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Sabine Frittella

Publications and source records attributed to Sabine Frittella.

At least 19 recordsLinked to original sources
arXiv2025

Vers une mod\'elisation de la confiance dans le renseignement sur les menaces cyber

Cyber threat intelligence (CTI) is essential for effective system defense. CTI is a collection of information about current or past threats to a computer system. This information is gathered by an agent through observation, or based on a set of sources. Building intelligence only makes sense if you have confidence in it. To achieve this, it is necessary to estimate the confidence in each piece of information gathered, taking into account the different dimensions that can make it up: reliability of the source, competence, plausibility of the information, credibility of the information, for example. The information gathered must then be combined with other information to consolidate an agent's knowledge. Recent advances have been made in the theory underlying the modeling of trust for decision-making based on uncertain information, notably by using multivalued logic. This approach makes it possible to deal with unknown values of trust-building parameters, or to easily integrate dimensions. In this article we present the problem of CTI and CTI information sharing, and the reasons that led us to use a logic-based solution for an initial implementation.

cs.CR↗
arXiv2025

Distributed Transition System with Tags and Value-wise Metric, for Privacy Analysis

We introduce a logical framework named Distributed Labeled Tagged Transition System (DLTTS), using concepts from Probabilistic Automata, Probabilistic Concurrent Systems, and Probabilistic labelled transition systems. We show that DLTTS can be used to formally model how a given piece of private information P (e.g., a set of tuples) stored in a given database D can get captured progressively by an adversary A repeatedly querying D, enhancing the knowledge acquired from the answers to these queries with relational deductions using certain additional non-private data. The database D is assumed protected with generalization mechanisms. We also show that, on a large class of databases, metrics can be defined 'value-wise', and more general notions of adjacency between data bases can be defined, based on these metrics. These notions can also play a role in differentially private protection mechanisms.

cs.LO↗
arXiv2024

Filter-induced entailment relations in paraconsistent G\"{o}del logics

We consider two expansions of G\"{o}del logic $\mathsf{G}$ with two versions of paraconsistent negation. The first one is $\mathsf{G_{inv}}$ -- the expansion of $\mathsf{G}$ with an involuitive negation ${\sim_\mathsf{i}}$ defined via $v({\sim_\mathsf{i}}\phi)=1-v(\phi)$. The second one is $\mathsf{G}^2_{(\rightarrow,-\!<)}$ -- an expansion with a so-called strong negation $\neg$. This logic utilises two independent valuations on $[0,1]$ -- $v_1$ (support of truth or positive support) and $v_2$ (support of falsity or negative support) that are connected with $\neg$. Two valuations in $\mathsf{G}^2_{(\rightarrow,-\!<)}$ can be combined into one valuation $v$ on $[0,1]^{\Join}$ -- the twisted product of $[0,1]$ with itself -- with two components $v_1$ and $v_2$. The two logics are closely connected as ${\sim_\mathsf{i}}$ and $\neg$ allow for similar definitions of co-implication -- $\phi-\!<\chi:={\sim_\mathsf{i}}({\sim_\mathsf{i}}\chi\rightarrow{\sim_\mathsf{i}}\phi)$ and $\phi-\!<\chi:=\neg(\neg\chi\rightarrow\neg\phi)$ -- but do not coincide since the set of values of $\mathsf{G}^2_{(\rightarrow,-\!<)}$ is not ordered linearly. Our main goal is to study different entailment relations in $\mathsf{G_{inv}}$ and $\mathsf{G}^2_{(\rightarrow,-\!<)}$ that are induced by filters on $[0,1]$ and $[0,1]^{\Join}$, respectively. In particular, we determine the exact number of such relations in both cases, establish whether any of them coincide with the entailment defined via the order on $[0,1]$ and $[0,1]^{\Join}$, and obtain their hierarchy. We also construct reductions of filter-induced entailment relations to the ones defined via the order.

