Coq Library of Undecidability Proofs

The Coq Library of Undecidability Proofs contains mechanised reductions to establish undecidability results in Coq. The undecidability proofs are based on a synthetic approach to undecidability. A problem P is considered undecidable if its decidability in Coq implies the enumerability of the complement of halting problem for Turing machines (SBTM_HALT in TM/SBTM.v). Since the Turing machine halting is enumerable (SBTM_HALT_enum in TM/SBTM_enum.v), enumerability of its complement would classically imply its decidability.

As in the traditional literature, undecidability of a problem P in the library is often established by constructing a many-one reduction from an undecidable problem to P.

For more information on the structure of the library, the synthetic approach, and included problems see Publications below, our Wiki, look at the slides or the recording of the talk on the Coq Library of Undecidability proofs at CoqPL '20.

The library is a collaborative effort, growing constantly and we invite everybody to contribute undecidability proofs!

Problems in the Library

The problems in the library can mostly be categorized into seed problems, advanced problems, and target problems.

Seed problems are simple to state and thus make for good starting points of undecidability proofs, often leading to easier reductions to other problems.

Advanced problems do not work well as seeds, but they highlight the potential of our library as a framework for mechanically checking pen&paper proofs of potentially hard undecidability results. Some advanced problems are proven decidable to contrast negative results.

Target problems are very expressive and thus work well as targets for reduction, with the aim of closing loops in the reduction graph to establish the inter-reducibility of problems.

Seed Problems

• Halting problem for single-tape two-symbol Turing machines (SBTM_HALT in TM/SBTM.v)
• Post correspondence problem (PCP in PCP/PCP.v)
• Halting problem for two counters Minsky machines (MM2_HALTING in MinskyMachines/MM2.v)
• Halting problem for FRACTRAN programs (FRACTRAN_REG_HALTING in FRACTRAN/FRACTRAN.v)
• Satisfiability of elementary Diophantine constraints of the form x = 1, x = y + z or x = y · z (H10C_SAT in DiophantineConstraints/H10C.v)
• Satisfiability of uniform Diophantine constraints of the form x = 1 + y + z · z (H10UC_SAT in DiophantineConstraints/H10C.v)
• Halting problem for one counter machines (CM1_HALT in CounterMachines/CM1.v)
• Solvability of finite multiset constraint (FMsetC_SAT in SetConstraints/FMsetC.v)
• Simple semi-Thue system 01 rewriting (SSTS01 in StringRewriting/SSTS.v)

Advanced Problems

Halting Problems for Traditional Models of Computation

• Halting problem for the weak call-by-value lambda-calculus (HaltL in L/L.v)
• Halting problem for multi-tape Turing machines (HaltMTM in TM/TM.v)
• Halting problem for single-tape Turing machines (HaltTM 1 in TM/TM.v)
• Halting problem for simple binary single-tape Turing machines (HaltSBTM in TM/SBTM.v)
• Halting problem for program counter based binary single-tape Turing machines (PCTM_HALT in TM/PCTM.v)
• Halting problem for Binary Stack Machines (BSM_HALTING in StackMachines/BSM.v)
• Halting problems for Minsky machines (MM_HALTING and MMA_HALTING n in MinskyMachines/MM.v)
• Halting problem for partial recursive functions (MUREC_HALTING in MuRec/recalg.v)
• Halting problem for the weak call-by-name lambda-calculus (wCBNclosed in LambdaCalculus/Lambda.v)

An equivalence proof that most of the mentioned models of computation compute the same n-ary functional relations over natural numbers is available in Models_Equivalent.v.

