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My research interests lie at the interface between computer science and logic, in particular proof theory. More specifically:
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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-null polynomials.
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.
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 Bohm(-like) tree and a theory of program approximation in the call-by-value setting. We prove that all lambda terms having the same Bohm tree are observationally equivalent, and characterize those Bohm-like trees arising as actual Bohm 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 Bohm tree. From this it follows that two lambda terms have the same Bohm tree if and only if the normal forms of their Taylor expansions coincide.
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.
Despite the fact that call-by-value lambda-calculus was defined by Plotkin in 1977, we believe that its theory of program approximation is still at the beginning. A problem that is often encountered when studying its operational semantics is that, during the reduction of a lambda-term, some redexes remain stuck (waiting for a value). Recently, Carraro and Guerrieri proposed to endow this calculus with permutation rules, naturally arising in the context of linear logic proof-nets, that succeed in unblocking a certain number of such redexes. In the present paper we introduce a new class of models of call-by-value lambda-calculus, arising from non-idempotent intersection type systems. Beside satisfying the usual properties as soundness and adequacy, these models validate the permutation rules mentioned above as well as some reductions obtained by contracting suitable lambda-I-redexes. Thanks to these (perhaps unexpected) features, we are able to demonstrate that every model living in this class satisfies an Approximation Theorem with respect to a refined notion of syntactic approximant. While this kind of results often require impredicative techniques like reducibility candidates, the quantitative information carried by type derivations in our system allows us to provide a combinatorial proof.
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.
We prove the conservation theorem for differential nets -- the graph-theoretical syntax of the differential extension of Linear Logic (Ehrhard and Regnier's DiLL). The conservation theorem states that the property of having infinite reductions (here infinite chains of cut elimination steps) is preserved by non-erasing steps. This turns the quest for strong normalization into one for non-erasing weak normalization, and indeed we use this result to prove strong normalization of simply typed DiLL (with promotion). Along the way to the theorem we achieve a number of additional results having their own interest, such as a standardization theorem and a slightly modified system of nets, DiLLdeltarho.
We give a geometric condition that characterizes the differential nets having a finitary interpretation in finiteness spaces: visible acyclicity. This is based on visible paths, an extension to differential nets of a class of paths we introduced in the framework of linear logic nets. The characterization is then carried out as follows: the differential nets having no visible cycles are exactly those whose interpretation is a finitary relation. Visible acyclicity discloses a new kind of correctness for the promotion rule of linear logic, which goes beyond sequent calculus correctness.
We give a semantic account of the execution time (i.e. the number of cut elimination steps leading to the normal form) of an untyped MELL net. We first prove that: 1) a net is head-normalizable (i.e. normalizable at depth 0) if and only if its interpretation in the multiset based relational semantics is not empty and 2) a net is normalizable if and only if its exhaustive interpretation (a suitable restriction of its interpretation) is not empty. We then give a semantic measure of execution time: we prove that we can compute the number of cut elimination steps leading to a cut free normal form of the net obtained by connecting two cut free nets by means of a cut-link, from the interpretations of the two cut free nets. These results are inspired by similar ones obtained by the first author for the untyped lambda-calculus.
We study the notion of solvability in the resource calculus, an extension of the lambda-calculus allowing to model resource consumption. Since this calculus is non-deterministic two different notions of solvability arise, one optimistic (angelical, may) and one pessimistic (demoniac, must). We give a syntactical, operational and logical characterization for the may-solvability and only a partial characterization of the the must-solvability. Finally we discuss the open problem of a complete characterization of the must-solvability.
The paper contains the first complete proof of strong normalization (SN) for full second order linear logic (LL): Girard's original proof uses a standardization theorem which is not proven. We introduce sliced pure structures (sps), a very general version of Girard's proof-nets, and we apply to sps Gandy's method to infer SN from weak normalization (WN). We prove a standardization theorem for sps: if WN without erasing steps holds for an sps, then it enjoys SN. A key step in our proof of standardization is a confluence theorem for sps obtained by using only a very weak form of correctness, namely acyclicity slice by slice. We conclude by showing how standardization for sps allows to prove SN of LL, using as usual Girard's reducibility candidates.
We study full completeness and syntactical separability of MLL proof nets with the mix rule. The general method we use consists first in addressing the two questions in the less restrictive framework of proof structures, and then in adapting the results to proof nets. At the level of proof structures, we find a semantical characterization of their interpretations in relational semantics, and we define an observational equivalence which is proved to be the equivalence induced by cut elimination. Hence, we obtain a semantical characterization (in coherent spaces) and an observational equivalence for the proof nets with the mix rule.
