arXiv subjects

Andrea De Domenico

Publications and source records attributed to Andrea De Domenico.

17 recordsLinked to original sources

arXiv2026

Refutation calculi for lattice-based logics: from display to tableaux

Refutation calculi are formal systems developed to derive the invalid formulas of a given logic. While the notion of refutation calculi has played a key role in the development of tableaux calculi, a refutation approach to display calculi has not yet been attempted. In this paper, we introduce refutation display calculi for basic LE-logics, i.e., those logics canonically associated with basic normal lattice expansions of any signature. In particular, we prove soundness and completeness via proof-analysis results on derivable sequents. Finally, we obtain terminating tableaux calculi from these refutation display calculi.

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arXiv2026

Modular constructive Lyndon interpolation for nondistributive logics

We establish the Lyndon interpolation property for basic lattice expansion logics (LE-logics) in arbitrary signatures using display calculi. Our approach is constructive, yielding interpolants algorithmically from derivations, and modular, in the sense that interpolation for axiomatic extensions can be obtained by verifying a local interpolation property for the analytic structural rules corresponding to the additional axioms. To this end, we identify a class of interpolation-safe structural rules preserving local Lyndon interpolation. As applications of the general framework, we show that the tense version of Holliday's fundamental modal logic enjoys the Lyndon interpolation property.

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arXiv2026

Inception Display Calculi

Display calculi were introduced by Nuel Belnap in `Display logic' (1982) as a natural extension of Gentzen's sequent calculi, as a uniform and modular framework capable of encompassing broad classes of logics. In `Unified correspondence as a proof-theoretic tool', the properly displayable (D)LE-logics are syntactically characterized as the logics axiomatised by analytic inductive axioms for any signature. We extend the framework of proper display calculi for LE-logics to include axiomatic extensions with axioms that are inductive but not necessarily analytic inductive. This class of axioms covers and properly extends all Sahlqvist axioms. The present framework takes inspiration from Schroeder-Heister's calculus of Higher-Level Rules and captures the whole acyclic fragment of the substructural hierarchy when generalized to arbitrary signatures. We apply unified correspondence theory and the algorithm ALBA to uniformly generate analytic rules for the aforementioned axiomatic extensions.

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arXiv2025

Algorithmic correspondence and analytic rules

We introduce the algorithm MASSA which takes classical modal formulas in input, and, when successful, effectively generates: (a) (analytic) geometric rules of the labelled calculus G3K, and (b) cut-free derivations (of a certain `canonical' shape) of each given input formula in the geometric labelled calculus obtained by adding the rule in output to G3K. We show that MASSA successfully terminates whenever its input formula is a (definite) analytic inductive formula, in which case, the geometric axiom corresponding to the output rule is, modulo logical equivalence, the first-order correspondent of the input formula. In proving the correctness of MASSA, we also show that the algorithm for the elimination of second-order quantifiers SCAN is complete with respect to the class of inductive analytic formulas. Finally, we show how our algorithm can be extended to the class of inductive formulas and to modal logic with quantifiers.

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arXiv2025

Normative implications

We continue to develop a research line initiated in \cite{wollic22}, studying I/O logic from an algebraic approach based on subordination algebras. We introduce the classes of slanted (co-)Heyting algebras as equivalent presentations of distributive lattices with subordination relations. Interpreting subordination relations as the algebraic counterparts of input/output relations on formulas yields (slanted) modal operations with interesting deontic interpretations. We study the theory of slanted and co-slanted Heyting algebras, develop algorithmic correspondence and inverse correspondence, and present some deontically meaningful axiomatic extensions and examples.

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arXiv2024

Correspondence and Inverse Correspondence for Input/Output Logic and Region-Based Theories of Space

