arXiv subjects

Charles Grellois

Publications and source records attributed to Charles Grellois.

13 recordsLinked to original sources

arXiv2026

Enhancing Automated Machine Learning via Homogeneous Train-Test Splitting Methods

Accurate model evaluation in machine learning depends critically on how datasets are split into training and testing subsets. Standard random splitting assumes that both partitions share the same underlying distribution, an assumption often violated in datasets with class imbalance, natural clustering, or spatial autocorrelation. This paper investigates the role of statistical similarity in train-test splitting and its consequences for AutoML model evaluation. Five established strategies are compared across fifteen UCI benchmark datasets: random splitting, stratified sampling, Kennard-Stone, Duplex, and SPXY. Similarity is assessed using chi-square, Kolmogorov-Smirnov, and Maximum Mean Discrepancy (MMD) tests. Geometry-based methods consistently produce near-zero MMD scores, introducing instability into downstream performance estimates. The proposed Optimised-Distribution method treats similarity as an explicit optimisation objective and achieves the highest mean MMD similarity, 89.0%, across all strategies evaluated.

cs.LG↗

arXiv2026

Phase Semantic Cut-elimination for Intuitionistic Linear Logic with Least and Greatest Fixed Points

This paper establishes the cut-elimination theorem for intuitionistic propositional multiplicative-additive linear logic with the least and greatest fixpoints ($\mu$IMALL) by means of its phase semantics. A classical first-order multiplicative-additive linear logic system with the least and greatest fixpoints was introduced by Baelde and Miller (2007). Its intuitionistic fragment was discussed in Baelde (2012), but the cut-elimination theorem for this fragment has not yet been proved. We introduce a propositional fragment of this system, $\mu$IMALL, and establish the cut-elimination theorem. To prove the theorem, we define phase semantics for $\mu$IMALL and show the following two statements: (1) Soundness: if a formula is provable in $\mu$IMALL, then it is true in all phase models, and (2) Cut-free Completeness: if a formula is true in all phase models, then it is provable in $\mu$IMALL without Cut. Okada (1999, 2002) employed a phase semantic method to prove the cut-elimination theorems for classical and intuitionistic linear logic systems. De et al. (2022) applied this method to a propositional fragment of classical propositional multiplicative-additive linear logic with the least and greatest fixpoints. We refine and apply their arguments to prove the cut-elimination theorem for $\mu$IMALL.

cs.LO↗

arXiv2026

A coalgebraic higher-order modal fixed-point logic

We introduce a coalgebraic extension of the higher-order modal fixed-point logic (HFL) which subsumes both HFL and its probabilistic extension. We show that the emptiness problem for non-deterministic finite automata as well as the value-1 problem for probabilistic automata reduce to model-checking problems for this coalgebraic formulation of HFL.

cs.LO↗

arXiv2026

Tree Automata Acceptance up to Measurable Defect

Automata acceptance can, in several situations of interest, be captured game-theoretically via acceptance games. The existence of a winning strategy for Verifier then captures the existence of a winning run-tree of a given automaton over a model. However, such acceptance is rigid, in that it does not allow a measurable defect budget, which can be a challenge in software verification. In this paper, we draw inspiration from how bisimulation distance can be defined as an extension of bisimilarity to define epsilon-acceptance games. Our main theorem shows that a tree T is epsilon-accepted iff there is a tree T' that is accepted in the traditional (rigid) sense and the bisimulation distance of T' and T is at most epsilon. Our work also suggests a strong connection with measure theory, of which we give a preliminary exploration via appropriate examples. Our framework is defined over binary trees with leaves and infinite branches, and strictly contains the case in which binary nodes are seen as probabilistic choice and the defect measures the probability of the set of rejected branches.

cs.LO↗

arXiv2026

Adequate Losses via Quantitative Linear Logic

As neural components are increasingly embedded in existing symbolic software -- including safety-critical systems -- the question arises of how to specify and enforce the safety of the newly introduced neural parts. Unlike traditional logical specifications, these must be amenable not only to the standard Boolean interpretation, but also to training and optimisation. The latter calls for a quantitative interpretation of the logical syntax, subject to further requirements such as smoothness and differentiability. Moreover, the qualitative and quantitative sides of the logic must share a unifying proof-theoretic and categorical semantics. Finally, the new logic should link cleanly to the substructural and program logics that underpin the verification of existing symbolic programs. In this paper, we present a logic that ticks all of these boxes. We introduce a family of calculi, pQLL, indexed by a hardness degree $p$, prove a cut-elimination theorem for them, and establish completeness with respect to enriched residuated `soft' lattices. At $p = \infty$, \pQLL reduces to multiplicative additive linear logic (MALL), and provability in pQLL converges to provability in MALL as $p \to \infty$. We express optimisation objectives in the syntax of this logic and prove the quantitative adequacy of neuro-symbolic loss functions -- a result that has eluded the neuro-symbolic machine learning community for nearly a decade.

