Proof Pearl: Faithful Computation and Extraction of ?-Recursive Algorithms in Coq

Basing on an original Coq implementation of unbounded linear search for partially decidable predicates, we study the computational contents of ?-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 ?-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 ?-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

Graph-based decision for Gödel-Dummett logics

International audienceWe present a graph-based decision procedure for Gödel-Dummett logics and an algorithm to compute counter-models. A formula is transformed into a conditional bi-colored graph in which we detect some specific cycles and alternating chains using matrix computations. From an instance graph containing no such cycle (resp. no (n+1)-alternating chain) we extract a counter-model in LC (resp. LCn)

Constructive Decision via Redundancy-Free Proof-Search

International audienceWe give a constructive account of Kripke-Curry's method which was used to establish the decidability of Implicational Relevance Logic (R →). To sustain our approach, we mechanize this method in axiom-free Coq, abstracting away from the specific features of R → to keep only the essential ingredients of the technique. In particular we show how to replace Kripke/Dickson's lemma by a constructive form of Ramsey's theorem based on the notion of almost full relation. We also explain how to replace König's lemma with an inductive form of Brouwer's Fan theorem. We instantiate our abstract proof to get a constructive decision procedure for R → and discuss potential applications to other logical decidability problems

An Alternative Direct Simulation of Minsky Machines into Classical Bunched Logics via Group Semantics

International audienceRecently, Brotherston & Kanovich, and independently Larchey-Wendling & Galmiche, proved the undecidability of the bunched implication logic BBI. Moreover, Brotherston & Kanovich also proved the undecidability of the related logic CBI, as well as its neighbours. All of the above results are based on encodings of two-counter Minsky machines, but are derived using different techniques. Here, we show that the technique of Larchey-Wendling & Galmiche can also be extended, via group Kripke semantics, to prove the undecidability of CBI. Hence, we propose an alternative direct simulation of Minsky machines into both BBI and CBI. We identify a fragment called elementary Boolean BI (eBBI) which is common to the BBI/CBI families of logics and we show that the problem of Minsky machine acceptance can be encoded into eBBI. The soundness of the encoding is derived from the soundness of a goal directed sequent calculus designed for eBBI. The faithfulness of the encoding is obtained from a Kripke model based on the free commutative group Zn

Hilbert's Tenth Problem in Coq (Extended Version)

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.Comment: submitted to LMC

The formal strong completeness of partial monoidal Boolean BI

International audienceThis article presents a self-contained proof of the strong completeness of the labelled tableaux method for partial monoidalBoolean 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

Simulating Induction-Recursion for Partial Algorithms

International audienceWe describe a generic method to implement and extract partial recursive algorithms in Coq in a purely constructive way, using L. Paulson's if-then-else normalization as a running example

Exploring the relation between intuitionistic bi and boolean bi: An unexpected embedding

International audienceThe logic of Bunched Implications, through its intuitionistic version (BI) as well as one of its classical versions called Boolean BI (BBI), serves as a logical basis to spatial or separation logic frameworks. In BI, the logical implication is interpreted intuitionistically whereas it is generally interpreted classically in spatial or separation logics like in BBI. In this paper, we aim at giving some new insights w.r.t. the semantic relations between BI and BBI. Then we propose a sound and complete syntactic constraints based framework for Kripke semantics of both BI and BBI, a sound labelled tableau proof system for BBI, and a representation theorem relating the syntactic models of BI to those of BBI. Finally we deduce, as main and unexpected result, a sound and faithful embedding of BI into BBI

Trakhtenbrot's Theorem in Coq: Finite Model Theory through the Constructive Lens

26 pages, extended version of the IJCAR 2020 paper. arXiv admin note: substantial text overlap with arXiv:2004.07390International audienceWe 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 Trakthenbrot'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

The Undecidability of Boolean BI through Phase Semantics

International audienceWe 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