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**Published in:** LIPIcs, Volume 237, 13th International Conference on Interactive Theorem Proving (ITP 2022)

We develop and mechanize compact proofs of the undecidability of various problems for dyadic first-order logic over a small logical fragment. In this fragment, formulas are restricted to only a single binary relation, and a minimal set of logical connectives. We show that validity, satisfiability, and provability, along with finite satisfiability and finite validity are undecidable, by directly reducing from a suitable binary variant of Diophantine constraints satisfiability. Our results improve upon existing work in two ways: First, the reductions are direct and significantly more compact than existing ones. Secondly, the undecidability of the small logic fragment of dyadic first-order logic was not mechanized before. We contribute our mechanization to the Coq Library of Undecidability Proofs, utilizing its synthetic approach to computability theory.

Johannes Hostert, Andrej Dudenhefner, and Dominik Kirst. Undecidability of Dyadic First-Order Logic in Coq. In 13th International Conference on Interactive Theorem Proving (ITP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 237, pp. 19:1-19:19, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)

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@InProceedings{hostert_et_al:LIPIcs.ITP.2022.19, author = {Hostert, Johannes and Dudenhefner, Andrej and Kirst, Dominik}, title = {{Undecidability of Dyadic First-Order Logic in Coq}}, booktitle = {13th International Conference on Interactive Theorem Proving (ITP 2022)}, pages = {19:1--19:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-252-5}, ISSN = {1868-8969}, year = {2022}, volume = {237}, editor = {Andronick, June and de Moura, Leonardo}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITP.2022.19}, URN = {urn:nbn:de:0030-drops-167280}, doi = {10.4230/LIPIcs.ITP.2022.19}, annote = {Keywords: undecidability, synthetic computability, first-order logic, Coq} }

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**Published in:** LIPIcs, Volume 228, 7th International Conference on Formal Structures for Computation and Deduction (FSCD 2022)

Two-counter machines, pioneered by Minsky in the 1960s, constitute a particularly simple, universal model of computation. Universality of reversible two-counter machines (having a right-unique step relation) has been shown by Morita in the 1990s. Therefore, the halting problem for reversible two-counter machines is undecidable. Surprisingly, this statement is specific to certain instruction sets of the underlying machine model.
In the present work we consider two-counter machines (CM2) with instructions inc_c (increment counter c, go to next instruction), dec_c q (if counter c is zero, then go to next instruction, otherwise decrement counter c and go to instruction q). While the halting problem for CM2 is undecidable, we give a decision procedure for the halting problem for reversible CM2, contrasting Morita’s result.
We supplement our result with decision procedures for uniform boundedness (is there a uniform bound on the number of reachable configurations?) and uniform mortality (is there a uniform bound on the number of steps in any run?) for CM2.
Termination and correctness of each presented decision procedure is certified using the Coq proof assistant. In fact, both the implementation and certification is carried out simultaneously using the tactic language of the Coq proof assistant. Building upon existing infrastructure, the mechanized decision procedures are contributed to the Coq library of undecidability proofs.

Andrej Dudenhefner. Certified Decision Procedures for Two-Counter Machines. In 7th International Conference on Formal Structures for Computation and Deduction (FSCD 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 228, pp. 16:1-16:18, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)

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@InProceedings{dudenhefner:LIPIcs.FSCD.2022.16, author = {Dudenhefner, Andrej}, title = {{Certified Decision Procedures for Two-Counter Machines}}, booktitle = {7th International Conference on Formal Structures for Computation and Deduction (FSCD 2022)}, pages = {16:1--16:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-233-4}, ISSN = {1868-8969}, year = {2022}, volume = {228}, editor = {Felty, Amy P.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2022.16}, URN = {urn:nbn:de:0030-drops-162978}, doi = {10.4230/LIPIcs.FSCD.2022.16}, annote = {Keywords: constructive mathematics, computability theory, decidability, counter automata, mechanization, Coq} }

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**Published in:** LIPIcs, Volume 216, 30th EACSL Annual Conference on Computer Science Logic (CSL 2022)

