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Join-preserving maps on the discrete time scale $\omega^+$, referred to as time warps, have been proposed as graded modalities that can be used to quantify the growth of information in the course of program execution. The set of time warps forms a simple distributive involutive residuated lattice -- called the time warp algebra -- that is equipped with residual operations relevant to potential applications. In this paper, we show that although the time warp algebra generates a variety that lacks the finite model property, it nevertheless has a decidable equational theory. We also describe an implementation of a procedure for deciding equations in this algebra, written in the OCaml programming language, that makes use of the Z3 theorem prover.

It was proved by Maksimova in 1977 that exactly eight varieties of Heyting algebras have the amalgamation property, and hence exactly eight axiomatic extensions of intuitionistic propositional logic have the deductive interpolation property. The prevalence of the deductive interpolation property for axiomatic extensions of substructural logics and the amalgamation property for varieties of pointed residuated lattices, their equivalent algebraic semantics, is far less well understood, however. Taking as our starting point a formulation of intuitionistic propositional logic as the full Lambek calculus with exchange, weakening, and contraction, we investigate the role of the exchange rule--algebraically, the commutativity law--in determining the scope of these properties. First, we show that there are continuum-many varieties of idempotent semilinear residuated lattices that have the amalgamation property and contain non-commutative members, and hence continuum-many axiomatic extensions of the corresponding logic that have the deductive interpolation property in which exchange is not derivable. We then show that, in contrast, exactly sixty varieties of commutative idempotent semilinear residuated lattices have the amalgamation property, and hence exactly sixty axiomatic extensions of the corresponding logic with exchange have the deductive interpolation property. From this latter result, it follows also that there are exactly sixty varieties of commutative idempotent semilinear residuated lattices whose first-order theories have a model completion.

The one-variable fragment of any first-order logic may be considered as a modal logic, where the universal and existential quantifiers are replaced by a box and diamond modality, respectively. In several cases, axiomatizations of algebraic semantics for these logics have been obtained: most notably, for the modal counterparts S5 and MIPC of the one-variable fragments of first-order classical logic and intuitionistic logic, respectively. Outside the setting of first-order intermediate logics, however, a general approach is lacking. This paper provides the basis for such an approach in the setting of first-order lattice-valued logics, where formulas are interpreted in algebraic structures with a lattice reduct. In particular, axiomatizations are obtained for modal counterparts of one-variable fragments of a broad family of these logics by generalizing a functional representation theorem of Bezhanishvili and Harding for monadic Heyting algebras. An alternative proof-theoretic proof is also provided for one-variable fragments of first-order substructural logics that have a cut-free sequent calculus and admit a certain bounded interpolation property.

Graded modalities have been proposed in recent work on programming languages as a general framework for refining type systems with intensional properties. In particular, continuous endomaps of the discrete time scale, or time warps, can be used to quantify the growth of information in the course of program execution. Time warps form a complete residuated lattice, with the residuals playing an important role in potential programming applications. In this paper, we study the algebraic structure of time warps, and prove that their equational theory is decidable, a necessary condition for their use in real-world compilers. We also describe how our universal-algebraic proof technique lends itself to a constraint-based implementation, establishing a new link between universal algebra and verification technology., Comment: 22 pages, version accepted at RAMICS 2021 plus appendices

Characterizations of admissible quasi-identities, which may be understood as quasi-identities holding in free algebras on countably infinitely many generators, are provided for classes of De Morgan algebras and lattices

A many-valued modal logic is introduced that combines the usual Kripke frame semantics of the modal logic K with connectives interpreted locally at worlds by lattice and group operations over the real numbers. A labelled tableau system is provided and a coNEXPTIME upper bound obtained for checking validity in the logic. Focussing on the modal-multiplicative fragment, the labelled tableau system is then used to establish completeness for a sequent calculus that admits cut-elimination and an axiom system that extends the multiplicative fragment of Abelian logic.

A new hierarchy of "exact" unification types is introduced, motivated by the study of admissible rules for equational classes and non-classical logics. In this setting, unifiers of identities in an equational class are preordered, not by instantiation, but rather by inclusion over the corresponding sets of unified identities. Minimal complete sets of unifiers under this new preordering always have a smaller or equal cardinality than those provided by the standard instantiation preordering, and in significant cases a dramatic reduction may be observed. In particular, the classes of distributive lattices, idempotent semigroups, and MV-algebras, which all have nullary unification type, have unitary or finitary exact type. These results are obtained via an algebraic interpretation of exact unification, inspired by Ghilardi's algebraic approach to equational unification., Comment: arXiv admin note: substantial text overlap with arXiv:1410.5583

Checking the admissibility of quasiequations in a finitely generated (i.e., generated by a finite set of finite algebras) quasivariety Q amounts to checking validity in a suitable finite free algebra of the quasivariety, and is therefore decidable. However, since free algebras may be large even for small sets of small algebras and very few generators, this naive method for checking admissibility in $\Q$ is not computationally feasible. In this paper, algorithms are introduced that generate a minimal (with respect to a multiset well-ordering on their cardinalities) finite set of algebras such that the validity of a quasiequation in this set corresponds to admissibility of the quasiequation in Q. In particular, structural completeness (validity and admissibility coincide) and almost structural completeness (validity and admissibility coincide for quasiequations with unifiable premises) can be checked. The algorithms are illustrated with a selection of well-known finitely generated quasivarieties, and adapted to handle also admissibility of rules in finite-valued logics.

The Craig interpolation property is investigated for substructural logics whose algebraic semantics are varieties of semilinear (subdirect products of linearly ordered) pointed commutative residuated lattices. It is shown that Craig interpolation fails for certain classes of these logics with weakening if the corresponding algebras are not idempotent. A complete characterization is then given of axiomatic extensions of the >R-mingle with unit> logic (corresponding to varieties of Sugihara monoids) that have the Craig interpolation property. This latter characterization is obtained using a model-theoretic quantifier elimination strategy to determine the varieties of Sugihara monoids admitting the amalgamation property. © 2012 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.

Analytic proof calculi are introduced for box and diamond fragments of basic modal fuzzy logics that combine the Kripke semantics of modal logic K with the many-valued semantics of G\"odel logic. The calculi are used to establish completeness and complexity results for these fragments.

Characterizations of admissible quasi-identities, which may be understood as quasi-identities holding in free algebras on countably infinitely many generators, are provided for classes of De Morgan algebras and lattices