arXiv subjects

Stéphane Graham-Lengrand

Publications and source records attributed to Stéphane Graham-Lengrand.

11 recordsLinked to original sources

arXiv2025

Boosting MCSat Modulo Nonlinear Integer Arithmetic via Local Search

The Model Constructing Satisfiability (MCSat) approach to the SMT problem extends the ideas of CDCL from the SAT level to the theory level. Like SAT, its search is driven by incrementally constructing a model by assigning concrete values to theory variables and performing theory-level reasoning to learn lemmas when conflicts arise. Therefore, the selection of values can significantly impact the search process and the solver's performance. In this work, we propose guiding the MCSat search by utilizing assignment values discovered through local search. First, we present a theory-agnostic framework to seamlessly integrate local search techniques within the MCSat framework. Then, we highlight how to use the framework to design a search procedure for (quantifier-free) Nonlinear Integer Arithmetic (NIA), utilizing accelerated hill-climbing and a new operation called feasible-sets jumping. We implement the proposed approach in the MCSat engine of the Yices2 solver, and empirically evaluate its performance over the N IA benchmarks of SMT-LIB.

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arXiv2024

MCSat-based Finite Field Reasoning in the Yices2 SMT Solver

This system description introduces an enhancement to the Yices2 SMT solver, enabling it to reason over non-linear polynomial systems over finite fields. Our reasoning approach fits into the model-constructing satisfiability (MCSat) framework and is based on zero decomposition techniques, which find finite basis explanations for theory conflicts over finite fields. As the MCSat solver within Yices2 can support (and combine) several theories via theory plugins, we implemented our reasoning approach as a new plugin for finite fields and extended Yices2's frontend to parse finite field problems, making our implementation the first MCSat-based reasoning engine for finite fields. We present its evaluation on finite field benchmarks, comparing it against cvc5. Additionally, our work leverages the modular architecture of the MCSat solver in Yices2 to provide a foundation for the rapid implementation of further reasoning techniques for this theory.

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arXiv2021

Machine-checked ZKP for NP-relations: Formally Verified Security Proofs and Implementations of MPC-in-the-Head

MPC-in-the-Head (MitH) is a general framework that allows constructing efficient Zero Knowledge protocols for general NP-relations from secure multiparty computation (MPC) protocols. In this paper we give the first machine-checked implementation of this transformation. We begin with an EasyCrypt formalization of MitH that preserves the modular structure of MitH and can be instantiated with arbitrary MPC protocols that satisfy standard notions of security, which allows us to leverage an existing machine-checked secret-sharing-based MPC protocol development. The resulting concrete ZK protocol is proved secure and correct in EasyCrypt. Using a recently developed code extraction mechanism for EasyCrypt we synthesize a formally verified implementation of the protocol, which we benchmark to get an indication of the overhead associated with our formalization choices and code extraction mechanism.

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arXiv2020

Solving bitvectors with MCSAT: explanations from bits and pieces (long version)

We present a decision procedure for the theory of fixed-sized bitvectors in the MCSAT framework. MCSAT is an alternative to CDCL(T) for SMT solving and can be seen as an extension of CDCL to domains other than the Booleans. Our procedure uses BDDs to record and update the sets of feasible values of bitvector variables. For explaining conflicts and propagations, we develop specialized word-level interpolation for two common fragments of the theory. For full generality, explaining conflicts outside of the covered fragments resorts to local bitblasting. The approach is implemented in the Yices 2 SMT solver and we present experimental results.

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arXiv2018

Tight Typings and Split Bounds

Multi types---aka non-idempotent intersection types---have been used to obtain quantitative bounds on higher-order programs, as pioneered by de Carvalho. Notably, they bound at the same time the number of evaluation steps and the size of the result. Recent results show that the number of steps can be taken as a reasonable time complexity measure. At the same time, however, these results suggest that multi types provide quite lax complexity bounds, because the size of the result can be exponentially bigger than the number of steps. Starting from this observation, we refine and generalise a technique introduced by Bernadet & Graham-Lengrand to provide exact bounds for the maximal strategy. Our typing judgements carry two counters, one measuring evaluation lengths and the other measuring result sizes. In order to emphasise the modularity of the approach, we provide exact bounds for four evaluation strategies, both in the lambda-calculus (head, leftmost-outermost, and maximal evaluation) and in the linear substitution calculus (linear head evaluation). Our work aims at both capturing the results in the literature and extending them with new outcomes. Concerning the literature, it unifies de Carvalho and Bernadet & Graham-Lengrand via a uniform technique and a complexity-based perspective. The two main novelties are exact split bounds for the leftmost strategy---the only known strategy that evaluates terms to full normal forms and provides a reasonable complexity measure---and the observation that the computing device hidden behind multi types is the notion of substitution at a distance, as implemented by the linear substitution calculus.

cs.PL↗

arXiv2015

Realisability semantics of abstract focussing, formalised

We present a sequent calculus for abstract focussing, equipped with proof-terms: in the tradition of Zeilberger's work, logical connectives and their introduction rules are left as a parameter of the system, which collapses the synchronous and asynchronous phases of focussing as macro rules. We go further by leaving as a parameter the operation that extends a context of hypotheses with new ones, which allows us to capture both classical and intuitionistic focussed sequent calculi. We then define the realisability semantics of (the proofs of) the system, on the basis of Munch-Maccagnoni's orthogonality models for the classical focussed sequent calculus, but now operating at the higher level of abstraction mentioned above. We prove, at that level, the Adequacy Lemma, namely that if a term is of type A, then in the model its denotation is in the (set-theoretic) interpretation of A. This exhibits the fact that the universal quantification involved when taking the orthogonal of a set, reflects in the semantics Zeilberger's universal quantification in the macro rule for the asynchronous phase. The system and its semantics are all formalised in Coq.

