Nonmonotonic Reasoning Research Articles

Probability propagation rules for Aristotelian syllogisms

We present a coherence-based probability semantics and probability propagation rules for (categorical) Aristotelian syllogisms. For framing the Aristotelian syllogisms as probabilistic inferences, we interpret basic syllogistic sentence types A, E, I, O by suitable precise and imprecise conditional probability assessments. Then, we define validity of probabilistic inferences and probabilistic notions of the existential import which is required, for the validity of the syllogisms. Based on a generalization of de Finetti's fundamental theorem to conditional probability, we investigate the coherent probability propagation rules of argument forms of the syllogistic Figures I, II, and III, respectively. These results allow to show, for all three figures, that each traditionally valid syllogism is also valid in our coherence-based probability semantics. Moreover, we interpret the basic syllogistic sentence types by suitable defaults and negated defaults. Thereby, we build a bridge from our probability semantics of Aristotelian syllogisms to nonmonotonic reasoning. Then we show that reductio by conversion does not work while reductio ad impossibile can be applied in our approach. Finally, we show how the proposed probability propagation rules can be used to analyze syllogisms involving generalized quantifiers (like Most).

Equivalence in Argumentation Frameworks with a Claim-centric View: Classical Results with Novel Ingredients

A common feature of non-monotonic logics is that the classical notion of equivalence does not preserve the intended meaning in light of additional information. Consequently, the term strong equivalence was coined in the literature and thoroughly investigated. In the present paper, the knowledge representation formalism under consideration is claimaugmented argumentation frameworks (CAFs) which provide a formal basis to analyze conclusion-oriented problems in argumentation by adapting a claim-focused perspective. CAFs extend Dung AFs by associating a claim to each argument representing its conclusion. In this paper, we investigate both ordinary and strong equivalence in CAFs. Thereby, we take the fact into account that one might either be interested in the actual arguments or their claims only. The former point of view naturally yields an extension of strong equivalence for AFs to the claim-based setting while the latter gives rise to a novel equivalence notion which is genuine for CAFs. We tailor, examine and compare these notions and obtain a comprehensive study of this matter for CAFs. We conclude by investigating the computational complexity of naturally arising decision problems.

Non-deterministic Approximation Operators: Ultimate Operators, Semi-equilibrium Semantics, and Aggregates

AbstractApproximation fixpoint theory (AFT) is an abstract and general algebraic framework for studying the semantics of non-monotonic logics. In recent work, AFT was generalized to non-deterministic operators, that is, operators whose range are sets of elements rather than single elements. In this paper, we make three further contributions to non-deterministic AFT: (1) we define and study ultimate approximations of non-deterministic operators, (2) we give an algebraic formulation of the semi-equilibrium semantics by Amendola et al., and (3) we generalize the characterizations of disjunctive logic programs to disjunctive logic programs with aggregates.

Embeddings as epistemic states: Limitations on the use of pooling operators for accumulating knowledge

Various neural network architectures rely on pooling operators to aggregate information coming from different sources. It is often implicitly assumed in such contexts that vectors encode epistemic states, i.e. that vectors capture the evidence that has been obtained about some properties of interest, and that pooling these vectors yields a vector that combines this evidence. We study, for a number of standard pooling operators, under what conditions they are compatible with this idea, which we call the epistemic pooling principle. While we find that all the considered pooling operators can satisfy the epistemic pooling principle, this only holds when embeddings are sufficiently high-dimensional and, for most pooling operators, when the embeddings satisfy particular constraints (e.g. having non-negative coordinates). We furthermore show that these constraints have important implications on how the embeddings can be used in practice. In particular, we find that when the epistemic pooling principle is satisfied, in most cases it is impossible to verify the satisfaction of propositional formulas using linear scoring functions, with two exceptions: (i) max-pooling with embeddings that are upper-bounded and (ii) Hadamard pooling with non-negative embeddings. This finding helps to clarify, among others, why Graph Neural Networks sometimes under-perform in reasoning tasks. Finally, we also study an extension of the epistemic pooling principle to weighted epistemic states, which are important in the context of non-monotonic reasoning, where max-pooling emerges as the most suitable operator.

