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Categorization of business processes is an important part of auditing. Large amounts of transactional 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. We use Dempster-Shafer mass functions to represent agendas showing different interest in different set of financial accounts. 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. We use this framework to describe a machine-leaning meta algorithm for outlier detection and classification which can provide local and global explanations of its result and demonstrate it through an outlier detection algorithm., Comment: arXiv admin note: substantial text overlap with arXiv:2210.17330

We generalize Kracht's theory of internal describability from classical modal logic to the family of all logics canonically associated with varieties of normal lattice expansions (LE algebras). We work in the purely algebraic setting of perfect LEs; the formulas playing the role of Kracht's formulas in this generalized setting pertain to a first order language whose atoms are special inequalities between terms of perfect algebras. Via duality, formulas in this language can be equivalently translated into first order conditions in the frame correspondence languages of several types of relational semantics for LE-logics.

Graph-based frames have been introduced as a logical framework which internalizes an inherent boundary to knowability. They also support the interpretation of lattice-based (modal) logics as hyper-constructive logics of evidential reasoning. Conceptually, the present paper proposes graph-based frames as a formal framework suitable for generalizing Pawlak's rough set theory to a setting in which inherent limits to knowability need to be considered. Technically, the present paper establishes systematic connections between the first-order correspondents of Sahlqvist modal reduction principles on Kripke frames, and on the more general relational environments of graph-based and polarity-based frames. This work is part of a research line aiming at: (a) comparing and inter-relating the various (first-order) conditions corresponding to a given (modal) axiom in different relational semantics (b) recognizing when first-order sentences in the frame-correspondence languages of different relational structures encode the same modal content (c) meaningfully transferring relational properties across different semantic contexts. The present paper develops these results for the graph-based semantics, polarity-based semantics, and all Sahlqvist modal reduction principles. As an application, we study well known modal axioms in rough set theory on graph-based frames and show that, although these axioms correspond to different first-order conditions on graph-based frames, their intuitive meaning is retained.This allows us to introduce the notion of hyperconstructivist approximation spaces as the subclass of graph-based frames defined by the first-order conditions corresponding to the same modal axioms defining classical generalized approximation spaces, and to transfer the properties and the intuitive understanding of different approximation spaces to graph-based frames.

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 consider precontact algebras as a suitable algebraic environment for negative permission, and we characterize properties of several types of permission (negative, static, dynamic), as well as their interactions with normative systems, by means of suitable modal languages encoding outputs., Comment: arXiv admin note: text overlap with arXiv:2205.13903

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., Comment: 21 pages

In general, providing an axiomatization for an arbitrary logic is a task that may require some ingenuity. In the case of logics defined by a finite logical matrix (three-valued logics being a particularly simple example), the generation of suitable finite axiomatizations can be completely automatized, essentially by expressing the matrix tables via inference rules. In this chapter we illustrate how two formalisms, the 3-labelled calculi of Baaz, Ferm\"uller and Zach and the multiple-conclusion (or Set-Set) Hilbert-style calculi of Shoesmith and Smiley, may be uniformly employed to axiomatize logics defined by a three-valued logical matrix. The generating procedure common to both formalisms can be described as follows: first (i) convert the matrix semantics into rule form (we refer to this step as the generating subprocedure) and then (ii) simplify the set of rules thus obtained, essentially relying on the defining properties of any Tarskian consequence relation (we refer to this step as the streamlining subprocedure). We illustrate through some examples that, if a minimal expressiveness assumption is met (namely, if the matrix defining the logic is monadic), then it is straightforward to define effective translations guaranteeing the equivalence between the 3-labelled and the Set-Set approach.

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.

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., Comment: In the new version, some of the proofs have been corrected and simplified, We also improved the exposition of the complexity results

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.

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.

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.

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., Comment: This is a revision of the previous version

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.

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., Comment: arXiv admin note: text overlap with arXiv:1604.08822 by other authors

We introduce proper display calculi for basic monotonic modal logic, the conditional logic CK and a number of their axiomatic extensions. These calculi are sound, complete, conservative and enjoy cut elimination and subformula property. Our proposal applies the multi-type methodology in the design of proper display calculi, starting from a semantic analysis which motivates syntactic translations from single-type non-normal modal logics to multi-type normal poly-modal logics., Comment: extended version. arXiv admin note: substantial text overlap with arXiv:1903.04868

We prove an algebraic canonicity theorem for normal LE-logics of arbitrary signature, in a generalized setting in which the non-lattice connectives are interpreted as operations mapping tuples of elements of the given lattice to closed or open elements of its canonical extension. Interestingly, the syntactic shape of LE-inequalities which guarantees their canonicity in this generalized setting turns out to coincide with the syntactic shape of analytic inductive inequalities, which guarantees LE-inequalities to be equivalently captured by analytic structural rules of a proper display calculus. We show that this canonicity result connects and strengthens a number of recent canonicity results in two different areas: subordination algebras, and transfer results via G\"odel-McKinsey-Tarski translations., Comment: arXiv admin note: text overlap with arXiv:1603.08515, arXiv:1603.08341

We discuss an ongoing line of research in the relational (non topological) semantics of non-distributive logics. The developments we consider are technically rooted in dual characterization results and insights from unified correspondence theory. However, they also have broader, conceptual ramifications for the intuitive meaning of non-distributive logics, which we explore., Comment: 36 pages

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., Comment: Fixed list of authors in metadata

In this paper, we generalize the basic notions and results of Dempster-Shafer theory from predicates to formal concepts. Results include the representation of conceptual belief functions as inner measures of suitable probability functions, and a Dempster-Shafer rule of combination on belief functions on formal concepts.

This paper continues the investigation of the logic of competing theories, be they scientific, social, political etc. We introduce a many-valued, multi-type modal language which we endow with relational semantics based on enriched reflexive graphs, inspired by Plo\v{s}\v{c}ica's representation of general lattices. We axiomatize the resulting many-valued, non-distributive modal logic of these structures and prove a completeness theorem. We illustrate the application of this logic through a case study in which we model competition among interacting political promises and social demands within an arena of political parties social groups., Comment: 24 pages

We introduce a complete many-valued semantics for basic normal lattice-based modal logic. This relational semantics is grounded on many-valued formal contexts from Formal Concept Analysis. We discuss an interpretation and possible applications of this logical framework for categorization theory to the formal analysis of multi-market competition., Comment: 24 pages

The present paper proposes a novel way to unify Rough Set Theory and Formal Concept Analysis. Our method stems from results and insights developed in the algebraic theory of modal logic, and is based on the idea that Pawlak's original approximation spaces can be seen as special instances of enriched formal contexts, i.e. relational structures based on formal contexts from Formal Concept Analysis.