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arXiv2015

Axiomatic constraint systems for proof search modulo theories

Goal-directed proof search in first-order logic uses meta-variables to delay the choice of witnesses; substitutions for such variables are produced when closing proof-tree branches, using first-order unification or a theory-specific background reasoner. This paper investigates a generalisation of such mechanisms whereby theory-specific constraints are produced instead of substitutions. In order to design modular proof-search procedures over such mechanisms, we provide a sequent calculus with meta-variables, which manipulates such constraints abstractly. Proving soundness and completeness of the calculus leads to an axiomatisation that identifies the conditions under which abstract constraints can be generated and propagated in the same way unifiers usually are. We then extract from our abstract framework a component interface and a specification for concrete implementations of background reasoners.

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arXiv2014

Polarities & Focussing: a journey from Realisability to Automated Reasoning

This dissertation explores the roles of polarities and focussing in various aspects of Computational Logic. These concepts play a key role in the the interpretation of proofs as programs, a.k.a. the Curry-Howard correspondence, in the context of classical logic. Arising from linear logic, they allow the construction of meaningful semantics for cut-elimination in classical logic, some of which relate to the Call-by-Name and Call-by-Value disciplines of functional programming. The first part of this dissertation provides an introduction to these interpretations, highlighting the roles of polarities and focussing. For instance: proofs of positive formulae provide structured data, while proofs of negative formulae consume such data; focussing allows the description of the interaction between the two kinds of proofs as pure pattern-matching. This idea is pushed further in the second part of this dissertation, and connected to realisability semantics, where the structured data is interpreted algebraically, and the consumption of such data is modelled with the use of an orthogonality relation. Most of this part has been proved in the Coq proof assistant. Polarities and focussing were also introduced with applications to logic programming in mind, where computation is proof-search. In the third part of this dissertation, we push this idea further by exploring the roles that these concepts can play in other applications of proof-search, such as theorem proving and more particularly automated reasoning. We use these concepts to describe the main algorithm of SAT-solvers and SMT-solvers: DPLL. We then describe the implementation of a proof-search engine called Psyche. Its architecture, based on the concept of focussing, offers a platform where smart techniques from automated reasoning (or a user interface) can safely and trustworthily be implemented via the use of an API.

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arXiv2013

Sequent Calculi with procedure calls

In this paper, we introduce two focussed sequent calculi, LKp(T) and LK+(T), that are based on Miller-Liang's LKF system for polarised classical logic. The novelty is that those sequent calculi integrate the possibility to call a decision procedure for some background theory T, and the possibility to polarise literals "on the fly" during proof-search. These features are used in our other works to simulate the DPLL(T) procedure as proof-search in the extension of LKp(T) with a cut-rule. In this report we therefore prove cut-elimination in LKp(T). Contrary to what happens in the empty theory, the polarity of literals affects the provability of formulae in presence of a theory T. On the other hand, changing the polarities of connectives does not change the provability of formulae, only the shape of proofs. In order to prove this, we introduce a second sequent calculus, LK+(T) that extends LKp(T) with a relaxed focussing discipline, but we then show an encoding of LK+(T) back into the more restrictive system LK(T). We then prove completeness of LKp(T) (and therefore of LK+(T)) with respect to first-order reasoning modulo the ground propositional lemmas of the background theory T .

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arXiv2013

Proceedings Sixth Workshop on Intersection Types and Related Systems

This volume contains the proceedings of the Sixth Workshop on Intersection Types and Related Systems (ITRS 2012). The workshop was held in Dubrovnik (Croatia) on June 29th, 2012, affiliated to Twenty-Seventh Annual ACM/IEEE Symposium on Logic in Computer Science (LICS 2012). The ITRS workshop aims to bring together researchers working on both the theory and practical applications of systems based on intersection types and related approaches (e.g., union types, refinement types, behavioural types, recursive types, and so on).

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arXiv2013

A simple presentation of the effective topos

We propose for the Effective Topos an alternative construction: a realisability framework composed of two levels of abstraction. This construction simplifies the proof that the Effective Topos is a topos (equipped with natural numbers), which is the main issue that this paper addresses. In this our work can be compared to Frey's monadic tripos-to-topos construction. However, no topos theory or even category theory is here required for the construction of the framework itself, which provides a semantics for higher-order type theories, supporting extensional equalities and the axiom of unique choice.

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