The undecidability of semi-unification (unification combined with matching) has been proven by Kfoury, Tiuryn, and Urzyczyn in the 1990s. The original argument is by Turing reduction from Turing machine immortality (existence of a diverging configuration).
There are several aspects of the existing work which can be improved upon. First, many-one completeness of semi-unification is not established due to the use of Turing reductions. Second, existing mechanizations do not cover a comprehensive reduction from Turing machine halting to semi-unification. Third, reliance on principles such as König’s lemma or the fan theorem does not support constructivity of the arguments.
Improving upon the above aspects, the present work gives a constructive many-one reduction from the Turing machine halting problem to semi-unification. This establishes many-one completeness of semi-unification. Computability of the reduction function, constructivity of the argument, and correctness of the argument is witnessed by an axiom-free mechanization in the Coq proof assistant. The mechanization is incorporated into the existing Coq library of undecidability proofs. Notably, the mechanization relies on a technique invented by Hooper in the 1960s for Turing machine immortality.
An immediate consequence of the present work is an alternative approach to the constructive many-one equivalence of System F typability and System F type checking, compared to the argument established in the 1990s by Wells.

Andrej Dudenhefner. Constructive Many-One Reduction from the Halting Problem to Semi-Unification. In 30th EACSL Annual Conference on Computer Science Logic (CSL 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 216, pp. 18:1-18:19, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)

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@InProceedings{dudenhefner:LIPIcs.CSL.2022.18, author = {Dudenhefner, Andrej}, title = {{Constructive Many-One Reduction from the Halting Problem to Semi-Unification}}, booktitle = {30th EACSL Annual Conference on Computer Science Logic (CSL 2022)}, pages = {18:1--18:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-218-1}, ISSN = {1868-8969}, year = {2022}, volume = {216}, editor = {Manea, Florin and Simpson, Alex}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2022.18}, URN = {urn:nbn:de:0030-drops-157380}, doi = {10.4230/LIPIcs.CSL.2022.18}, annote = {Keywords: constructive mathematics, undecidability, mechanization, semi-unification} }

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**Published in:** LIPIcs, Volume 167, 5th International Conference on Formal Structures for Computation and Deduction (FSCD 2020)

Semi-unification (unification combined with matching) has been proven undecidable by Kfoury, Tiuryn, and Urzyczyn in the 1990s. The original argument reduces Turing machine immortality via Turing machine boundedness to semi-unification. The latter part is technically most challenging, involving several intermediate models of computation.
This work presents a novel, simpler reduction from Turing machine boundedness to semi-unification. In contrast to the original argument, we directly translate boundedness to solutions of semi-unification and vice versa. In addition, the reduction is mechanized in the Coq proof assistant, relying on a mechanization-friendly stack machine model that corresponds to space-bounded Turing machines. Taking advantage of the simpler proof, the mechanization is comparatively short and fully constructive.

Andrej Dudenhefner. Undecidability of Semi-Unification on a Napkin. In 5th International Conference on Formal Structures for Computation and Deduction (FSCD 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 167, pp. 9:1-9:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)

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@InProceedings{dudenhefner:LIPIcs.FSCD.2020.9, author = {Dudenhefner, Andrej}, title = {{Undecidability of Semi-Unification on a Napkin}}, booktitle = {5th International Conference on Formal Structures for Computation and Deduction (FSCD 2020)}, pages = {9:1--9:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-155-9}, ISSN = {1868-8969}, year = {2020}, volume = {167}, editor = {Ariola, Zena M.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2020.9}, URN = {urn:nbn:de:0030-drops-123311}, doi = {10.4230/LIPIcs.FSCD.2020.9}, annote = {Keywords: undecidability, semi-unification, mechanization} }

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**Published in:** LIPIcs, Volume 130, 24th International Conference on Types for Proofs and Programs (TYPES 2018)

Provability in the intuitionistic second-order propositional logic (resp. inhabitation in the polymorphic lambda-calculus) was shown by Löb to be undecidable in 1976. Since the original proof is heavily condensed, Arts in collaboration with Dekkers provided a fully unfolded argument in 1992 spanning approximately fifty pages. Later in 1997, Urzyczyn developed a different, syntax oriented proof. Each of the above approaches embeds (an undecidable fragment of) first-order predicate logic into second-order propositional logic.
In this work, we develop a simpler undecidability proof by reduction from solvability of Diophantine equations (is there an integer solution to P(x_1, ..., x_n) = 0 where P is a polynomial with integer coefficients?). Compared to the previous approaches, the given reduction is more accessible for formalization and more comprehensible for didactic purposes. Additionally, we formalize soundness and completeness of the reduction in the Coq proof assistant under the banner of "type theory inside type theory".