Problems from Logic

• Provability in Minimal, Intuitionistic, and Classical First-Order Logic (FOL*_prv_intu, FOL_prv_intu, FOL_prv_class in FOL/FOL.v), including a formulation for the minimal binary signature (FOL/binFOL.v)
• Validity in Minimal and Intuitionistic First-Order Logic (FOL*_valid, FOL_valid_intu in FOL/FOL.v)
• Satisfiability in Minimal and Intuitionistic First-Order Logic (FOL*_satis, FOL_satis_intu in FOL/FOL.v)
• Finite satisfiability in First-Order Logic, known as "Trakhtenbrot's theorem" (FSAT in FOL/FSAT.v based on TRAKHTENBROT/red_utils.v)
• Semantic and deductive entailment in first-order ZF set-theory with (FOL/ZF.v) and without (FOL/minZF.v and FOL/binZF.v) function symbols for set operations
• Semantic and deductive entailment in Peano arithmetic (FOL/PA.v)
• Entailment in Elementary Intuitionistic Linear Logic (EILL_PROVABILITY in ILL/EILL.v)
• Entailment in Intuitionistic Linear Logic (ILL_PROVABILITY in ILL/ILL.v)
• Entailment in Classical Linear Logic (CLL_cf_PROVABILITY in ILL/CLL.v)
• Entailment in Intuitionistic Multiplicative Sub-Exponential Linear Logic (IMSELL_cf_PROVABILITY3 in ILL/IMSELL.v)
• Entailment in the (bunched) sequent calculus for the logic of Bunched Implications (BI) (BI_SEQ_PROVABLE) and of provability in its Hilbert calculus (BI_HILBERT_PROVABLE), to be found in BI/BI.v; see also the undecidability of BI
• Provability in Hilbert-style calculi (HSC_PRV in HilbertCalculi/HSC.v)
• Recognizing axiomatizations of Hilbert-style calculi (HSC_AX in HilbertCalculi/HSC.v)
• Satisfiability in Separation Logic (SLSAT in SeparationLogic/SL.v and MSLSAT in SeparationLogic/MSL.v)
• Validity and satisfiability in Second-Order Peano Arithmetic (SOL/PA2.v)
• Validity and satisfiability in Second-Order Logic in the empty signature (SOL/SOL.v)

Problems in Formal Languages and Automata

• Context-Free Grammar intersection inhabitedness (CFI in CFG/CFG.v)
• Presence of palindromes in the language of a Context-Free Grammar (CFP in CFG/CFG.v)

Other Problems

• Acceptance problem for two counters non-deterministic Minsky machines (ndMM2_ACCEPT in MinskyMachines/ndMM2.v)
• Acceptance problem for two counters And branching machines (ACM2_ACCEPT in MinskyMachines/ACM2.v), see also the undecidability of BI
• String rewriting in Semi-Thue and Thue-systems (SR and TSR in StringRewriting/SR.v)
• String rewriting in Post canonical systems in normal form (PCSnf in StringRewriting/PCSnf.v)
• Hilbert's 10th problem, i.e. solvability of a single diophantine equation (H10 in H10/H10.v)
• Solvability of linear polynomial (over N) constraints of the form x = 1, x = y + z, x = X · y (LPolyNC_SAT in PolynomialConstraints/LPolyNC.v)
• One counter machine halting problem (CM1_HALT in CounterMachines/CM1.v)
• Finite multiset constraint solvability (FMsetC_SAT in SetConstraints/FMsetC.v)
• Uniform boundedness of deterministic, length-preserving stack machines (SMNdl_UB in StackMachines/SMN.v)
• Semi-unification (SemiU in SemiUnification/SemiU.v)
• Halting problem for Krivine machines (KrivineM_HALT in LambdaCalculus/Krivine.v)
• Strong normalization for given terms in the strong call-by-name lambda-calculus (SNclosed in LambdaCalculus/Lambda.v)
• System F Inhabitation (SysF_INH in SystemF/SysF.v), System F Typability (SysF_TYP in SystemF/SysF.v), System F Type Checking (SysF_TC in SystemF/SysF.v)
• Intersection Type Inhabitation (CD_INH in IntersectionTypes/CD.v), Intersection Type Typability (CD_TYP in IntersectionTypes/CD.v), Intersection Type Type Checking (CD_TC in IntersectionTypes/CD.v)
• Higher-order matching wrt. beta-equivalence (HOMbeta in LambdaCalculus/HOMatching.v)

Decidable Problems

• Two-counter Minsky Program Machine Halting (MPM2_HALT in MinskyMachines/Deciders/MPM2_HALT_dec.v)
  The definition follows exactly Minsky¹ (Chapter 11, Table 11.1-1), and is different from MM2_HALTING in MinskyMachines/MM2.v.
• Reversible Two-counter Machine Halting (MM2_REV_HALT in MinskyMachines/MM2.v)
• Two-counter Machine Uniform Mortality (MM2_UMORTAL in MinskyMachines/MM2.v)
• Two-counter Machine Uniform Boundedness (MM2_UBOUNDED in MinskyMachines/MM2.v)

Target Problems

• Halting problem for the call-by-value lambda-calculus (HaltL in L/L.v)
• Validity, provability, and satisfiability in First-Order Logic (all problems in FOL/FOL.v)

Manual Installation Instructions

If you can use opam 2 on your system, you can follow the instructions here.