cs.LO↗

arXiv2019

Intuitionistic Non-Normal Modal Logics: A general framework

We define a family of intuitionistic non-normal modal logics; they can bee seen as intuitionistic counterparts of classical ones. We first consider monomodal logics, which contain only one between Necessity and Possibility. We then consider the more important case of bimodal logics, which contain both modal operators. In this case we define several interactions between Necessity and Possibility of increasing strength, although weaker than duality. For all logics we provide both a Hilbert axiomatisation and a cut-free sequent calculus, on its basis we also prove their decidability. We then give a semantic characterisation of our logics in terms of neighbourhood models. Our semantic framework captures modularly not only our systems but also already known intuitionistic non-normal modal logics such as Constructive K (CK) and the propositional fragment of Wijesekera's Constructive Concurrent Dynamic Logic.

cs.LO↗

arXiv2018

On the Termination Problem for Probabilistic Higher-Order Recursive Programs

In the last two decades, there has been much progress on model checking of both probabilistic systems and higher-order programs. In spite of the emergence of higher-order probabilistic programming languages, not much has been done to combine those two approaches. In this paper, we initiate a study on the probabilistic higher-order model checking problem, by giving some first theoretical and experimental results. As a first step towards our goal, we introduce PHORS, a probabilistic extension of higher-order recursion schemes (HORS), as a model of probabilistic higher-order programs. The model of PHORS may alternatively be viewed as a higher-order extension of recursive Markov chains. We then investigate the probabilistic termination problem -- or, equivalently, the probabilistic reachability problem. We prove that almost sure termination of order-2 PHORS is undecidable. We also provide a fixpoint characterization of the termination probability of PHORS, and develop a sound (but possibly incomplete) procedure for approximately computing the termination probability. We have implemented the procedure for order-2 PHORSs, and confirmed that the procedure works well through preliminary experiments that are reported at the end of the article.

cs.PL↗

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↗

arXiv2015

Relational semantics of linear logic and higher-order model-checking

In this article, we develop a new and somewhat unexpected connection between higher-order model-checking and linear logic. Our starting point is the observation that once embedded in the relational semantics of linear logic, the Church encoding of any higher-order recursion scheme (HORS) comes together with a dual Church encoding of an alternating tree automata (ATA) of the same signature. Moreover, the interaction between the relational interpretations of the HORS and of the ATA identifies the set of accepting states of the tree automaton against the infinite tree generated by the recursion scheme. We show how to extend this result to alternating parity automata (APT) by introducing a parametric version of the exponential modality of linear logic, capturing the formal properties of colors (or priorities) in higher-order model-checking. We show in particular how to reunderstand in this way the type-theoretic approach to higher-order model-checking developed by Kobayashi and Ong. We briefly explain in the end of the paper how his analysis driven by linear logic results in a new and purely semantic proof of decidability of the formulas of the monadic second-order logic for higher-order recursion schemes.

cs.LO↗

arXiv2015

Finitary semantics of linear logic and higher-order model-checking

In this paper, we explain how the connection between higher-order model-checking and linear logic recently exhibited by the authors leads to a new and conceptually enlightening proof of the selection problem originally established by Carayol and Serre using collapsible pushdown automata. The main idea is to start from an infinitary and colored relational semantics of the lambdaY-calculus already formulated, and to replace it by its finitary counterpart based on finite prime-algebraic lattices. Given a higher-order recursion scheme G, the finiteness of its interpretation in the model enables us to associate to any MSO formula phi a new higher-order recursion scheme G_phi resolving the selection problem.

cs.LO↗

arXiv2015

Indexed linear logic and higher-order model checking

In recent work, Kobayashi observed that the acceptance by an alternating tree automaton A of an infinite tree T generated by a higher-order recursion scheme G may be formulated as the typability of the recursion scheme G in an appropriate intersection type system associated to the automaton A. The purpose of this article is to establish a clean connection between this line of work and Bucciarelli and Ehrhard's indexed linear logic. This is achieved in two steps. First, we recast Kobayashi's result in an equivalent infinitary intersection type system where intersection is not idempotent anymore. Then, we show that the resulting type system is a fragment of an infinitary version of Bucciarelli and Ehrhard's indexed linear logic. While this work is very preliminary and does not integrate key ingredients of higher-order model-checking like priorities, it reveals an interesting and promising connection between higher-order model-checking and linear logic.

cs.LO↗

arXiv2015

An infinitary model of linear logic

In this paper, we construct an infinitary variant of the relational model of linear logic, where the exponential modality is interpreted as the set of finite or countable multisets. We explain how to interpret in this model the fixpoint operator Y as a Conway operator alternatively defined in an inductive or a coinductive way. We then extend the relational semantics with a notion of color or priority in the sense of parity games. This extension enables us to define a new fixpoint operator Y combining both inductive and coinductive policies. We conclude the paper by sketching the connection between the resulting model of lambda-calculus with recursion and higher-order model-checking.

cs.LO↗

arXiv2011

Algebraic theories, monads, and arities

Monads are of interest both in semantics and in higher dimensional algebra. It turns out that the idea behind usual notion finitary monads (whose values on all sets can be computed from their values on finite sets) extends to a more general class of monads called monads with arities, so that not only algebraic theories can be computed from a proper set of arities, but also more general structures like n-categories, the computing process being realized using Kan extensions. This Master thesis compiles the required material in order to understand this question of arities and reconstruction of monads, and tries to give some examples of relevant interest from both semantics and higher category theory. A discussion on the promising field of operads is then provided as appendix.

math.CT↗