Certified undecidability of intuitionistic linear logic via binary stack machines and Minsky machines

Forster, Yannick; Larchey-Wendling, Dominique

doi:10.1145/3293880.3294096

Cited by 15 publications

(38 citation statements)

References 35 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We review the main ingredients of our synthetic approach to decidability and undecidability [7,8,10,11,13,19], based on the computability of all functions definable in constructive type theory. 2 We first introduce standard notions of computability theory without referring to a formal model of computation, e.g.…”

Section: Synthetic (Un-)decidabilitymentioning

confidence: 99%

“…We also provide a trivial proof of the equivalence of two definitions of BPCP. See [7,10] for details on the reduction from the halting problem to BPCP.…”

Section: Synthetic (Un-)decidabilitymentioning

confidence: 99%

“…the length of the enumerating list of X 9. Hence lr denotes a list of representatives of equivalence classes 10. For a given X, the value n (usually called cardinal) is unique by the PHP.…”

mentioning

confidence: 99%

See 2 more Smart Citations

Trakhtenbrot’s Theorem in Coq

Kirst¹,

Larchey-Wendling²

2020

Automated Reasoning

Self Cite

View full text Add to dashboard Cite

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. All our results are mechanised in the framework of a growing Coq library of synthetic undecidability proofs.

show abstract

Section: Synthetic (Un-)decidabilitymentioning

confidence: 99%

“…We also provide a trivial proof of the equivalence of two definitions of BPCP. See [7,10] for details on the reduction from the halting problem to BPCP.…”

Section: Synthetic (Un-)decidabilitymentioning

confidence: 99%

See 1 more Smart Citation

Trakhtenbrot’s Theorem in Coq

Kirst¹,

Larchey-Wendling²

2020

Automated Reasoning

Self Cite

View full text Add to dashboard Cite

show abstract

“…In contrast, Forster and Smolka [2017] used a call-by-value lambda calculus as computational model. Forster and Larchey-Wendling [2019] proved undecibility of intuitionistic linear logic by reducing from PCP.…”

Section: Related Workmentioning

confidence: 99%

Undecidability of d _<: and its decidable fragments

Lhoták

2019

Proc. ACM Program. Lang.

View full text Add to dashboard Cite

Dependent Object Types (DOT) is a calculus with path dependent types, intersection types, and object selfreferences, which serves as the core calculus of Scala 3. Although the calculus has been proven sound, it remains open whether type checking in DOT is decidable. In this paper, we establish undecidability proofs of type checking and subtyping of D <: , a syntactic subset of DOT. It turns out that even for D <: , undecidability is surprisingly difficult to show, as evidenced by counterexamples for past attempts. To prove undecidability, we discover an equivalent definition of the D <: subtyping rules in normal form. Besides being easier to reason about, this definition makes the phenomenon of subtyping reflection explicit as a single inference rule. After removing this rule, we discover two decidable fragments of D <: subtyping and identify algorithms to decide them. We prove soundness and completeness of the algorithms with respect to the fragments, and we prove that the algorithms terminate. Our proofs are mechanized in a combination of Coq and Agda. CCS Concepts: • Theory of computation → Type theory; • Software and its engineering → Functional languages.

show abstract

“…More precisely, he conjectured that if A ⊆ N k is PCP: Post correspondence problem, see e.g. [FL19]. (matching) MM: Given n : N, a Minsky machine P : L I n with n registers, and v : N n , does (1, P ) terminate from input state (1, v)?…”

Section: Introductionmentioning

confidence: 99%

Hilbert's Tenth Problem in Coq (Extended Version)

Larchey-Wendling¹,

Forster²

2022

Logical Methods in Computer Science

Self Cite

View full text Add to dashboard Cite

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.

show abstract

Certified undecidability of intuitionistic linear logic via binary stack machines and Minsky machines

Cited by 15 publications

References 35 publications

Trakhtenbrot’s Theorem in Coq

Trakhtenbrot’s Theorem in Coq

Undecidability of d _<: and its decidable fragments

Hilbert's Tenth Problem in Coq (Extended Version)

Contact Info

Product

Resources

About

Certified undecidability of intuitionistic linear logic via binary stack machines and Minsky machines

Cited by 15 publications

References 35 publications

Trakhtenbrot’s Theorem in Coq

Trakhtenbrot’s Theorem in Coq

Undecidability of d <: and its decidable fragments

Hilbert's Tenth Problem in Coq (Extended Version)

Contact Info

Product

Resources

About

Undecidability of d _<: and its decidable fragments