Install from git via opam

You can use opam to install the current state of this branch as follows.

We recommend creating a fresh opam switch:

opam switch create coq-library-undecidability --packages=ocaml-variants.4.14.2+options,ocaml-option-flambda
eval $(opam env)

Then the following commands install the library:

opam repo add coq-released https://coq.inria.fr/opam/released
opam update
opam pin add coq-library-undecidability.dev+9.2 "https://github.com/uds-psl/coq-library-undecidability.git#rocq-9.2"

Manual installation

You need Rocq 9.2 built on OCAML >= 4.09.1 (but we recommend and test OCaml version 4.14.2+flambda) and the Template-Coq part of the MetaCoq package for Coq. If you are using opam 2 you can use the following commands to install the dependencies on a new switch:

opam switch create coq-library-undecidability --packages=ocaml-variants.4.14.2+options,ocaml-option-flambda
eval $(opam env)
opam repo add rocq-released https://rocq-prover.org/opam/released
opam update
opam install . --deps-only

Building the undecidability library

• make all builds the library
• make TM/TM.vo compiles only the file theories/TM/TM.v and its dependencies
• make html generates clickable coqdoc .html in the website subdirectory
• make clean removes all build files in theories and .html files in the website directory

Compiled interfaces

The library is compatible with Coq's compiled interfaces (vos) for quick infrastructural access.

• make vos builds compiled interfaces for the library
• make vok checks correctness of the library

Troubleshooting

Coq version

Be careful that this branch only compiles under Rocq 9.2. If you want to use a different Coq version you have to change to a different branch. Due to compatibility issues, not every branch contains exactly the same problems. We recommend to use the newest branch if possible.

Publications

Papers and abstracts on the library

A Coq Library of Undecidable Problems. Yannick Forster, Dominique Larchey-Wendling, Andrej Dudenhefner, Edith Heiter, Dominik Kirst, Fabian Kunze, Gert Smolka, Simon Spies, Dominik Wehr, Maximilian Wuttke. CoqPL '20. https://popl20.sigplan.org/details/CoqPL-2020-papers/5/A-Coq-Library-of-Undecidable-Problems

Computability in Constructive Type Theory. Yannick Forster. PhD thesis. https://dx.doi.org/10.22028/D291-35758