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 by Dal Lago et al. 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 context 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 context equivalence. Our proof is based on Leventis Separation Theorem, using probabilistic Nakajima trees as a tree-like representation of the context equivalence classes. Finally, we prove also that the inequality full abstraction fails, showing that the probabilistic applicative similarity is strictly contained in the context preorder.
We consider the notion of probabilistic applicative bisimilarity (PAB), recently introduced as a behavioural equivalence over a probabilistic extension of the untyped lambda-calculus. Alberti, Dal Lago and Sangiorgi have shown that PAB is not fully abstract with respect to the context equivalence induced by the lazy call-by-name evaluation strategy. We prove that extending this calculus with a let-in operator allows for achieving the full abstraction. In particular, we recall Larsen and Skou's testing language, which is known to correspond with PAB, and we prove that every test is representable by a context of our calculus.
Probabilistic coherence spaces yield a model of linear logic and lambda-calculus with a linear algebra flavor. Formulas/types are associated with convex sets of real valued vectors, linear logic proofs with linear functions and lambda-terms with entire functions, both mapping the convex set of their domain into the one of their codomain. Previous results show that this model is particularly precise in describing the observational equivalences between probabilistic functional programs. We prove here that the exponential modality is the free commutative comonad, giving a further mark of canonicity to the model.
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 lambda-calculus by introducing a finiteness structure on resource terms, which is such that a lambda-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 lambda-term.
Various typing system have been recently introduced giving a parametric version of the exponential modality of linear logic. The parameters are taken from a semi-ring, and allow to express coeffects -- i.e. specific requirements of a program with respect to the environment (availability of a resource, some prerequisite of the input, etc.). We show that all these systems can be interpreted in the relational category (REL) of sets and relations. This is possible because of the notion of multiplicity semi-ring, introduced by Carraro et al. and allowing a great variety of exponential comonads in REL. The interpretation of a particular typing system corresponds then to give a suitable notion of stratification of the exponential comonad associated with the semi-ring parametrising the exponential modality.
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.
Probabilistic coherence spaces (PCOH) yield a semantics of higher-order probabilistic computation, interpreting types as convex sets and programs as power series. We prove that the equality of interpretations in PCOH characterizes the operational indistinguishability of programs in PCF with a random primitive. This is the first result of full abstraction for a semantics of probabilistic PCF. The key ingredient relies on the regularity of power series and introduces, in denotational semantics, techniques coming from Calculus. Along the way to the theorem, we design a weighted intersection type assignment system giving a logical presentation of PCOH.
The Taylor expansion of lambda-terms, as introduced by Ehrhard and Regnier, expresses a lambda-term as a series of multi-linear terms, called simple terms, which capture bounded computations. Normal forms of Taylor expansions give a notion of infinitary normal forms, refining the notion of Böhm trees in a quantitative setting. We give the algebraic conditions over a set of normal simple terms which characterize the property of being the normal form of the Taylor expansion of a lambda-term. From this full completeness result, we give further conditions which semantically describe normalizable and total lambda-terms.
The category Rel of sets and relations yields one among the simplest denotational semantics of Linear Logic (LL). It is known that Rel is the biproduct completion of the Boolean ring. We consider the generalization of this construction to an arbitrary continuous semiring R, producing a cpo-enriched category which is a semantics of LL, and its (co)Kleisli category is an adequate model of an extension of PCF, parametrized by R. Specific instances of R allow us to compare programs not only with respect to "what they can do", but also "in how many steps" or "in how many different ways" (for non-deterministic PCF) or even "with what probability" (for probabilistic PCF).
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.
We study the probabilistic coherence spaces — a denotational semantics interpreting programs by power series with non negative real coefficients. We prove that this semantics is adequate for a probabilistic extension of the untyped $\lambda$-calculus: the probability that a term reduces to a head normal form is equal to its denotation computed on a suitable set of values. The result gives, in a probabilistic setting, a quantitative refinement to the adequacy of Scott's model for untyped $\lambda$-calculus.
We study the resource calculus, an extension of the $\lambda$-calculus allowing to model resource consumption. We achieve an internal separation result, in analogy with Bohm's theorem of $\lambda$-calculus. We define an equivalence relation on the terms, which we prove to be the maximal non-trivial congruence on normalizable terms respecting $\beta$-reduction and sum idempotency. It is significant that this equivalence extends the usual $\eta$-equivalence and is related to Ehrhard's Taylor expansion — a translation mapping terms into series of finite resources.