We further develop the algebraic approach to input/output logic initiated in \cite{wollic22}, where subordination algebras and a family of their generalizations were proposed as a semantic environment of various input/output logics. In particular: we extend the modal characterizations of a finite number of well known conditions on normative and permission systems, as well as on subordination, precontact, and dual precontact algebras developed in \cite{de2024obligations}, to those corresponding to the infinite class of {\em clopen-analytic inequalities} in a modal language consisting both of positive and of negative unary modal operators; we characterize the syntactic shape of first-order conditions on algebras endowed with subordination, precontact, and dual precontact relations which guarantees these conditions to be the first-order correspondents of axioms in the modal language above; we introduce algorithms for computing the first-order correspondents of modal axioms on algebras endowed with subordination, precontact, and dual precontact relations, and conversely, for computing the modal axioms of which the conditions satisfying the suitable syntactic shape are the first-order correspondents; finally, we extend Celani's dual characterization results between subordination lattices and subordination spaces to a wider environment which also encompasses precontact and dual precontact relations, and relative to an infinite class of first order conditions relating subordination, precontact and dual precontact relations on distributive lattices. The modal characterizations established in the present paper pave the way to establishing faithful embeddings for infinite classes of input/output logics, and hence to their implementation in LogiKEy, Isabelle/HOL, Lean, or other interactive systems.

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arXiv2024

Non-distributive description logic

We define LE-ALC, a generalization of the description logic ALC based on the propositional logic of general (i.e. not necessarily distributive) lattices, and semantically interpreted on relational structures based on formal contexts from Formal Concept Analysis (FCA). The description logic LE-ALC allows us to formally describe databases with objects, features, and formal concepts, represented according to FCA as Galois-stable sets of objects and features. We describe ABoxes and TBoxes in LE-ALC, provide a tableaux algorithm for checking the consistency of LE-ALC knowledge bases with acyclic TBoxes, and show its termination, soundness and completeness. Interestingly, consistency checking for LE-ALC is in PTIME for acyclic and completely unravelled TBoxes, while the analogous problem in the classical ALC setting is PSPACE-complete.

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arXiv2024

Obligations and permissions, algebraically

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arXiv2024

Obligations and permissions on selfextensional logics

We further develop the abstract algebraic logic approach to input/output logic initiated in \cite{wollic22}, where the family of selfextensional logics was proposed as a general background environment for input/output logics. In this paper, we introduce and discuss the generalizations of several types of permission (negative, dual negative, static, dynamic), as well as their interactions with normative systems, to various families of selfextensional logics, thereby proposing a systematic approach to the definition of normative and permission systems on nonclassical propositional bases.

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arXiv2024

Labelled calculi for lattice-based modal logics

We introduce labelled sequent calculi for the basic normal non-distributive modal logic L and 31 of its axiomatic extensions, where the labels are atomic formulas of a first order language which is interpreted on the canonical extensions of the algebras in the variety corresponding to the logic L. Modular proofs are presented that these calculi are all sound, complete and conservative w.r.t. L, and enjoy cut elimination and the subformula property. The introduction of these calculi showcases a general methodology for introducing labelled calculi for the class of LE-logics and their analytic axiomatic extensions in a principled and uniform way.

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arXiv2024

Labelled calculi for the logics of rough concepts

We introduce sound and complete labelled sequent calculi for the basic normal non-distributive modal logic L and some of its axiomatic extensions, where the labels are atomic formulas of the first order language of enriched formal contexts, i.e., relational structures based on formal contexts which provide complete semantics for these logics. We also extend these calculi to provide a proof system for the logic of rough formal contexts.

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arXiv2023

Modal reduction principles across relational semantics

The present paper establishes systematic connections among the first-order correspondents of Sahlqvist modal reduction principles in various relational semantic settings which include crisp and many-valued Kripke frames, and crisp and many-valued polarity-based frames (aka enriched formal contexts). Building on unified correspondence theory, we aim at introducing a theoretical environment which makes it possible to: (a) compare and inter-relate the various frame correspondents (in different relational settings) of any given Sahlqvist modal reduction principle; (b) recognize when first-order sentences in the frame-correspondence languages of different types of relational structures encode the same "modal content"; (c) meaningfully transfer and represent well known relational properties such as reflexivity, transitivity, symmetry, seriality, confluence, density, across different semantic contexts. These results can be understood as a first step in a research program aimed at making correspondence theory not just (methodologically) unified, but also (effectively) parametric.

