arXiv subjects

François Fages

Publications and source records attributed to François Fages.

17 recordsLinked to original sources

arXiv2026

On the Trap Space Semantics of Normal Logic Programs

The logical semantics of normal logic programs has traditionally been based on the notions of Clark's completion and two-valued or three-valued canonical models, including supported, stable, regular, and well-founded models. Two-valued interpretations can also be seen as states evolving under a program's update operator, producing a transition graph whose fixed points and cycles capture stable and oscillatory behaviors, respectively. We refer to this view as dynamical semantics since it characterizes the program's meaning in terms of state-space trajectories, as first introduced in the stable (supported) class semantics. Recently, we have established a formal connection between Datalog^\neg programs (i.e., normal logic programs without function symbols) and Boolean networks, leading to the introduction of the trap space concept for Datalog^\neg programs. In this paper, we generalize the trap space concept to arbitrary normal logic programs, introducing trap space semantics as a new approach to their interpretation. This new semantics admits both model-theoretic and dynamical characterizations, providing a comprehensive approach to understanding program behavior. We establish the foundational properties of the trap space semantics and systematically relate it to the established model-theoretic semantics, including the stable (supported), stable (supported) partial, regular, and L-stable model semantics, as well as to the dynamical stable (supported) class semantics. Our results demonstrate that the trap space semantics offers a unified and precise framework for proving the existence of supported classes, strict stable (supported) classes, and regular models, in addition to uncovering and formalizing deeper relationships among the existing semantics of normal logic programs.

cs.LO↗

arXiv2025

On the Boolean Network Theory of Datalog$^\neg$

Datalog$^\neg$ is a central formalism used in a variety of domains ranging from deductive databases and abstract argumentation frameworks to answer set programming. Its model theory is the finite counterpart of the logical semantics developed for normal logic programs, mainly based on the notions of Clark's completion and two-valued or three-valued canonical models including supported, stable, regular and well-founded models. In this paper we establish a formal link between Datalog$^\neg$ and Boolean network theory first introduced for gene regulatory networks. We show that in the absence of odd cycles in a Datalog$^\neg$ program, the regular models coincide with the stable models, which entails the existence of stable models, and in the absence of even cycles, we prove the uniqueness of stable partial models and regular models. This connection also gives new upper bounds on the numbers of stable partial, regular, and stable models of a Datalog$^\neg$ program using the cardinality of a feedback vertex set in its atom dependency graph. Interestingly, our connection to Boolean network theory also points us to the notion of trap spaces. In particular we show the equivalence between subset-minimal stable trap spaces and regular models.

cs.LO↗

arXiv2025

Graphical Conditions for the Existence, Unicity and Number of Regular Models

The regular models of a normal logic program are a particular type of partial (i.e. 3-valued) models which correspond to stable partial models with minimal undefinedness. In this paper, we explore graphical conditions on the dependency graph of a finite ground normal logic program to analyze the existence, unicity and number of regular models for the program. We show three main results: 1) a necessary condition for the existence of non-trivial (i.e. non-2-valued) regular models, 2) a sufficient condition for the unicity of regular models, and 3) two upper bounds for the number of regular models based on positive feedback vertex sets. The first two conditions generalize the finite cases of the two existing results obtained by You and Yuan (1994) for normal logic programs with well-founded stratification. The third result is also new to the best of our knowledge. Key to our proofs is a connection that we establish between finite ground normal logic programs and Boolean network theory.

cs.LO↗

arXiv2024

Graphical Conditions ensuring Equality between Differential and Mean Stochastic Dynamics