The resource calculus is an extension of the lambda-calculus allowing to model resource consumption. Namely, the argument of a function comes as a finite multiset of resources, which in turn can be either linear or reusable, giving rise to non-deterministic choices, expressed by a formal sum. Using the lambda-calculus terminology, we call solvable a term that can interact with the environment: solvable terms represent meaningful programs. Because of the non-determinism, different definitions of solvability are possible in the resource calculus. Here we study the optimistic (angelical, or may) notion, and so we define a term solvable whenever there is a simple head context reducing the term into a sum where at least one addend is the identity. We give a syntactical, operational and logical characterization of this kind of solvability.
We study the resource calculus — the non-lazy version of Boudol's lambda-calculus with resources. In such a calculus arguments may be finitely available and mixed, giving rise to nondeterminism, modelled by a formal sum. We define parallel reduction in resource calculus and we apply, in such a nondeterministic setting, the technique by Tait and Martin-Löf to achieve confluence. Then, slightly generalizing a technique by Takahashi, we obtain a standardization result.
Recently Ehrhard and Regnier have introduced Differential Linear Logic, DILL for short — an extension of the Multiplicative Exponential fragment of Linear Logic that is able to express non-deterministic computations. The authors have examined the cut-elimination of the promotion-free fragment of DILL by means of a proofnet-like calculus: differential interaction nets. We extend this analysis to exponential boxes and prove the Cut-Elimination Theorem for the whole DILL: every differential net that is sequentializable can be reduced to a cut-free net.
Linear Logic is based on the analogy between algebraic linearity (i.e. commutation with sums and with products with scalars) and the computer science linearity (i.e. calling inputs only once). Keeping on this analogy, Ehrhard and Regnier introduced Differential Linear Logic (DILL) — an extension of Multiplicative Exponential Linear Logic with differential constructions. In this setting, promotion (the logical exponentiation) can be approximated by a sum of promotion-free proofs of DILL, via Taylor expansion. We present a constructive way to revert Taylor expansion. Precisely, we define merging reduction — a rewriting system which merges a finite sum of DILL proofs into a proof with promotion whenever the sum is an approximation of the Taylor expansion of this proof. We prove that this algorithm is sound, complete and can be run in non-deterministic polynomial time.
Differential interaction nets (DIN) have been introduced by Thomas Ehrhard and Laurent Regnier as an extension of linear logic proof-nets. We prove that DIN enjoy an internal separation property: given two different normal nets, there exists a dual net separating them, in analogy with Böhm's theorem for the lambda-calculus. Our result implies in particular the faithfulness of every non-trivial denotational model of DIN (such as Ehrhard's finiteness spaces). We also observe that internal separation does not hold for linear logic proof-nets: our work points out that this failure is due to the fundamental asymmetry of linear logic exponential modalities, which are instead completely symmetric in DIN.
We give a geometric condition that characterizes the MELL proof structures whose interpretation is a clique in the non-uniform coherent spaces: visible acyclicity. We define the visible paths and we prove that the proof structures which have no visible cycles are exactly those whose interpretation is a clique. It turns out that visible acyclicity has also nice computational properties, especially it is stable under cut reduction.
Lambdamu-calculus has been built as an untyped extension of Parigot's lambdamu-calculus in order to recover Böhm theorem which was known to fail in lambdamu-calculus. An essential computational feature of Lambdamu-calculus for separation to hold is the unrestricted use of abstractions over continuations that provides the calculus with a construction of streams. Based on the Curry-Howard paradigm Laurent has defined a translation of lambdamu-calculus in polarized proof-nets. Unfortunately, this translation cannot be immediately extended to Lambdamu-calculus: the type system on which it is based freezes Lambdamu-calculus's stream mechanism. We introduce stream associative nets (SANE), a notion of nets which is between Laurent's polarized proof-nets and the usual linear logic proof-nets. SANE have two kinds of par (hence of tensor), one is linear while the other one allows free structural rules (as in polarized proof-nets). We prove confluence for SANE and give a reduction preserving encoding of Lambdamu-calculus in SANE, based on a new type system introduced by the second author. It turns out that the stream mechanism at work in Lambdamu-calculus can be explained by the associativity of the two different kinds of par of SANE. At last, we achieve a Böhm theorem for SANE. This result follows Girard's program to put into the fore the separation as a key property of logic.