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This paper studies the relationship between labelled and nested calculi for propositional intuitionistic logic, first-order intuitionistic logic with non-constant domains and first-order intuitionistic logic with constant domains. It is…
Injectives in several classes of structures associated with logic are characterized. Among the classes considered are residuated lattices, MTL-algebras, IMTL-algebras, BL-algebras, NM-algebras and bounded hoops.
We apply the machinery of projection lattices and von Neumann algebras to analyze the question of how modal interpretations can (and do) circumvent von Neumann's infamous 'no-hidden-variables' theorem.
We construct new families of U_q(gl_n)-modules by continuation from finite dimensional representations. Each such module is associated with a combinatorial object - admissible set of relations defined in \cite{FRZ}. More precisely, we prove…
Differentiable logics are a family of quantitative logics originated in the machine learning literature. Because of their origin, differentiable logics often come equipped with analytic properties that guarantee that they are…
We introduce a novel learning and planning framework that replaces traditional reward-based optimisation with constructive logical inference. In our model, actions, transitions, and goals are represented as logical propositions, and…
In this paper we consider a general way of constructing profinite struc- tures based on a given framework - a countable family of objects and a countable family of recognisers (e.g. formulas). The main theorem states: A subset of a family…
The main goal of this paper is to formulate a constructive analogue of Ackermann's observation about finite set theory and arithmetic. We will see that Heyting arithmetic is bi-interpretable with $\mathsf{CZF^{fin}}$, the finitary version…
By using two different invariants for the Rubik's Magic puzzle, one of metric type, the other of topological type, we can dramatically reduce the universe of constructible configurations of the puzzle. Finding the set of actually…
Martin-L\"of's Intuitionistic Theory of Types is becoming popular for formal reasoning about computer programs. To handle recursion schemes other than primitive recursion, a theory of well-founded relations is presented. Using primitive…
Valuation based systems verifying an idempotent property are studied. A partial order is defined between the valuations giving them a lattice structure. Then, two different strategies are introduced to represent valuations: as infimum of…
It is shown that propositional calculuses of both quantum and classical logics are non-categorical. We find that quantum logic is in addition to an orthomodular lattice also modeled by a weakly orthomodular lattice and that classical logic…
Our paper is the first study of what one might call "reverse mathematics of explicit fixpoints". We study two methods of constructing such fixpoints for formulas whose principal connective is the intuitionistic Lewis arrow. Our main…
We begin with a context more general than set theory. The basic ingredients are essentially the object and functor primitives of category theory, and the logic is weak, requiring neither the Law of Excluded Middle nor quantification. Inside…
Interpretable classification models are built with the purpose of providing a comprehensible description of the decision logic to an external oversight agent. When considered in isolation, a decision tree, a set of classification rules, or…
We study the problem of fitting ontologies and constraints to positive and negative examples that take the form of a finite relational structure. As ontology and constraint languages, we consider the description logics $\mathcal{E\mkern-2mu…
We characterize commutative idempotent involutive residuated lattices as disjoint unions of Boolean algebras arranged over a distributive lattice. We use this description to introduce a new construction, called gluing, that allows us to…
A logic-enriched type theory (LTT) is a type theory extended with a primitive mechanism for forming and proving propositions. We construct two LTTs, named LTTO and LTTO*, which we claim correspond closely to the classical predicative…
This paper investigates the possibility of constructive extraction of measurable selector from set-valued maps which may commonly arise in viability theory, optimal control, discontinuous systems etc. For instance, existence of solutions to…
The P versus NP problem is addressed in a context of provability and limitations on the possibility of finding sound axioms for formal theories. It is shown that if the term "constructible theory" is defined in a way which satisfies certain…