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Cécilia Pradic

Publications and source records attributed to Cécilia Pradic.

18 recordsLinked to original sources
arXiv2026

The equational theory of the Weihrauch lattice with (iterated) composition

We study the equational theory of the Weihrauch lattice with composition and iteration, meaning the collection of equations between terms built from variables, the lattice operations $\sqcup,\sqcap$, the composition operator $\star$ and its iteration $(-)^\diamond$, which are true however we substitute (partial) Weihrauch degrees for the variables. We characterize them using Büchi games on finite graphs and give a complete axiomatization that derives them. The term signature and the axiomatization are reminiscent of Kleene algebras, except that we additionally have meets and the lattice operations do not fully distribute over composition. The game characterization gives a variant of the notion of simulation for alternating automata. It also implies that it is decidable whether an equation is universally valid. We give some complexity bounds; in particular, the problem is PSPACE-hard in general and we conjecture that it is solvable in PSPACE. We expect that the axiomatizations and games can also be applicable as-is to characterize the existence of strong natural transformation between polynomial functor expressions.

cs.LO↗
arXiv2026

Problems with fixpoints of polynomials of polynomials

Motivated by applications in computable analysis, we study fixpoints of certain endofunctors over categories of containers. More specifically, we focus on fibred endofunctors over the fibrewise opposite of the codomain fibration that can be themselves be represented by families of polynomial endofunctors. In this setting, we show how to compute initial algebras, terminal coalgebras and another kind of fixpoint $ζ$. We then explore a number of examples of derived operators inspired by Weihrauch complexity and the usual construction of the free polynomial monad. We introduce $ζ$-expressions as the syntax of $μ$-bicomplete categories, extended with $ζ$-binders and parallel products, which thus have a natural denotation in containers. By interpreting certain $ζ$-expressions in a category of type 2 computable maps, we are able to capture a number of meaningful Weihrauch degrees, ranging from closed choice on $\{0, 1\}$ to determinacy of infinite parity games, via an "answerable part" operator.

cs.LO↗
arXiv2025

Weihrauch problems as containers

We note that Weihrauch problems can be regarded as containers over the category of projective represented spaces and that Weihrauch reductions correspond exactly to container morphisms. We also show that Bauer's extended Weihrauch degrees and the posetal reflection of containers over partition assemblies are equivalent. Using this characterization, we show how a number of operators over Weihrauch degrees, such as the composition product, also arise naturally from the abstract theory of polynomial functors.

cs.LO↗
arXiv2025

The Myhill isomorphism theorem does not generalize much

The Myhill isomorphism is a variant of the Cantor-Bernstein theorem. It states that, from two injections that reduces two subsets of $\mathbb{N}$ to each other, there exists a bijection $\mathbb{N} \to \mathbb{N}$ that preserves them. This theorem can be proven constructively. We investigate to which extent the theorem can be extended to other infinite sets other than $\mathbb{N}$. We show that, assuming Markov's principle, the theorem can be extended to the conatural numbers $\mathbb{N}_{\infty}$ provided that we only require that bicomplemented sets are preserved by the bijection. This restriction is essential. Otherwise, the picture is overall negative: among other things, it is impossible to extend that result to either $2 \times \mathbb{N}_{\infty}$, $\mathbb{N} + \mathbb{N}_{\infty}$, $\mathbb{N} \times \mathbb{N}_{\infty}$, $\mathbb{N}_{\infty}^2$, $2^{\mathbb{N}}$ or $\mathbb{N}^{\mathbb{N}}$.

math.LO↗
arXiv2025

Computably discrete represented spaces

In computable topology, a represented space is called computably discrete if its equality predicate is semidecidable. While any such space is classically isomorphic to an initial segment of the natural numbers, the computable-isomorphism types of computably discrete represented spaces exhibit a rich structure. We show that the widely studied class of computably enumerable equivalence relations (ceers) corresponds precisely to the computably Quasi-Polish computably discrete spaces. We employ computably discrete spaces to exhibit several separating examples in computable topology. We construct a computably discrete computably Quasi-Polish space admitting no decidable properties, a computably discrete and computably Hausdorff precomputably Quasi-Polish space admitting no computable injection into the natural numbers, a two-point space which is computably Hausdorff but not computably discrete, and a two-point space which is computably discrete but not computably Hausdorff. We further expand an example due to Weihrauch that separates computably regular spaces from computably normal spaces.

math.LO↗
arXiv2024

Implicit automata in λ-calculi III: affine planar string-to-string functions

We prove a characterization of first-order string-to-string transduction via $λ$-terms typed in non-commutative affine logic that compute with Church encoding, extending the analogous known characterization of star-free languages. We show that every first-order transduction can be computed by a $λ$-term using a known Krohn-Rhodes-style decomposition lemma. The converse direction is given by compiling $λ$-terms into two-way reversible planar transducers. The soundness of this translation involves showing that the transition functions of those transducers live in a monoidal closed category of diagrams in which we can interpret purely affine $λ$-terms. One challenge is that the unit of the tensor of the category in question is not a terminal object. As a result, our interpretation does not identify $β$-equivalent terms, but it does turn $β$-reductions into inequalities in a poset-enrichment of the category of diagrams.

