# Simon Guilloud

**EPFL IC IINFCOM LARA **

INR 318 (Bâtiment INR)

Station 14

1015 Lausanne

Web site: Web site: https://lara.epfl.ch

### Fields of expertise

My main domains of research are proof assistants and algorithms for logic-related problems.

I also have interests in various areas of computer science and mathematics and their applicability in proof systems such as probabilistic proofs, type theory, quantum computing and more.

### Current work

My main current work consists in designing and developping the LISA proof assistant (work in progress).I also study applicability of decision procedures and datastructures for ortholattices as partial solutions for propositional logic problems.

### Biography

I am a PhD student at EPFL working in the Laboratory of Automated Reasoning and Analysis with Prof. Viktor Kunčak.I have previously completed my Bachelor and Master degree at EPFL and made an internship in quantum computing.

## Publications

### Infoscience publications

# 2024

## Conference Papers

#### * Mechanized HOL Reasoning in Set Theory

We present a mechanized embedding of higher-order logic (HOL) and algebraic data types (ADT) into first-order logic with ZFC axioms. We implement this in the Lisa proof assistant for schematic first-order logic and its library based on axiomatic set theory. HOL proof steps are implemented as proof producing tactics in Lisa, and the types are interpreted as sets, with function (or arrow) types coinciding with set-theoretic function spaces. % The embedded HOL proofs, as opposed to being a layer over the existing proofs, are interoperable with the existing library. This yields a form of soft type system supporting top-level polymorphism and ADTs over set theory, and offer tools to reason about functions in set theory.

2024.#### * Interpolation and Quantifiers in Ortholattices

We study quantifiers and interpolation properties in orthologic, a non-distributive weakening of classical logic that is sound for formula validity with respect to classical logic, yet has a quadratic-time decision procedure. We present a sequent-based proof system for quantified orthologic, which we prove sound and complete for the class of all complete ortholattices. We show that orthologic does not admit quantifier elimination in general. Despite that, we show that interpolants always exist in orthologic. We give an algorithm to compute interpolants efficiently. We expect our result to be useful to quickly establish unreachability as a component of verification algorithms.

2024-01-01. 25th International Conference on Verification, Model Checking, and Abstract Interpretation (VMCAI), London, ENGLAND, JAN 15-16, 2024. p. 235-257. DOI : 10.1007/978-3-031-50524-9_11.#### * Interpolation and Quantifiers in Ortholattices

We study quantifiers and interpolation properties in ortho- logic, a non-distributive weakening of classical logic that is sound for formula validity with respect to classical logic, yet has a quadratic-time decision procedure. We present a sequent-based proof system for quantified orthologic, which we prove sound and complete for the class of all complete ortholattices. We show that orthologic does not admit quantifier elimination in general. Despite that, we show that interpolants always exist in orthologic. We give an algorithm to compute interpolants efficiently. We expect our result to be useful to quickly establish unreachability as a component of verification algorithms.

2024. 25th International Conference on Verification, Model Checking, and Abstract Interpretation (VMCAI 2024), London, United Kingdom, January 15-16, 2024.#### * Orthologic with Axioms

We study the proof theory and algorithms for orthologic, a logical system based on ortholattices, which have shown practical relevance in simplification and normalization of verification conditions. Ortholattices weaken Boolean algebras while having polynomial-time equivalence checking that is sound with respect to Boolean algebra semantics. We generalize ortholattice reasoning and obtain an algorithm for proving a larger class of classically valid formulas. As the key result, we analyze a proof system for orthologic augmented with axioms. An important feature of the system is that it limits the number of formulas in a sequent to at most two, which makes the extension with axioms non-trivial. We show a generalized form of cut elimination for this system, which implies a sub-formula property. From there we derive a cubic-time algorithm for provability from axioms, or equivalently, for validity in finitely presented ortholattices. We further show that propositional resolution of width 5 proves all formulas provable in orthologic with axioms. We show that orthologic system subsumes resolution of width 2 and arbitrarily wide unit resolution and is complete for reasoning about generalizations of propositional Horn clauses. Moving beyond ground axioms, we introduce effectively propositional orthologic (by analogy with EPR for classical logic), presenting its semantics as well as a sound and complete proof system. Our proof system implies the decidability of effectively propositional orthologic, as well as its fixed-parameter tractability for a bounded maximal number of variables in each axiom. As a special case, we obtain a generalization of Datalog with negation and disjunction.