Papers and abstracts on problems and proofs included in the library

• Certified Decision Procedures for Two-Counter Machines. Andrej Dudenhefner. FSCD 2022. Subdirectory MinskyMachines/Deciders. https://drops.dagstuhl.de/opus/volltexte/2022/16297/
• Constructive Many-One Reduction from the Halting Problem to Semi-Unification. Andrej Dudenhefner. CSL 2022. Subdirectory SemiUnification. https://drops.dagstuhl.de/opus/volltexte/2022/15738/
• Undecidability, Incompleteness, and Completeness of Second-Order Logic in Coq. Mark Koch and Dominik Kirst. CPP 2022. Subdirectory SOL. https://www.ps.uni-saarland.de/extras/cpp22-sol/
• A Mechanised Proof of the Time Invariance Thesis for the Weak Call-By-Value λ-Calculus. Yannick Foster, Fabian Kunze, Gert Smolka, Maximilian Wuttke. ITP 2021. Subdirectory TM/L. https://drops.dagstuhl.de/opus/volltexte/2021/13914/
• Synthetic Undecidability of MSELL via FRACTRAN. Dominique Larchey-Wendling. FSCD 2021. File ILL/IMSELL.v. Also documents the undecidability proof for 2-counters Minsky machines MinskyMachines/MM2.v via FRACTRAN. https://github.com/uds-psl/coq-library-undecidability/releases/tag/FSCD-2021/
• The Undecidability of System F Typability and Type Checking for Reductionists. Andrej Dudenhefner. LICS 2021. Subdirectory SystemF. https://ieeexplore.ieee.org/document/9470520
• Trakhtenbrot's Theorem in Coq - A Constructive Approach to Finite Model Theory. Dominik Kirst and Dominique Larchey-Wendling. IJCAR 2020. Subdirectory TRAKTHENBROT. https://www.ps.uni-saarland.de/extras/fol-trakh/
• Undecidability of Semi-Unification on a Napkin. Andrej Dudenhefner. FSCD 2020. Subdirectory SemiUnification. https://www.ps.uni-saarland.de/Publications/documents/Dudenhefner_2020_Semi-unification.pdf
• Mechanized Undecidability Results for Propositional Calculi. TYPES 2020. Subdirectory HilbertCalculi. https://types2020.di.unito.it/abstracts/BookOfAbstractsTYPES2020.pdf#page=94
• Undecidability of Higher-Order Unification Formalised in Coq. Simon Spies and Yannick Forster. Technical report. Subdirectory HOU. https://www.ps.uni-saarland.de/Publications/details/SpiesForster:2019:UndecidabilityHOU.html
• Verified Programming of Turing Machines in Coq. Yannick Forster, Fabian Kunze, Maximilian Wuttke. Technical report. Subdirectory TM. https://github.com/uds-psl/tm-verification-framework/
• Hilbert's Tenth Problem in Coq. Dominique Larchey-Wendling and Yannick Forster. FSCD '19. Subdirectory H10. https://uds-psl.github.io/H10
• A certifying extraction with time bounds from Coq to call-by-value lambda-calculus. ITP '19. Subdirectory L. https://github.com/uds-psl/certifying-extraction-with-time-bounds
• Certified Undecidability of Intuitionistic Linear Logic via Binary Stack Machines and Minsky Machines. Yannick Forster and Dominique Larchey-Wendling. CPP '19. Subdirectory ILL. http://uds-psl.github.io/ill-undecidability/
• On Synthetic Undecidability in Coq, with an Application to the Entscheidungsproblem. Yannick Forster, Dominik Kirst, and Gert Smolka. CPP '19. Subdirectory FOL. https://www.ps.uni-saarland.de/extras/fol-undec
• Formal Small-step Verification of a Call-by-value Lambda Calculus Machine. Fabian Kunze, Gert Smolka, and Yannick Forster. APLAS 2018. Subdirectory L/AbstractMachines. https://www.ps.uni-saarland.de/extras/cbvlcm2/
• Towards a library of formalised undecidable problems in Coq: The undecidability of intuitionistic linear logic. Yannick Forster and Dominique Larchey-Wendling. LOLA 2018. Subdirectory ILL. https://www.ps.uni-saarland.de/~forster/downloads/LOLA-2018-coq-library-undecidability.pdf
• Verification of PCP-Related Computational Reductions in Coq. Yannick Forster, Edith Heiter, and Gert Smolka. ITP 2018. Subdirectory PCP. https://ps.uni-saarland.de/extras/PCP
• Call-by-Value Lambda Calculus as a Model of Computation in Coq. Yannick Forster and Gert Smolka. Journal of Automated Reasoning (2018) Subdirectory L. https://www.ps.uni-saarland.de/extras/L-computability/

How to contribute

Fork the project on GitHub, add a new subdirectory for your project and your files, then file a pull request. We have guidelines for the directory structure of projects.

Contributors

• Yannick Forster
• Dominique Larchey-Wendling
• Andrej Dudenhefner
• Edith Heiter
• Marc Hermes
• Johannes Hostert
• Dominik Kirst
• Mark Koch
• Fabian Kunze
• Gert Smolka
• Simon Spies
• Dominik Wehr
• Maximilian Wuttke

Parts of the Coq Library of Undecidability Proofs reuse generic code initially developed as a library for the lecture "Introduction to Computational Logics" at Saarland University, which was written by a subset of the above contributors, Sigurd Schneider, and Jan Christian Menz.

Footnotes

1. Minsky, Marvin Lee. Computation. Englewood Cliffs: Prentice-Hall, 1967. ↩