most past approaches to this problem fell short in one way or another, either limiting the language to an unusably small finitary fragment, or giving up important features of quantum physics such as entanglement. In this paper, we propose a denotational semantics for a quantum lambda calculus with recursion and an infinite data type, using constructions from quantitative semantics of linear logic.

cs.LO↗