2024. 51st ACM SIGPLAN Symposium on Principles of Programming Languages, London, United Kingdom, January 17-19, 2024.# 2023

## Conference Papers

#### * LISA – A Modern Proof System

We present LISA, a proof system and proof assistant for constructing proofs in schematic first-order logic and axiomatic set theory. The logical kernel of the system is a proof checker for first-order logic with equality and schematic predicate and function symbols. It implements polynomial-time proof checking and uses the axioms of ortholattices (which implies the irrelevance of the order of conjuncts and disjuncts and additional propositional laws). The kernel supports the notion of theorems (whose proofs are not expanded), as well as definitions of predicate symbols and objects whose unique existence is proven. A domain-specific language enables construction of proofs and development of proof tactics with user-friendly tools and presentation, while remaining within the general-purpose language, Scala. We describe the LISA proof system and illustrate the flavour and the level of abstraction of proofs written in LISA. This includes a proof-generating tactic for propositional tautologies, leveraging the ortholattice properties to reduce the size of proofs. We also present early formalization of set theory in LISA, including Cantor's theorem.

2023-06-30. 14th Conference on Interactive Theorem Proving, Białystok, Poland, 2023, July 31 - August 4, 2023. DOI : 10.4230/LIPIcs.ITP.2023.17.#### * Formula Normalizations in Verification

We propose a new approach for normalization and simplification of logical formulas. Our approach is based on algorithms for lattice-like structures. Specifically, we present two efficient algorithms for computing a normal form and deciding the word problem for two subtheories of Boolean algebra, giving a sound procedure for propositional logical equivalence that is incomplete in general but complete with respect to a subset of Boolean algebra axioms. We first show a new algorithm to produce a normal form for expressions in the theory of ortholattices (OL) in time O(n^2). We also consider an algorithm, recently presented but never evaluated in practice, producing a normal form for a slightly weaker theory, orthocomplemented bisemilattices (OCBSL), in time O(n log(n)^2). For both algorithms, we present an implementation and show efficiency in two domains. First, we evaluate the algorithms on large propositional expressions, specifically combinatorial circuits from a benchmark suite, as well as on large random formulas. Second, we implement and evaluate the algorithms in the Stainless verifier, a tool for verifying the correctness of Scala programs. We used these algorithms as a basis for a new formula simplifier, which is applied before valid verification conditions are saved into a persistent cache. The results show that normalization substantially increases cache hit ratio in large benchmarks.

2023-06-16. 35th International Conference on Computer Aided Verification, Paris, France.# 2022

## Conference Papers

#### * Equivalence Checking for Orthocomplemented Bisemilattices in Log-Linear Time

We present a quasilinear time algorithm to decide the word problem on a natural algebraic structures we call orthocomplemented bisemilattices, a subtheory of boolean algebra. We use as a base a variation of Hopcroft, Ullman and Aho algorithm for tree isomorphism which we combine with a term rewriting system to decide equivalence of two terms. We prove that the rewriting system is terminating and confluent and hence the existence of a normal form, and that our algorithm is computing it. We also discuss applications and present an effective implementation in Scala.

2022. 28th International Conference on Tools and Algorithms for the Construction and Analysis of Systems (ETAPS 2022), Munich, Germany, April 2-7, 2022. p. 196-214. DOI : 10.1007/978-3-030-99527-0_11.