math.LO↗
arXiv2024

Qualitative reasoning in a two-layered framework

The reasoning with qualitative uncertainty measures involves comparative statements about events in terms of their likeliness without necessarily assigning an exact numerical value to these events. The paper is divided into two parts. In the first part, we formalise reasoning with the qualitative counterparts of capacities, belief functions, and probabilities, within the framework of two-layered logics. Namely, we provide two-layered logics built over the classical propositional logic using a unary belief modality $\Be$ that connects the inner layer to the outer one where the reasoning is formalised by means of Gödel logic. We design their Hilbert-style axiomatisations and prove their completeness. In the second part, we discuss the paraconsistent generalisations of the logics for qualitative uncertainty that take into account the case of the available information being contradictory or inconclusive.

math.LO↗
arXiv2024

Fuzzy bi-Gödel modal logic and its paraconsistent relatives

We present the axiomatisation of the fuzzy bi-Gödel modal logic (formulated in the language containing $\triangle$ and treating the coimplication as a defined connective) and establish its PSpace-completeness. We also consider its paraconsistent relatives defined on fuzzy frames with two valuations $e_1$ and $e_2$ standing for the support of truth and falsity, respectively, and equipped with \emph{two fuzzy relations} $R^+$ and $R^-$ used to determine supports of truth and falsity of modal formulas. We establish embeddings of these paraconsistent logics into the fuzzy bi-Gödel modal logic and use them to prove their PSpace-completeness and obtain the characterisation of definable frames.

math.LO↗
arXiv2024

Two-layered logics for probabilities and belief functions over Belnap--Dunn logic

This paper is an extended version of an earlier submission to WoLLIC 2023. We discuss two-layered logics formalising reasoning with probabilities and belief functions that combine the Lukasiewicz $[0,1]$-valued logic with Baaz $\triangle$ operator and the Belnap--Dunn logic. We consider two probabilistic logics that present two perspectives on the probabilities in the Belnap--Dunn logic: $\pm$-probabilities and $\mathbf{4}$-probabilities. In the first case, every event $\phi$ has independent positive and negative measures that denote the likelihoods of $\phi$ and $\neg\phi$, respectively. In the second case, the measures of the events are treated as partitions of the sample into four exhaustive and mutually exclusive parts corresponding to pure belief, pure disbelief, conflict and uncertainty of an agent in $\phi$. In addition to that, we discuss two logics for the paraconsistent reasoning with belief and plausibility functions. They equip events with two measures (positive and negative) with their main difference being whether the negative measure of $\phi$ is defined as the belief in $\neg\phi$ or treated independently as the plausibility of $\neg\phi$. We provide a sound and complete Hilbert-style axiomatisation of the logic of $\mathbf{4}$-probabilities and establish faithful translations between it and the logic of $\pm$-probabilities. We also show that the satisfiability problem in all logics is $\mathsf{NP}$-complete.

math.LO↗
arXiv2023

Crisp bi-Gödel modal logic and its paraconsistent expansion

In this paper, we provide a Hilbert-style axiomatisation for the crisp bi-Gödel modal logic $\KbiG$. We prove its completeness w.r.t.\ crisp Kripke models where formulas at each state are evaluated over the standard bi-Gödel algebra on $[0,1]$. We also consider a paraconsistent expansion of $\KbiG$ with a De Morgan negation $\neg$ which we dub $\KGsquare$. We devise a Hilbert-style calculus for this logic and, as a~con\-se\-quence of a~conservative translation from $\KbiG$ to $\KGsquare$, prove its completeness w.r.t.\ crisp Kripke models with two valuations over $[0,1]$ connected via $\neg$. For these two logics, we establish that their decidability and validity are $\mathsf{PSPACE}$-complete. We also study the semantical properties of $\KbiG$ and $\KGsquare$. In particular, we show that Glivenko theorem holds only in finitely branching frames. We also explore the classes of formulas that define the same classes of frames both in $\mathbf{K}$ (the classical modal logic) and the crisp Gödel modal logic $\KG^c$. We show that, among others, all Sahlqvist formulas and all formulas $ϕ\rightarrowχ$ where $ϕ$ and $χ$ are monotone, define the same classes of frames in $\mathbf{K}$ and $\KG^c$.