Andrej Dudenhefner and Jakob Rehof. A Simpler Undecidability Proof for System F Inhabitation. In 24th International Conference on Types for Proofs and Programs (TYPES 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 130, pp. 2:1-2:11, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)

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@InProceedings{dudenhefner_et_al:LIPIcs.TYPES.2018.2, author = {Dudenhefner, Andrej and Rehof, Jakob}, title = {{A Simpler Undecidability Proof for System F Inhabitation}}, booktitle = {24th International Conference on Types for Proofs and Programs (TYPES 2018)}, pages = {2:1--2:11}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-106-1}, ISSN = {1868-8969}, year = {2019}, volume = {130}, editor = {Dybjer, Peter and Esp{\'\i}rito Santo, Jos\'{e} and Pinto, Lu{\'\i}s}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TYPES.2018.2}, URN = {urn:nbn:de:0030-drops-114061}, doi = {10.4230/LIPIcs.TYPES.2018.2}, annote = {Keywords: System F, Lambda Calculus, Inhabitation, Propositional Logic, Provability, Undecidability, Coq, Formalization} }

Document

**Published in:** LIPIcs, Volume 104, 23rd International Conference on Types for Proofs and Programs (TYPES 2017)

We show that recognizing axiomatizations of the Hilbert-style calculus containing only the axiom a -> (b -> a) is undecidable (a reduction from the Post correspondence problem is formalized in the Lean theorem prover). Interestingly, the problem remains undecidable considering only axioms which, when seen as simple types, are principal for some lambda-terms in beta-normal form. This problem is closely related to type-based composition synthesis, i.e. finding a composition of given building blocks (typed terms) satisfying a desired specification (goal type).
Contrary to the above result, axiomatizations of the Hilbert-style calculus containing only the axiom a -> (b -> b) are recognizable in linear time.

Andrej Dudenhefner and Jakob Rehof. Lower End of the Linial-Post Spectrum. In 23rd International Conference on Types for Proofs and Programs (TYPES 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 104, pp. 2:1-2:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)

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@InProceedings{dudenhefner_et_al:LIPIcs.TYPES.2017.2, author = {Dudenhefner, Andrej and Rehof, Jakob}, title = {{Lower End of the Linial-Post Spectrum}}, booktitle = {23rd International Conference on Types for Proofs and Programs (TYPES 2017)}, pages = {2:1--2:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-071-2}, ISSN = {1868-8969}, year = {2019}, volume = {104}, editor = {Abel, Andreas and Nordvall Forsberg, Fredrik and Kaposi, Ambrus}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TYPES.2017.2}, URN = {urn:nbn:de:0030-drops-100503}, doi = {10.4230/LIPIcs.TYPES.2017.2}, annote = {Keywords: combinatory logic, lambda calculus, type theory, simple types, inhabitation, principal type} }

Document

**Published in:** LIPIcs, Volume 84, 2nd International Conference on Formal Structures for Computation and Deduction (FSCD 2017)

It is shown that in the simply typed lambda-calculus the following decision problem of principal inhabitation is Pspace-complete: Given a simple type tau, is there a lambda-term N in beta-normal form such that tau is the principal type of N?
While a Ben-Yelles style algorithm was presented by Broda and Damas in 1999 to count normal principal inhabitants (thereby answering a question posed by Hindley), it does not induce a polynomial space upper bound for principal inhabitation. Further, the standard construction of the polynomial space lower bound for simple type inhabitation does not carry over immediately.
We present a polynomial space bounded decision procedure based on a characterization of principal inhabitation using path derivation systems over subformulae of the input type, which does not require candidate inhabitants to be constructed explicitly. The lower bound is shown by reducing a restriction of simple type inhabitation to principal inhabitation.

Andrej Dudenhefner and Jakob Rehof. The Complexity of Principal Inhabitation. In 2nd International Conference on Formal Structures for Computation and Deduction (FSCD 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 84, pp. 15:1-15:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017)

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@InProceedings{dudenhefner_et_al:LIPIcs.FSCD.2017.15, author = {Dudenhefner, Andrej and Rehof, Jakob}, title = {{The Complexity of Principal Inhabitation}}, booktitle = {2nd International Conference on Formal Structures for Computation and Deduction (FSCD 2017)}, pages = {15:1--15:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-047-7}, ISSN = {1868-8969}, year = {2017}, volume = {84}, editor = {Miller, Dale}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2017.15}, URN = {urn:nbn:de:0030-drops-77359}, doi = {10.4230/LIPIcs.FSCD.2017.15}, annote = {Keywords: Lambda Calculus, Type Theory, Simple Types, Inhabitation, Principal Type, Complexity} }