Complex systems can be advantageously modeled by formal reaction systems (RS), a.k.a. chemical reaction networks in chemistry. Reaction-based models can indeed be interpreted in a hierarchy of semantics, depending on the question at hand, most notably by Ordinary Differential Equations (ODEs), Continuous Time Markov Chains (CTMCs), discrete Petri nets and asynchronous Boolean transition systems. The last three semantics can be easily related in the framework of abstract interpretation. The first two are classically related by Kurtz's limit theorem which states that if reactions are density-dependent families, then, as the volume goes to infinity, the mean reactant concentrations of the CTMC tends towards the solution of the ODE. In the more realistic context of bounded volumes, it is easy to show, by moment closure, that the restriction to reactions with at most one reactant ensures similarly that the mean of the CTMC trajectories is equal to the solution of the ODE at all time points. In this paper, we generalize that result in presence of polyreactant reactions, by introducing the Stoichiometric Influence and Modification Graph (SIMG) of an RS, and by showing that the equality between the two interpretations holds for the variables that belong to distinct SIMG ancestors of polyreactant reactions. We illustrate this approach with several examples. Evaluation on BioModels reveals that the condition for all variables is satisfied on models with no polymolecular reaction only. However, our theorem can be applied selectively to certain variables of the model to provide insights into their behaviour within more complex systems. Interestingly, we also show that the equality holds for a basic oscillatory RS implementing the sine and cosine functions of time.

q-bio.MN↗

arXiv2024

A Constraint-based Mathematical Modeling Library in Prolog with Answer Constraint Semantics

Constraint logic programming emerged in the late 80's as a highly declarative class of programming languages based on first-order logic and theories with decidable constraint languages, thereby subsuming Prolog restricted to equality constraints over the Herbrand's term domain. This approach has proven extremely successfull in solving combinatorial problems in the industry which quickly led to the development of a variety of constraint solving libraries in standard programming languages. Later came the design of a purely declarative front-end constraint-based modeling language, MiniZinc, independent of the constraint solvers, in order to compare their performances and create model benchmarks. Beyond that purpose, the use of a high-level modeling language such as MiniZinc to develop complete applications, or to teach constraint programming, is limited by the impossibility to program search strategies, or new constraint solvers, in a modeling language, as well as by the absence of an integrated development environment for both levels of constraint-based modeling and constraint solving. In this paper, we propose to solve those issues by taking Prolog with its constraint solving libraries, as a unified relation-based modeling and programming language. We present a Prolog library for high-level constraint-based mathematical modeling, inspired by MiniZinc, using subscripted variables (arrays) in addition to lists and terms, quantifiers and iterators in addition to recursion, together with a patch of constraint libraries in order to allow array functional notations in constraints. We show that this approach does not come with a significant computation time overhead, and presents several advantages in terms of the possibility of focussing on mathematical modeling, getting answer constraints in addition to ground solutions, programming search or constraint solvers if needed, and debugging models within a unique modeling and programming environment.

cs.LO↗

arXiv2023

On a model of online analog computation in the cell with absolute functional robustness: algebraic characterization, function compiler and error control

The Turing completeness of continuous Chemical Reaction Networks (CRNs) states that any computable real function can be computed by a continuous CRN on a finite set of molecular species, possibly restricted to elementary reactions, i.e. with at most two reactants and mass action law kinetics. In this paper, we introduce a more stringent notion of robust online analog computation, and Absolute Functional Robustness (AFR), for the CRNs that stabilize the concentration values of some output species to the result of one function of the input species concentrations, in a perfectly robust manner with respect to perturbations of both intermediate and output species. We prove that the set of real functions stabilized by a CRN with mass action law kinetics is precisely the set of real algebraic functions. Based on this result, we present a compiler which takes as input any algebraic function (defined by one polynomial and one point for selecting one branch of the algebraic curve defined by the polynomial) and generates an abstract CRN to stabilize it. Furthermore, we provide error bounds to estimate and control the error of an unperturbed system, under the assumption that the environment inputs are driven by k-Lipschitz functions.

q-bio.QM↗

arXiv2023

A Skin Microbiome Model with AMP interactions and Analysis of Quasi-Stability vs Stability in Population Dynamics