math.LO↗
arXiv2023

Reasoning with belief functions over Belnap--Dunn logic

We design an expansion of Belnap--Dunn logic with belief and plausibility functions that allow non-trivial reasoning with inconsistent and incomplete probabilistic information. We also formalise reasoning with non-standard probabilities and belief functions in two ways. First, using a calculus of linear inequalities, akin to the one presented in~\cite{FaginHalpernMegiddo1990}. Second, as a two-layered modal logic wherein reasoning with evidence (the outer layer) utilises paraconsistent expansions of Łukasiewicz logic. The second approach is inspired by~\cite{BaldiCintulaNoguera2020}. We prove completeness for both kinds of calculi and show their equivalence by establishing faithful translations in both directions.

math.LO↗
arXiv2023

Presumptive Reasoning in a Paraconsistent Setting

We explore presumptive reasoning in the paraconsistent case. Specifically, we provide semantics for non-trivial reasoning with presumptive arguments with contradictory assumptions or conclusions. We adapt the case models proposed by Verheij and define the paraconsistent analogues of the three types of validity defined therein: coherent, presumptively valid, and conclusive ones. To formalise the reasoning, we define case models that use the expansion of the Belnap-Dunn logic with the Baaz Delta operator. We also show how to recover presumptive reasoning in the original, classical context from our paraconsistent version of case models. Finally, we construct a two-layered logic and obtain a faithful translation of presumptive arguments into formulas.

math.LO↗
arXiv2023

Paraconsistent Gödel modal logic on bi-relational frames

We further develop the paraconsistent Gödel modal logic. In this paper, we consider its version endowed with Kripke semantics on $[0,1]$-valued frames with two fuzzy relations $R^+$ and $R^-$ (degrees of trust in assertions and denials) and two valuations $v_1$ and $v_2$ (support of truth and support of falsity) linked with a De Morgan negation $\neg$. We demonstrate that it \emph{does not} extend Gödel modal logic and that $\Box$ and $\lozenge$ are not interdefinable. We also show that several important classes of frames are $\birelKGsquare$ definable (in particular, crisp, mono-relational, and finitely branching). For $\birelKGsquare$ over finitely branching frames, we create a sound and complete constraint tableaux calculus and a decision procedure based upon it. Using the decision procedure we show that $\birelKGsquare$ satisfiability and validity are in PSPACE.

math.LO↗
arXiv2023

Non-standard modalities in paraconsistent Gödel logic

We introduce a paraconsistent expansion of the Gödel logic with a De Morgan negation $\neg$ and modalities $\blacksquare$ and $\blacklozenge$. We equip it with Kripke semantics on frames with two (possibly fuzzy) relations: $R^+$ and $R^-$ (interpreted as the degree of trust in affirmations and denials by a given source) and valuations $v_1$ and $v_2$ (positive and negative support) ranging over $[0,1]$ and connected via $\neg$. We motivate the semantics of $\blacksquareϕ$ (resp., $\blacklozengeϕ$) as infima (suprema) of both positive and negative supports of $ϕ$ in $R^+$- and $R^-$-accessible states, respectively. We then prove several instructive semantical properties of the logic. Finally, we devise a tableaux system for branching fragment and establish the complexity of satisfiability and validity.

