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MOST SIMPLE EXTENSIONS OF $\textbf{FL}_{\textbf{e}}$ ARE UNDECIDABLE

Part of: Algebraic logic Ordered structures Varieties General logic

Published online by Cambridge University Press:  10 June 2021

NIKOLAOS GALATOS  and
GAVIN ST. JOHN
NIKOLAOS GALATOS
Affiliation:
DEPARTMENT OF MATHEMATICS UNIVERSITY OF DENVERDENVER, CO 80210, USAE-mail:Nikolaos.Galatos@du.edu
GAVIN ST. JOHN
Affiliation:
DIPARTIMENTO DI PEDAGOGIA, PSICOLOGIA, & FILOSOFIA UNIVERSITÀ DI CAGLIARI09123CAGLIARI, ITALYE-mail:Gavin.StJohn@gmail.com
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Abstract

All known structural extensions of the substructural logic $\textbf{FL}_{\textbf{e}}$ , the Full Lambek calculus with exchange/commutativity (corresponding to subvarieties of commutative residuated lattices axiomatized by $\{\vee , \cdot , 1\}$ -equations), have decidable theoremhood; in particular all the ones defined by knotted axioms enjoy strong decidability properties (such as the finite embeddability property). We provide infinitely many such extensions that have undecidable theoremhood, by encoding machines with undecidable halting problem. An even bigger class of extensions is shown to have undecidable deducibility problem (the corresponding varieties of residuated lattices have undecidable word problem); actually with very few exceptions, such as the knotted axioms and the other prespinal axioms, we prove that undecidability is ubiquitous. Known undecidability results for non-commutative extensions use an encoding that fails in the presence of commutativity, so and-branching counter machines are employed. Even these machines provide encodings that fail to capture proper extensions of commutativity, therefore we introduce a new variant that works on an exponential scale. The correctness of the encoding is established by employing the theory of residuated frames.

Keywords

substructural logicfull Lambek calculussimple extensionsresiduated latticeequational theoryword problemdecidability

MSC classification

Primary: 03B47: Substructural logics (including relevance, entailment, linear logic, Lambek calculus, BCK and BCI logics)
Secondary: 08B15: Lattices of varieties 03G10: Lattices and related structures 06F05: Ordered semigroups and monoids
Type
Article
Information
The Journal of Symbolic Logic , Volume 87 , Issue 3 , September 2022 , pp. 1156 - 1200
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Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

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