cs.LO↗
arXiv2024

The equational theory of the Weihrauch lattice with multiplication

We study the equational theory of the Weihrauch lattice with multiplication, meaning the collection of equations between terms built from variables, the lattice operations $\sqcup$, $\sqcap$, the product $\times$, and the finite parallelization $(-)^*$ which are true however we substitute Weihrauch degrees for the variables. We provide a combinatorial description of these in terms of a reducibility between finite graphs, and moreover, show that deciding which equations are true in this sense is complete for the third level of the polynomial hierarchy.

cs.LO↗
arXiv2024

Synthesizing nested relational queries from implicit specifications: via model theory and via proof theory

Derived datasets can be defined implicitly or explicitly. An implicit definition (of dataset O in terms of datasets I) is a logical specification involving two distinguished sets of relational symbols. One set of relations is for the "source data" I, and the other is for the "interface data" O. Such a specification is a valid definition of O in terms of I, if any two models of the specification agreeing on I agree on O. In contrast, an explicit definition is a transformation (or "query" below) that produces O from I. Variants of Beth's theorem state that one can convert implicit definitions to explicit ones. Further, this conversion can be done effectively given a proof witnessing implicit definability in a suitable proof system. We prove the analogous implicit-to-explicit result for nested relations: implicit definitions, given in the natural logic for nested relations, can be converted to explicit definitions in the nested relational calculus (NRC). We first provide a model-theoretic argument for this result, which makes some additional connections that may be of independent interest, between NRC queries, interpretations, a standard mechanism for defining structure-to-structure translation in logic, and between interpretations and implicit to definability "up to unique isomorphism". The latter connection uses a variation of a result of Gaifman concerning "relatively categorical" theories. We also provide a proof-theoretic result that provides an effective argument: from a proof witnessing implicit definability, we can efficiently produce an NRC definition. This will involve introducing the appropriate proof system for reasoning with nested sets, along with some auxiliary Beth-type results for this system. As a consequence, we can effectively extract rewritings of NRC queries in terms of NRC views, given a proof witnessing that the query is determined by the views.

cs.LO↗
arXiv2023

On the Weihrauch degree of the additive Ramsey theorem

We characterize the strength, in terms of Weihrauch degrees, of certain problems related to Ramsey-like theorems concerning colourings of the rationals and of the natural numbers. The theorems we are chiefly interested in assert the existence of almost-homogeneous sets for colourings of pairs of rationals respectively natural numbers satisfying properties determined by some additional algebraic structure on the set of colours. In the context of reverse mathematics, most of the principles we study are equivalent to $Σ^0_2$-induction over $\mathrm{RCA}_0$. The associated problems in the Weihrauch lattice are related to $\mathrm{TC}_\mathbb{N}^*$, $(\mathrm{LPO}')^*$ or their product, depending on their precise formalizations.

cs.LO↗
arXiv2023

Refutations of pebble minimization via output languages

Polyregular functions are the class of string-to-string functions definable by pebble transducers, an extension of finite-state automata with outputs and multiple two-way reading heads (pebbles) with a stack discipline. If a polyregular function can be computed with $k$ pebbles, then its output length is bounded by a polynomial of degree $k$ in the input length. But Bojańczyk has shown that the converse fails. In this paper, we provide two alternative easier proofs. The first establishes by elementary means that some quadratic polyregular function requires 3 pebbles. The second proof - just as short, albeit less elementary - shows a stronger statement: for every $k$, there exists some polyregular function with quadratic growth whose output language differs from that of any $k$-fold composition of macro tree transducers (and which therefore cannot be computed by a $k$-pebble transducer). Along the way, we also refute a conjectured logical characterization of polyblind functions.

cs.FL↗
arXiv2023

Comparison-free polyregular functions

This paper introduces a new automata-theoretic class of string-to-string functions with polynomial growth. Several equivalent definitions are provided: a machine model which is a restricted variant of pebble transducers, and a few inductive definitions that close the class of regular functions under certain operations. Our motivation for studying this class comes from another characterization, which we merely mention here but prove elsewhere, based on a $λ$-calculus with a linear type system.As their name suggests, these comparison-free polyregular functions form a subclass of polyregular functions; we prove that the inclusion is strict. We also show that they are incomparable with HDT0L transductions, closed under usual function composition -- but not under a certain ``map'' combinator -- and satisfy a comparison-free version of the pebble minimization theorem.On the broader topic of polynomial growth transductions, we also consider the recently introduced layered streaming string transducers (SSTs), or equivalently k-marble transducers. We prove that a function can be obtained by composing such transducers together if and only if it is polyregular, and that k-layered SSTs (or k-marble transducers) are closed under ``map'' and equivalent to a corresponding notion of (k+1)-layered HDT0L systems.