math.LO↗
arXiv2022

Distributed Transition Systems with Tags for Privacy Analysis

We present a logical framework that formally models how a given private information P stored on a given database D, can get captured progressively, by an agent/adversary querying the database repeatedly. Named DLTTS (Distributed Labeled Tagged Transition System), the framework borrows ideas from several domains: Probabilistic Automata of Segala, Probabilistic Concurrent Systems, and Probabilistic labelled transition systems. To every node on a DLTTS is attached a tag that represents the 'current' knowledge of the adversary, acquired from the responses of the answering mechanism of the DBMS to his/her queries, at the nodes traversed earlier, along any given run; this knowledge is completed at the same node, with further relational deductions, possibly in combination with 'public' information from other databases given in advance. A 'blackbox' mechanism is also part of a DLTTS, and it is meant as an oracle; its role is to tell if the private information has been deduced by the adversary at the current node, and if so terminate the run. An additional special feature is that the blackbox also gives information on how 'close', or how 'far', the knowledge of the adversary is, from the private information P , at the current node. A metric is defined for that purpose, on the set of all 'type compatible' tuples from the given database, the data themselves being typed with the headers of the base. Despite the transition systems flavor of our framework, this metric is not 'behavioral' in the sense presented in some other works. It is exclusively database oriented, and allows to define new notions of adjacency and of indistinguishabilty between databases, more generally than those usually based on the Hamming metric (and a restricted notion of adjacency). Examples are given all along to illustrate how our framework works. Keywords:Database, Privacy, Transition System, Probability, Distribution.

cs.CL↗
arXiv2022

Updating belief functions over Belnap--Dunn logic

Belief and plausibility are weaker measures of uncertainty than that of probability. They are motivated by the situations when full probabilistic information is not available. However, information can also be contradictory. Therefore, the framework of classical logic is not necessarily the most adequate. Belnap-Dunn logic was introduced to reason about incomplete and contradictory information. Klein et al and Bilkova et al generalize the notion of probability measures and belief functions to Belnap-Dunn logic, respectively. In this article, we study how to update belief functions with new pieces of information. We present a first approach via a frame semantics of Belnap-Dunn logic.

cs.AI↗
arXiv2022

Paraconsistent Gödel modal logic

We introduce a~paraconsistent modal logic $\mathbf{K}\mathsf{G}^2$, based on Gödel logic with coimplication (bi-Gödel logic) expanded with a De Morgan negation $\neg$. We use the logic to formalise reasoning with graded, incomplete and inconsistent information. Semantics of $\mathbf{K}\mathsf{G}^2$ is two-dimensional: we interpret $\mathbf{K}\mathsf{G}^2$ on crisp frames with two valuations $v_1$ and $v_2$, connected via $\neg$, that assign to each formula two values from the real-valued interval $[0,1]$. The first (resp., second) valuation encodes the positive (resp., negative) information the state gives to a~statement. We obtain that $\mathbf{K}\mathsf{G}^2$ is strictly more expressive than the classical modal logic $\mathbf{K}$ by proving that finitely branching frames are definable and by establishing a faithful embedding of $\mathbf{K}$ into $\mathbf{K}\mathsf{G}^2$. We also construct a~constraint tableau calculus for $\mathbf{K}\mathsf{G}^2$ over finitely branching frames, establish its decidability and provide a~complexity evaluation.

math.LO↗
arXiv2021

Constraint tableaux for two-dimensional fuzzy logics

We introduce two-dimensional logics based on Łukasiewicz and Gödel logics to formalize paraconsistent fuzzy reasoning. The logics are interpreted on matrices, where the common underlying structure is the bi-lattice (twisted) product of the $[0,1]$ interval. The first (resp.\ second) coordinate encodes the positive (resp.\ negative) information one has about a statement. We propose constraint tableaux that provide a modular framework to address their completeness and complexity.

math.LO↗
arXiv2020

How to reason with inconsistent probabilistic information?

A recent line of research has developed around logics of belief based on evidence. The approach of Bílková et al understands belief as based on information confirmed by a reliable source. We propose a finer analysis of how belief can be based on information, where the confirmation comes from multiple possibly conflicting sources and is of a probabilistic nature. We use Belnap-Dunn logic and its probabilistic extensions to account for potentially contradictory information on which belief is grounded. We combine it with an extension of Lukasiewicz logic, or a bilattice logic, within a two-layer modal logical framework to account for belief.

cs.LO↗
arXiv2020

Rough concepts

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.

cs.LO↗