Document

**Published in:** LIPIcs, Volume 52, 1st International Conference on Formal Structures for Computation and Deduction (FSCD 2016)

The intersection type unification problem is an important component in
proof search related to several natural decision problems in
intersection type systems. It is unknown and remains open whether the
unification problem is decidable. We give the first nontrivial lower
bound for the problem by showing (our main result) that it is
exponential time hard. Furthermore, we show that this holds even under
rank 1 solutions (substitutions whose codomains are restricted to
contain rank 1 types). In addition, we provide a fixed-parameter
intractability result for intersection type matching (one-sided
unification), which is known to be NP-complete.
We place the intersection type unification problem in the context of
unification theory. The equational theory of intersection types can
be presented as an algebraic theory with an ACI (associative,
commutative, and idempotent) operator (intersection type) combined
with distributivity properties with respect to a second operator
(function type). Although the problem is algebraically natural and
interesting, it appears to occupy a hitherto unstudied place in the
theory of unification, and our investigation of the problem suggests
that new methods are required to understand the problem. Thus, for the
lower bound proof, we were not able to reduce from known results in
ACI-unification theory and use game-theoretic methods for two-player
tiling games.

Andrej Dudenhefner, Moritz Martens, and Jakob Rehof. The Intersection Type Unification Problem. In 1st International Conference on Formal Structures for Computation and Deduction (FSCD 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 52, pp. 19:1-19:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2016)

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@InProceedings{dudenhefner_et_al:LIPIcs.FSCD.2016.19, author = {Dudenhefner, Andrej and Martens, Moritz and Rehof, Jakob}, title = {{The Intersection Type Unification Problem}}, booktitle = {1st International Conference on Formal Structures for Computation and Deduction (FSCD 2016)}, pages = {19:1--19:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-010-1}, ISSN = {1868-8969}, year = {2016}, volume = {52}, editor = {Kesner, Delia and Pientka, Brigitte}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2016.19}, URN = {urn:nbn:de:0030-drops-59955}, doi = {10.4230/LIPIcs.FSCD.2016.19}, annote = {Keywords: Intersection Type, Equational Theory, Unification, Tiling, Complexity} }

Document

**Published in:** LIPIcs, Volume 38, 13th International Conference on Typed Lambda Calculi and Applications (TLCA 2015)

We present a method for synthesizing compositions of mixins using type inhabitation in intersection types. First, recursively defined classes and mixins, which are functions over classes, are expressed as terms in a lambda calculus with records. Intersection types with records and record-merge are used to assign meaningful types to these terms without resorting to recursive types. Second, typed terms are translated to a repository of typed combinators. We show a relation between record types with record-merge and intersection types with constructors. This relation is used to prove soundness and partial completeness of the translation with respect to mixin composition synthesis. Furthermore, we demonstrate how a translated repository and goal type can be used as input to an existing framework for composition synthesis in bounded combinatory logic via type inhabitation. The computed result corresponds to a mixin composition typed by the goal type.

Jan Bessai, Andrej Dudenhefner, Boris Düdder, Tzu-Chun Chen, Ugo de’Liguoro, and Jakob Rehof. Mixin Composition Synthesis Based on Intersection Types. In 13th International Conference on Typed Lambda Calculi and Applications (TLCA 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 38, pp. 76-91, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2015)

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@InProceedings{bessai_et_al:LIPIcs.TLCA.2015.76, author = {Bessai, Jan and Dudenhefner, Andrej and D\"{u}dder, Boris and Chen, Tzu-Chun and de’Liguoro, Ugo and Rehof, Jakob}, title = {{Mixin Composition Synthesis Based on Intersection Types}}, booktitle = {13th International Conference on Typed Lambda Calculi and Applications (TLCA 2015)}, pages = {76--91}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-87-3}, ISSN = {1868-8969}, year = {2015}, volume = {38}, editor = {Altenkirch, Thorsten}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TLCA.2015.76}, URN = {urn:nbn:de:0030-drops-51563}, doi = {10.4230/LIPIcs.TLCA.2015.76}, annote = {Keywords: Record Calculus, Combinatory Logic, Type Inhabitation, Mixin, Intersection Type} }

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