cs.FL↗
arXiv2022

Synthesizing Nested Relational Queries from Implicit Specifications

Derived datasets can be defined implicitly or explicitly. An implicit definition (of dataset $O$ in terms of datasets $\vec{I}$) is a logical specification involving the source data $\vec{I}$ and the interface data $O$. It is a valid definition of $O$ in terms of $\vec{I}$, if any two models of the specification agreeing on $\vec{I}$ agree on $O$. In contrast, an explicit definition is a query that produces $O$ from $\vec{I}$. Variants of Beth's theorem state that one can convert implicit definitions to explicit ones. Further, this conversion can be done effectively given a proof witnessing implicit definability in a suitable proof system. We prove the analogous effective implicit-to-explicit result for nested relations: implicit definitions, given in the natural logic for nested relations, can be effectively converted to explicit definitions in the nested relational calculus NRC. As a consequence, we can effectively extract rewritings of NRC queries in terms of NRC views, given a proof witnessing that the query is determined by the views.

cs.DB↗
arXiv2022

Cantor-Bernstein implies Excluded Middle

We prove in constructive logic that the statement of the Cantor-Bernstein theorem implies excluded middle. This establishes that the Cantor-Bernstein theorem can only be proven assuming the full power of classical logic. The key ingredient is a theorem of Martín Escardó stating that quantification over a particular subset of the Cantor space $2^{\mathbb{N}}$, the so-called one-point compactification of $\mathbb{N}$, preserves decidable predicates.

math.LO↗
arXiv2021

Implicit automata in typed $λ$-calculi II: streaming transducers vs categorical semantics

We characterize regular string transductions as programs in a linear $λ$-calculus with additives. One direction of this equivalence is proved by encoding copyless streaming string transducers (SSTs), which compute regular functions, into our $λ$-calculus. For the converse, we consider a categorical framework for defining automata and transducers over words, which allows us to relate register updates in SSTs to the semantics of the linear $λ$-calculus in a suitable monoidal closed category. To illustrate the relevance of monoidal closure to automata theory, we also leverage this notion to give abstract generalizations of the arguments showing that copyless SSTs may be determinized and that the composition of two regular functions may be implemented by a copyless SST. Our main result is then generalized from strings to trees using a similar approach. In doing so, we exhibit a connection between a feature of streaming tree transducers and the multiplicative/additive distinction of linear logic. Keywords: MSO transductions, implicit complexity, Dialectica categories, Church encodings

cs.LO↗
arXiv2020

Generating collection transformations from proofs

Nested relations, built up from atomic types via product and set types, form a rich data model. Over the last decades the nested relational calculus, NRC, has emerged as a standard language for defining transformations on nested collections. NRC is a strongly-typed functional language which allows building up transformations using tupling and projections, a singleton-former, and a map operation that lifts transformations on tuples to transformations on sets. In this work we describe an alternative declarative method of describing transformations in logic. A formula with distinguished inputs and outputs gives an implicit definition if one can prove that for each input there is only one output that satisfies it. Our main result shows that one can synthesize transformations from proofs that a formula provides an implicit definition, where the proof is in an intuitionistic calculus that captures a natural style of reasoning about nested collections. Our polynomial time synthesis procedure is based on an analog of Craig's interpolation lemma, starting with a provable containment between terms representing nested collections and generating an NRC expression that interpolates between them. We further show that NRC expressions that implement an implicit definition can be found when there is a classical proof of functionality, not just when there is an intuitionistic one. That is, whenever a formula implicitly defines a transformation, there is an NRC expression that implements it.

cs.LO↗
arXiv2019

A Curry-Howard Approach to Church's Synthesis

Church's synthesis problem asks whether there exists a finite-state stream transducer satisfying a given input-output specification. For specifications written in Monadic Second-Order Logic (MSO) over infinite words, Church's synthesis can theoretically be solved algorithmically using automata and games. We revisit Church's synthesis via the Curry-Howard correspondence by introducing SMSO, an intuitionistic variant of MSO over infinite words, which is shown to be sound and complete w.r.t. synthesis thanks to an automata-based realizability model.

cs.LO↗
arXiv2019

The logical strength of Büchi's decidability theorem

We study the strength of axioms needed to prove various results related to automata on infinite words and Büchi's theorem on the decidability of the MSO theory of $(N, {\le})$. We prove that the following are equivalent over the weak second-order arithmetic theory $RCA_0$: (1) the induction scheme for $Σ^0_2$ formulae of arithmetic, (2) a variant of Ramsey's Theorem for pairs restricted to so-called additive colourings, (3) Büchi's complementation theorem for nondeterministic automata on infinite words, (4) the decidability of the depth-$n$ fragment of the MSO theory of $(N, {\le})$, for each $n \ge 5$. Moreover, each of (1)-(4) implies McNaughton's determinisation theorem for automata on infinite words, as well as the "bounded-width" version of König's Lemma, often used in proofs of McNaughton's theorem.

cs.LO↗