arXiv subjects

Apostolos Tzimoulis

Publications and source records attributed to Apostolos Tzimoulis.

At least 19 recordsLinked to original sources

arXiv2026

A proof-theoretic approach to abstract interpretation

This paper develops a proof-theoretic framework for abstract interpretation by systematically associating logical systems with finite abstractions. Building on earlier work on the internal logics of abstractions, we propose a general procedure for generating a logic whose Lindenbaum-Tarski algebra is isomorphic to a given abstract lattice. The approach identifies logical connectives preserved by the concretization map and derives corresponding proof rules and axioms. The paper establishes soundness and completeness results under suitable conditions, extends the framework to Cartesian products and multi-variable settings, and investigates the logical structure of non-Cartesian abstractions such as octagons. These observations suggest new connections between abstract interpretation, proof theory, and algebraic logic, providing a foundation for a systematic logical analysis of program abstractions.

cs.LO↗

arXiv2026

Canonical Extensions of Quantale-Enriched Categories

Drawing on well-known results from the theory of canonical extensions and the theory of categories enriched over a quantale, we define canonical extensions of quantale-enriched categories and establish their basic properties.

math.CT↗

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.

math.LO↗

arXiv2025

Questions as cognitive filters

In this paper, we develop a logico-algebraic framework for modeling decision-making through deliberation in multi-agent settings. The central concept in this framework is that of interrogative agendas, which represent the cognitive stances of agents regarding which features should be considered relevant in the final decision. We formalize an agent's interrogative agenda as an equivalence relation that identifies outcomes differing only in aspects the agent deems irrelevant. Moreover, we characterize the sublattices of the resulting lattice that correspond to relevant interrogative agendas for deliberation scenarios governed by different ``winning rules." We then introduce a two-sorted logico-algebraic structure-comprising the lattice of relevant interrogative agendas and the Boolean algebras of agent coalitions-to model the interaction between agents and agendas during deliberation. Finally, we discuss which interaction conditions can and cannot be defined within this framework.

cs.LO↗

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.

math.LO↗

arXiv2024

Flexible categorization using formal concept analysis and Dempster-Shafer theory

The framework developed in the present paper provides a formal ground to generate and study explainable categorizations of sets of entities, based on the epistemic attitudes of individual agents or groups thereof. Based on this framework, we discuss a machine-leaning meta-algorithm for outlier detection and classification which provides local and global explanations of its results.

cs.AI↗

arXiv2024

Game semantics for lattice-based modal μ-calculus

In this paper, we generalize modal $μ$-calculus to the non-distributive (lattice-based) modal $μ$-calculus and formalize some scenarios regarding categorization using it. We also provide a game semantics for the developed logic. The proof of adequacy of this game semantics proceeds by generalizing the unfolding games on the power-set algebras to the arbitrary lattices and showing that these games can be used to determine the least and the greatest fixed points of a monotone operator on a lattice. Finally, we define a notion of bisimulations on the polarities and show invariance of non-distributive modal $μ$-calculus under them.

math.LO↗

arXiv2024

Algebraic proof theory for LE-logics

In this paper we extend the research programme in algebraic proof theory from axiomatic extensions of the full Lambek calculus to logics algebraically captured by certain varieties of normal lattice expansions (normal LE-logics). Specifically, we generalise the residuated frames in [34] to arbitrary signatures of normal lattice expansions (LE). Such a generalization provides a valuable tool for proving important properties of LE-logics in full uniformity. We prove semantic cut elimination for the display calculi D.LE associated with the basic normal LE-logics and their axiomatic extensions with analytic inductive axioms. We also prove the finite model property (FMP) for each such calculus D.LE, as well as for its extensions with analytic structural rules satisfying certain additional properties.

math.LO↗

arXiv2023

Outlier detection using flexible categorisation and interrogative agendas

Categorization is one of the basic tasks in machine learning and data analysis. Building on formal concept analysis (FCA), the starting point of the present work is that different ways to categorize a given set of objects exist, which depend on the choice of the sets of features used to classify them, and different such sets of features may yield better or worse categorizations, relative to the task at hand. In their turn, the (a priori) choice of a particular set of features over another might be subjective and express a certain epistemic stance (e.g. interests, relevance, preferences) of an agent or a group of agents, namely, their interrogative agenda. In the present paper, we represent interrogative agendas as sets of features, and explore and compare different ways to categorize objects w.r.t. different sets of features (agendas). We first develop a simple unsupervised FCA-based algorithm for outlier detection which uses categorizations arising from different agendas. We then present a supervised meta-learning algorithm to learn suitable (fuzzy) agendas for categorization as sets of features with different weights or masses. We combine this meta-learning algorithm with the unsupervised outlier detection algorithm to obtain a supervised outlier detection algorithm. We show that these algorithms perform at par with commonly used algorithms for outlier detection on commonly used datasets in outlier detection. These algorithms provide both local and global explanations of their results.

cs.AI↗

arXiv2023

Causal Kripke Models

This work extends Halpern and Pearl's causal models for actual causality to a possible world semantics environment. Using this framework we introduce a logic of actual causality with modal operators, which allows for reasoning about causality in scenarios involving multiple possibilities, temporality, knowledge and uncertainty. We illustrate this with a number of examples, and conclude by discussing some future directions for research.

cs.AI↗

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.

cs.LO↗

arXiv2022

Flexible categorization for auditing using formal concept analysis and Dempster-Shafer theory