The skin microbiome plays an important role in the maintenance of a healthy skin. It is an ecosystem, composed of several species, competing for resources and interacting with the skin cells. Imbalance in the cutaneous microbiome, also called dysbiosis, has been correlated with several skin conditions, including acne and atopic dermatitis. Generally, dysbiosis is linked to colonization of the skin by a population of opportunistic pathogenic bacteria. Treatments consisting in non-specific elimination of cutaneous microflora have shown conflicting results. In this article, we introduce a mathematical model based on ordinary differential equations, with 2 types of bacteria populations (skin commensals and opportunistic pathogens) and including the production of antimicrobial peptides to study the mechanisms driving the dominance of one population over the other. By using published experimental data, assumed to correspond to the observation of stable states in our model, we reduce the number of parameters of the model from 13 to 5. We then use a formal specification in quantitative temporal logic to calibrate our model by global parameter optimization and perform sensitivity analyses. On the time scale of 2 days of the experiments, the model predicts that certain changes of the environment, like the elevation of skin surface pH, create favorable conditions for the emergence and colonization of the skin by the opportunistic pathogen population, while the production of human AMPs has non-linear effect on the balance between pathogens and commensals. Surprisingly, simulations on longer time scales reveal that the equilibrium reached around 2 days can in fact be a quasi-stable state followed by the reaching of a reversed stable state after 12 days or more. We analyse the conditions of quasi-stability observed in this model using tropical algebraic methods, and show their non-generic character in contrast to slow-fast systems. These conditions are then generalized to a large class of population dynamics models over any number of species.

q-bio.QM↗

arXiv2023

On Estimating Derivatives of Input Signals in Biochemistry

The online estimation of the derivative of an input signal is widespread in control theory and engineering. In the realm of chemical reaction networks (CRN), this raises however a number of specific issues on the different ways to achieve it. A CRN pattern for implementing a derivative block has already been proposed for the PID control of biochemical processes, and proved correct using Tikhonov's limit theorem. In this paper, we give a detailed mathematical analysis of that CRN, thus clarifying the computed quantity and quantifying the error done as a function of the reaction kinetic parameters. In a synthetic biology perspective, we show how this can be used to design error correcting terms to compute online functions involving derivatives with CRNs. In the systems biology perspective, we give the list of models in BioModels containing (in the sense of subgraph epimorphisms) the core derivative CRN, most of which being models of oscillators and control systems in the cell, and discuss in detail two such examples: one model of the circadian clock and one model of a bistable switch.

q-bio.QM↗

arXiv2023

Neural-based classification rule learning for sequential data

Discovering interpretable patterns for classification of sequential data is of key importance for a variety of fields, ranging from genomics to fraud detection or more generally interpretable decision-making. In this paper, we propose a novel differentiable fully interpretable method to discover both local and global patterns (i.e. catching a relative or absolute temporal dependency) for rule-based binary classification. It consists of a convolutional binary neural network with an interpretable neural filter and a training strategy based on dynamically-enforced sparsity. We demonstrate the validity and usefulness of the approach on synthetic datasets and on an open-source peptides dataset. Key to this end-to-end differentiable method is that the expressive patterns used in the rules are learned alongside the rules themselves.

cs.LG↗

arXiv2023

Reactmine: a statistical search algorithm for inferring chemical reactions from time series data

Inferring chemical reaction networks (CRN) from concentration time series is a challenge encouragedby the growing availability of quantitative temporal data at the cellular level. This motivates thedesign of algorithms to infer the preponderant reactions between the molecular species observed ina given biochemical process, and build CRN structure and kinetics models. Existing ODE-basedinference methods such as SINDy resort to least square regression combined with sparsity-enforcingpenalization, such as Lasso. However, we observe that these methods fail to learn sparse modelswhen the input time series are only available in wild type conditions, i.e. without the possibility toplay with combinations of zeroes in the initial conditions. We present a CRN inference algorithmwhich enforces sparsity by inferring reactions in a sequential fashion within a search tree of boundeddepth, ranking the inferred reaction candidates according to the variance of their kinetics on theirsupporting transitions, and re-optimizing the kinetic parameters of the CRN candidates on the wholetrace in a final pass. We show that Reactmine succeeds both on simulation data by retrievinghidden CRNs where SINDy fails, and on two real datasets, one of fluorescence videomicroscopyof cell cycle and circadian clock markers, the other one of biomedical measurements of systemiccircadian biomarkers possibly acting on clock gene expression in peripheral organs, by inferringpreponderant regulations in agreement with previous model-based analyses. The code is available athttps://gitlab.inria.fr/julmarti/crninf/ together with introductory notebooks.