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arXiv2023

A Meta-Learning Algorithm for Interrogative Agendas

Explainability is a key challenge and a major research theme in AI research for developing intelligent systems that are capable of working with humans more effectively. An obvious choice in developing explainable intelligent systems relies on employing knowledge representation formalisms which are inherently tailored towards expressing human knowledge e.g., interrogative agendas. In the scope of this work, we focus on formal concept analysis (FCA), a standard knowledge representation formalism, to express interrogative agendas, and in particular to categorize objects w.r.t. a given set of features. Several FCA-based algorithms have already been in use for standard machine learning tasks such as classification and outlier detection. These algorithms use a single concept lattice for such a task, meaning that the set of features used for the categorization is fixed. Different sets of features may have different importance in that categorization, we call a set of features an agenda. In many applications a correct or good agenda for categorization is not known beforehand. In this paper, we propose a meta-learning algorithm to construct a good interrogative agenda explaining the data. Such algorithm is meant to call existing FCA-based classification and outlier detection algorithms iteratively, to increase their accuracy and reduce their sample complexity. The proposed method assigns a measure of importance to different set of features used in the categorization, hence making the results more explainable.

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arXiv2022

Complexity assessments for decidable fragments of Set Theory. IV: A quadratic reduction of constraints over nested sets to Boolean formulae

As a contribution to quantitative set-theoretic inferencing, a translation is proposed of conjunctions of literals of the forms $x=y\setminus z$, $x \neq y\setminus z$, and $z =\{x\}$, where $x,y,z$ stand for variables ranging over the von Neumann universe of sets, into unquantified Boolean formulae of a rather simple conjunctive normal form. The formulae in the target language involve variables ranging over a Boolean ring of sets, along with a difference operator and relators designating equality, non-disjointness and inclusion. Moreover, the result of each translation is a conjunction of literals of the forms $x=y\setminus z$, $x\neq y\setminus z$ and of implications whose antecedents are isolated literals and whose consequents are either inclusions (strict or non-strict) between variables, or equalities between variables. Besides reflecting a simple and natural semantics, which ensures satisfiability-preservation, the proposed translation has quadratic algorithmic time-complexity, and bridges two languages both of which are known to have an NP-complete satisfiability problem.

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arXiv2022

Subordination Algebras as Semantic Environment of Input/Output Logic

We establish a novel connection between two research areas in non-classical logics which have been developed independently of each other so far: on the one hand, input/output logic, introduced within a research program developing logical formalizations of normative reasoning in philosophical logic and AI; on the other hand, subordination algebras, investigated in the context of a research program integrating topological, algebraic, and duality-theoretic techniques in the study of the semantics of modal logic. Specifically, we propose that the basic framework of input/output logic, as well as its extensions, can be given formal semantics on (slight generalizations of) subordination algebras. The existence of this interpretation brings benefits to both research areas: on the one hand, this connection allows for a novel conceptual understanding of subordination algebras as mathematical models of the properties and behaviour of norms; on the other hand, thanks to the well developed connection between subordination algebras and modal logic, the output operators in input/output logic can be given a new formal representation as modal operators, whose properties can be explicitly axiomatised in a suitable language, and be systematically studied by means of mathematically established and powerful tools.

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arXiv2022

Unified inverse correspondence for DLE-Logics

By exploiting the algebraic and order theoretic mechanisms behind Sahlqvist correspondence, the theory of unified correspondence provides powerful tools for correspondence and canonicity across different semantics and signatures, covering all the logics whose algebraic semantics are given by normal (distributive) lattice expansions (referred to as (D)LEs). In particular, the algorithm ALBA, parametric in each (D)LE, effectively computes the first order correspondents of (D)LE-inductive formulas. We present an algorithm that makes use of ALBA's rules and algebraic language to invert its steps in the DLE setting; therefore effectively computing an inductive formula starting from its first order correspondent.

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arXiv2021

On the Convexity of a Fragment of Pure Set Theory with Applications within a Nelson-Oppen Framework

The Satisfiability Modulo Theories (SMT) issue concerns the satisfiability of formulae from multiple background theories, usually expressed in the language of first-order predicate logic with equality. SMT solvers are often based on variants of the Nelson-Oppen combination method, a solver for the quantifier-free fragment of the combination of theories with disjoint signatures, via cooperation among their decision procedures. When each of the theories to be combined by the Nelson-Oppen method is convex (that is, any conjunction of its literals can imply a disjunction of equalities only when it implies at least one of the equalities) and decidable in polynomial time, the running time of the combination procedure is guaranteed to be polynomial in the size of the input formula. In this paper, we prove the convexity of a fragment of Zermelo-Fraenkel set theory, called Multi-Level Syllogistic, most of whose polynomially decidable fragments we have recently characterized.

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