In constructive commutative algebra, we revive the bar inductive characterization of Noetherian rings. We contribute the first constructive (axiom free) implementation of Hilbert's basis theorem, in the Rocq proof assistant. We show that the polynomial ring R[X] is Noetherian when the ring R is Noetherian, without assuming any additional condition on R, like coherence or else strong discreteness. We also contribute and implement a new result, that Noetherian rings are closed under direct products, again without assuming any supplementary condition on rings. We study induction principles for Noetherian rings, and relate bar Noetherianity with some other constructive characterizations.

Basing on an original Coq implementation of unbounded linear search for partially decidable predicates, we study the computational contents of μ-recursive functions via their syntactic representation, and a correct by construction Coq interpreter for this abstract syntax. When this interpreter is extracted, we claim the resulting OCaml code to be the natural combination of the implementation of the μ-recursive schemes of composition, primitive recursion and unbounded minimization of partial (i.e., possibly non-terminating) functions. At the level of the fully specified Coq terms, this implies the representation of higher-order functions of which some of the arguments are themselves partial functions. We handle this issue using some techniques coming from the Braga method. Hence we get a faithful embedding of μ-recursive algorithms into Coq preserving not only their extensional meaning but also their intended computational behavior. We put a strong focus on the quality of the Coq artifact which is both self contained and with a line of code count of less than 1k in total.

We study finite first-order satisfiability (FSAT) in the constructive setting of dependent type theory. Employing synthetic accounts of enumerability and decidability, we give a full classification of FSAT depending on the first-order signature of non-logical symbols. On the one hand, our development focuses on Trakhtenbrot's theorem, stating that FSAT is undecidable as soon as the signature contains an at least binary relation symbol. Our proof proceeds by a many-one reduction chain starting from the Post correspondence problem. On the other hand, we establish the decidability of FSAT for monadic first-order logic, i.e. where the signature only contains at most unary function and relation symbols, as well as the enumerability of FSAT for arbitrary enumerable signatures. To showcase an application of Trakhtenbrot's theorem, we continue our reduction chain with a many-one reduction from FSAT to separation logic. All our results are mechanised in the framework of a growing Coq library of synthetic undecidability proofs.

We formalise the undecidability of solvability of Diophantine equations, i.e. polynomial equations over natural numbers, in Coq's constructive type theory. To do so, we give the first full mechanisation of the Davis-Putnam-Robinson-Matiyasevich theorem, stating that every recursively enumerable problem -- in our case by a Minsky machine -- is Diophantine. We obtain an elegant and comprehensible proof by using a synthetic approach to computability and by introducing Conway's FRACTRAN language as intermediate layer. Additionally, we prove the reverse direction and show that every Diophantine relation is recognisable by $\mu$-recursive functions and give a certified compiler from $\mu$-recursive functions to Minsky machines.

We present an alternate undecidability proof for entailment in (intuitionistic) multiplicative sub-exponential linear logic (MSELL). We contribute the result and its mechanised proof to the Coq library of synthetic undecidability. The result crucially relies on the undecidability of the halting problem for two counters Minsky machines, which we also hand out to the library. As a seed of undecidability, we start from FRACTRAN halting which we (many-one) reduce to Minsky machines termination by implementing Euclidean division using two counters only. We then give an alternate presentation of those two counters machines as sequent rules, where computation is performed by proof-search, and halting reduced to provability. We use this system called non-deterministic two counters Minsky machines to describe and compare both the legacy reduction to linear logic, and the more recent reduction to MSELL. In contrast with that former MSELL undecidability proof, our correctness argument for the reduction uses trivial phase semantics in place of a focused calculus.

We formally prove the undecidability of entailment in intuitionistic linear logic in Coq. We reduce the Post correspondence problem (PCP) via binary stack machines and Minsky machines to intuitionistic linear logic. The reductions rely on several technically involved formalisations, amongst them a binary stack machine simulator for PCP, a verified low-level compiler for instruction-based languages and a soundness proof for intuitionistic linear logic with respect to trivial phase semantics. We exploit the computability of all functions definable in constructive type theory and thus do not have to rely on a concrete model of computation, enabling the reduction proofs to focus on correctness properties.

This article presents a self-contained proof of the strong completeness of the labelled tableaux method for partial monoidal Boolean BI: if a formula has no tableau proof then there exists a counter-model for it which is simple. Simple counter-models are those which are generated from the specific constraints that occur during the tableaux proof-search process. As a companion to this article, we provide a complete formalization of this result in Coq and discuss some of its implementation details. The Coq code is distributed under a free software license and is accessible at http://www.loria.fr/~larchey/BBI .

We solve the open problem of the decidability of Boolean BI logic (BBI), which can be considered the core of separation and spatial logics. For this, we define a complete phase semantics suitable for BBI and characterize it as trivial phase semantics. We deduce an embedding between trivial phase semantics for intuitionistic linear logic (ILL) and Kripke semantics for BBI. We single out the elementary fragment of ILL, which is both undecidable and complete for trivial phase semantics. Thus, we obtain the undecidability of BBI.

We solve the open problem of the decidability of Boolean BI logic (BBI), which can be considered as the core of separation and spatial logics. For this, we define a complete phase semantics for BBI and characterize it as trivial phase semantics. We deduce an embedding between trivial phase semantics for intuitionistic linear logic (ILL) and Kripke semantics for BBI. We single out a fragment of ILL which is both undecidable and complete for trivial phase semantics. Therefore, we obtain the undecidability of BBI.

The logic of Bunched Implications, through both its intuitionistic version (BI) and one of its classical versions, called BooleanBI(BBI), serves as a logical basis to spatial or separation logic frameworks. InBI, the logical implication is interpreted intuitionistically whereas it is generally interpreted classically in spatial or separation logics, as inBBI. In this paper, we aim to give some new insights into the semantic relations betweenBIandBBI. Then we propose a sound and complete syntactic constraints based framework for the Kripke semantics of bothBIandBBI, a sound labelled tableau proof system forBBI, and a representation theorem relating the syntactic models ofBIto those ofBBI. Finally, we deduce as our main, and unexpected, result, a sound and faithful embedding ofBIintoBBI.