q-bio.QM↗

arXiv2022

Stability versus Meta-stability in a Skin Microbiome Model

The skin microbiome plays an important role in the maintenance of a healthy skin. It is an ecosystem, composed of several species, competing for resources and interacting with the skin cells. Imbalance in the cutaneous microbiome, also called dysbiosis, has been correlated with several skin conditions, including acne and atopic dermatitis. Generally, dysbiosis is linked to colonization of the skin by a population of opportunistic pathogenic bacteria (for example C. acnes in acne or S. aureus in atopic dermatitis). Treatments consisting in non-specific elimination of cutaneous microflora have shown conflicting results. It is therefore necessary to understand the factors influencing shifts of the skin microbiome composition. In this work, we introduce a mathematical model based on ordinary differential equations, with 2 types of bacteria populations (skin commensals and opportunistic pathogens) to study the mechanisms driving the dominance of one population over the other. By using published experimental data, assumed to correspond to the observation of stable states in our model, we derive constraints that allow us to reduce the number of parameters of the model from 13 to 5. Interestingly, a meta-stable state settled at around 2 days following the introduction of bacteria in the model, is followed by a reversed stable state after 300 hours. On the time scale of the experiments, we show that certain changes of the environment, like the elevation of skin surface pH, create favorable conditions for the emergence and colonization of the skin by the opportunistic pathogen population. Such predictions help identifying potential therapeutic targets for the treatment of skin conditions involving dysbiosis of the microbiome, and question the importance of meta-stable states in mathematical models of biological processes.

q-bio.QM↗

arXiv2022

Algebraic Biochemistry: a Framework for Analog Online Computation in Cells

The Turing completeness of continuous chemical reaction networks (CRNs) states that any computable real function can be computed by a continuous CRN on a finite set of molecular species, possibly restricted to elementary reactions, i.e. with at most two reactants and mass action law kinetics. In this paper, we introduce a notion of online analog computation for the CRNs that stabilize the concentration of their output species to the result of some function of the concentration values of their input species, whatever changes are operated on the inputs during the computation. We prove that the set of real functions stabilized by a CRN with mass action law kinetics is precisely the set of real algebraic functions.

q-bio.MN↗

arXiv2021

Compiling Elementary Mathematical Functions into Finite Chemical Reaction Networks via a Polynomialization Algorithm for ODEs

The Turing completeness result for continuous chemical reaction networks (CRN) shows that any computable function over the real numbers can be computed by a CRN over a finite set of formal molecular species using at most bimolecular reactions with mass action law kinetics. The proof uses a previous result of Turing completeness for functions defined by polynomial ordinary differential equations (PODE), the dualrail encoding of real variables by the difference of concentration between two molecular species, and a back-end quadratization transformation to restrict to elementary reactions with at most two reactants. In this paper, we present a polynomialization algorithm of quadratic time complexity to transform a system of elementary differential equations in PODE. This algorithm is used as a front-end transformation to compile any elementary mathematical function, either of time or of some input species, into a finite CRN. We illustrate the performance of our compiler on a benchmark of elementary functions relevant to CRN design problems in synthetic biology specified by mathematical functions. In particular, the abstract CRN obtained by compilation of the Hill function of order 5 is compared to the natural CRN structure of MAPK signalling networks.

q-bio.QM↗

arXiv2020

On the Complexity of Quadratization for Polynomial Differential Equations

Chemical reaction networks (CRNs) are a standard formalism used in chemistry and biology to reason about the dynamics of molecular interaction networks. In their interpretation by ordinary differential equations, CRNs provide a Turing-complete model of analog computattion, in the sense that any computable function over the reals can be computed by a finite number of molecular species with a continuous CRN which approximates the result of that function in one of its components in arbitrary precision. The proof of that result is based on a previous result of Bournez et al. on the Turing-completeness of polyno-mial ordinary differential equations with polynomial initial conditions (PIVP). It uses an encoding of real variables by two non-negative variables for concentrations, and a transformation to an equivalent quadratic PIVP (i.e. with degrees at most 2) for restricting ourselves to at most bimolecular reactions. In this paper, we study the theoretical and practical complexities of the quadratic transformation. We show that both problems of minimizing either the number of variables (i.e., molecular species) or the number of monomials (i.e. elementary reactions) in a quadratic transformation of a PIVP are NP-hard. We present an encoding of those problems in MAX-SAT and show the practical complexity of this algorithm on a benchmark of quadratization problems inspired from CRN design problems.