Categorization of business processes is an important part of auditing. Large amounts of transnational data in auditing can be represented as transactions between financial accounts using weighted bipartite graphs. We view such bipartite graphs as many-valued formal contexts, which we use to obtain explainable categorization of these business processes in terms of financial accounts involved in a business process by using methods in formal concept analysis. The specific explainability feature of the methodology introduced in the present paper provides several advantages over e.g.~non-explainable machine learning techniques, and in fact, it can be taken as a basis for the development of algorithms which perform the task of clustering on transparent and accountable principles. Here, we focus on obtaining and studying different ways to categorize according to different extents of interest in different financial accounts, or interrogative agendas, of various agents or sub-tasks in audit. We use Dempster-Shafer mass functions to represent agendas showing different interest in different set of financial accounts. We propose two new methods to obtain categorizations from these agendas. We also model some possible deliberation scenarios between agents with different interrogative agendas to reach an aggregated agenda and categorization. The framework developed in this paper provides a formal ground to obtain and study explainable categorizations from the data represented as bipartite graphs according to the agendas of different agents in an organization (e.g.~an audit firm), and interaction between these through deliberation.

cs.AI↗

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.

math.LO↗

arXiv2021

First order logic properly displayed

We introduce a proper display calculus for first-order logic, of which we prove soundness, completeness, conservativity, subformula property and cut elimination via a Belnap-style metatheorem. All inference rules are closed under uniform substitution and are without side conditions.

math.LO↗

arXiv2021

Neighbourhood semantics for graded modal logic

We introduce a class of neighbourhood frames for graded modal logic embedding Kripke frames into neighbourhood frames. This class of neighbourhood frames is shown to be first-order definable but not modally definable. We also obtain a new definition of graded bisimulation with respect to Kripke frames by modifying the definition of monotonic bisimulation.

math.LO↗

arXiv2021

Vector spaces as Kripke frames

In recent years, the compositional distributional approach in computational linguistics has opened the way for an integration of the \emph{lexical} aspects of meaning into Lambek's type-logical grammar program. This approach is based on the observation that a sound semantics for the associative, commutative and unital Lambek calculus can be based on vector spaces by interpreting fusion as the tensor product of vector spaces. In this paper, we build on this observation and extend it to a `vector space semantics' for the \emph{general} Lambek calculus, based on \emph{algebras over a field} $\mathbb{K}$ (or $\mathbb{K}$-algebras), i.e. vector spaces endowed with a bilinear binary product. Such structures are well known in algebraic geometry and algebraic topology, since they are important instances of Lie algebras and Hopf algebras. Applying results and insights from duality and representation theory for the algebraic semantics of nonclassical logics, we regard $\mathbb{K}$-algebras as `Kripke frames' the complex algebras of which are complete residuated lattices. This perspective makes it possible to establish a systematic connection between vector space semantics and the standard Routley-Meyer semantics of (modal) substructural logics.

cs.LO↗

arXiv2021

Syntactic completeness of proper display calculi

A recent strand of research in structural proof theory aims at exploring the notion of analytic calculi (i.e. those calculi that support general and modular proof-strategies for cut elimination), and at identifying classes of logics that can be captured in terms of these calculi. In this context, Wansing introduced the notion of proper display calculi as one possible design framework for proof calculi in which the analiticity desiderata are realized in a particularly transparent way. Recently, the theory of properly displayable logics (i.e. those logics that can be equivalently presented with some proper display calculus) has been developed in connection with generalized Sahlqvist theory (aka unified correspondence). Specifically, properly displayable logics have been syntactically characterized as those axiomatized by analytic inductive axioms, which can be equivalently and algorithmically transformed into analytic structural rules so that the resulting proper display calculi enjoy a set of basic properties: soundness, completeness, conservativity, cut elimination and subformula property. In this context, the proof that the given calculus is complete w.r.t. the original logic is usually carried out syntactically, i.e. by showing that a (cut free) derivation exists of each given axiom of the logic in the basic system to which the analytic structural rules algorithmically generated from the given axiom have been added. However, so far this proof strategy for syntactic completeness has been implemented on a case-by-case base, and not in general. In this paper, we address this gap by proving syntactic completeness for properly displayable logics in any normal (distributive) lattice expansion signature. Specifically, we show that for every analytic inductive axiom a cut free derivation can be effectively generated which has a specific shape, referred to as pre-normal form.

cs.LO↗

arXiv2020

Lambek-Grishin Calculus: Focusing, Display and Full Polarization

\emph{Focused sequent calculi} are a refinement of sequent calculi, where additional side-conditions on the applicability of inference rules force the implementation of a proof search strategy. Focused cut-free proofs exhibit a special normal form that is used for defining identity of sequent calculi proofs. We introduce a novel focused display calculus fD.LG and a fully polarized algebraic semantics FP.LG for Lambek-Grishin logic by generalizing the theory of \emph{multi-type calculi} and their algebraic semantics with \emph{heterogenous consequence relations}. The calculus fD.LG has \emph{strong focalization} and it is \emph{sound and complete} w.r.t. FP.LG. This completeness result is in a sense stronger than completeness with respect to standard polarized algebraic semantics (see e.g. the phase semantics of Bastenhof for Lambek-Grishin logic or Hamano and Takemura for linear logic), insofar we do not need to quotient over proofs with consecutive applications of shifts over the same formula. We plan to investigate the connections, if any, between this completeness result and the notion of \emph{full completeness} introduced by Abramsky et al. We also show a number of additional results. fD.LG is sound and complete w.r.t. LG-algebras: this amounts to a semantic proof of the so-called \emph{completeness of focusing}, given that the standard (display) sequent calculus for Lambek-Grishin logic is complete w.r.t. LG-algebras. fD.LG and the focused calculus f.LG of Moortgat and Moot are equivalent with respect to proofs, indeed there is an effective translation from f.LG-derivations to fD.LG-derivations and vice versa: this provides the link with operational semantics, given that every f.LG-derivation is in a Curry-Howard correspondence with a directional $\overlineλμ\widetildeμ$-term.

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