q-bio.QM↗

arXiv2020

Graphical Conditions for Rate Independence in Chemical Reaction Networks

Chemical Reaction Networks (CRNs) provide a useful abstraction of molecular interaction networks in which molecular structures as well as mass conservation principles are abstracted away to focus on the main dynamical properties of the network structure. In their interpretation by ordinary differential equations, we say that a CRN with distinguished input and output species computes a positive real function $f : R+ $\rightarrow$ R+$, if for any initial concentration x of the input species, the concentration of the output molecular species stabilizes at concentration f (x). The Turing-completeness of that notion of chemical analog computation has been established by proving that any computable real function can be computed by a CRN over a finite set of molecular species. Rate-independent CRNs form a restricted class of CRNs of high practical value since they enjoy a form of absolute robustness in the sense that the result is completely independent of the reaction rates and depends solely on the input concentrations. The functions computed by rate-independent CRNs have been characterized mathematically as the set of piecewise linear functions from input species. However, this does not provide a mean to decide whether a given CRN is rate-independent. In this paper, we provide graphical conditions on the Petri Net structure of a CRN which entail the rate-independence property either for all species or for some output species. We show that in the curated part of the Biomodels repository, among the 590 reaction models tested, 2 reaction graphs were found to satisfy our rate-independence conditions for all species, 94 for some output species, among which 29 for some non-trivial output species. Our graphical conditions are based on a non-standard use of the Petri net notions of place-invariants and siphons which are computed by constraint programming techniques for efficiency reasons.

q-bio.MN↗

arXiv2018

Graphical Requirements for Multistationarity in Reaction Networks and their Verification in BioModels

Thomas's necessary conditions for the existence of multiple steady states in gene networks have been proved by Soulé with high generality for dynamical systems defined by differential equations. When applied to (protein) reaction networks however, those conditions do not provide information since they are trivially satisfied as soon as there is a bimolecular or a reversible reaction. Refined graphical requirements have been proposed to deal with such cases. In this paper, we present for the first time a graph rewriting algorithm for checking the refined conditions given by Soliman, and evaluate its practical performance by applying it systematically to the curated branch of the BioModels repository. This algorithm analyzes all reaction networks (of size up to 430 species) in less than 0.05 second per network, and permits to conclude to the absence of multistationarity in 160 networks over 506. The short computation times obtained in this graphical approach are in sharp contrast to the Jacobian-based symbolic computation approach. We also discuss the case of one extra graphical condition by arc rewiring that allows us to conclude on 20 more networks of this benchmark but with a high computational cost. Finally, we study with some details the case of phosphorylation cycles and MAPK signalling models which show the importance of modelling the intermediate complexations with the enzymes in order to correctly analyze the multistationarity capabilities of such biochemical reaction networks.

q-bio.QM↗

arXiv2013

Un modèle booléen pour l'énumération des siphons et des pièges minimaux dans les réseaux de Petri

Petri-nets are a simple formalism for modeling concurrent computation. Recently, they have emerged as a powerful tool for the modeling and analysis of biochemical reaction networks, bridging the gap between purely qualitative and quantitative models. These networks can be large and complex, which makes their study difficult and computationally challenging. In this paper, we focus on two structural properties of Petri-nets, siphons and traps, that bring us information about the persistence of some molecular species. We present two methods for enumerating all minimal siphons and traps of a Petri-net by iterating the resolution of a boolean model interpreted as either a SAT or a CLP(B) program. We compare the performance of these methods with a state-of-the-art dedicated algorithm of the Petri-net community. We show that the SAT and CLP(B) programs are both faster. We analyze why these programs perform so well on the models of the repository of biological models biomodels.net, and propose some hard instances for the problem of minimal siphons enumeration.

cs.CE↗