Conditional Syntax Splitting for Non-monotonic Inference Operators

Syntax splitting is a property of inductive inference operators that ensures we can restrict our attention to parts of the conditional belief base that share atoms with a given query. To apply syntax splitting, a conditional belief base needs to consist of syntactically disjoint conditionals. This requirement is often too strong in practice, as conditionals might share atoms. In this paper we introduce the concept of conditional syntax splitting, inspired by the notion of conditional independence as known from probability theory. We show that lexicographic inference and system W satisfy conditional syntax splitting, and connect conditional syntax splitting to several known properties from the literature on non-monotonic reasoning, including the drowning effect.

Permissive and regulative norms in deontic logic

Abstract This article provides a systematic analysis of the well-known notions of weak and strong permission in input/output (I/O) logic. We extend the account of permission initially put forward by Makinson and Van der Torre to the whole family of I/O systems developed during the last two decades. The main contribution is a series of characterization results for strong permission, based on establishing the so-called non-repetition property. We also study an input/output logic not yet covered in the literature. It supports reasoning by cases—a natural feature of human reasoning. The output is not closed under logical entailment. At the same time, it avoids excess output using a consistency check—a technique familiar from non-monotonic logic. This makes it well suited for contrary-to-duty reasoning. The axiomatic characterization is in terms of a generalized OR rule. We discuss the implications of all this for our understanding of the notion of the coherence of a normative system. Topics for future research are identified.1

On conditional probabilities and their canonical extensions to Boolean algebras of compound conditionals

In this paper we investigate canonical extensions of conditional probabilities to Boolean algebras of conditionals. Before entering into the probabilistic setting, we first prove that the lattice order relation of every Boolean algebra of conditionals can be characterized in terms of the well-known order relation given by Goodman and Nguyen. Then, as an interesting methodological tool, we show that canonical extensions behave well with respect to conditional subalgebras. As a consequence, we prove that a canonical extension and its original conditional probability agree on basic conditionals. Moreover, we verify that the probability of conjunctions and disjunctions of conditionals in a recently introduced framework of Boolean algebras of conditionals are in full agreement with the corresponding operations of conditionals as defined in the approach developed by two of the authors to conditionals as three-valued objects, with betting-based semantics, and specified as suitable random quantities. Finally we discuss relations of our approach with nonmonotonic reasoning based on an entailment relation among conditionals.

Observational Equivalence of Conditional Belief Bases

In nonmonotonic reasoning, a conditional of the form ‘If A then usually B’ is typically accepted if a situation where both A and B hold is deemed to be more plausible, more probable, or less surprising, etc., than a situation where A holds, but B does not hold. In a propositional setting, this leads to a relation on the propositional interpretations, also called worlds, by comparing worlds according to their plausibility. In this paper, we address the question of which kind of relations on the set of worlds can be obtained by completing a conditional belief base via an inductive inference operator. As a key concept for our investigations, we introduce and employ the notion of observational equivalence of belief bases that takes an inductive inference operator and a set of queries into account. This leads to the notion of the observational normal form (ONF) and, by focussing on so-called base conditionals, to the base conditional normal form (BCNF). The acceptance of base conditionals corresponds to the plausibility ordering on possible worlds induced by an inference method. Both normal forms ONF and BCNF are combined with renamings as an additional dimension. We establish the interrelationships among the normal forms and evaluate them empirically with respect to systematically generated belief bases.

Nonmonotonic inferences: Classical conclusions in an intuitionistic modal framework

Abstract We introduce a conceptual foundation for nonmonotonic reasoning which integrates an intuitionistic plurimodal logic with classical logic. The need for a multilogical system arises once we understand that to legitimately infer less than certain conclusions, we make two different kinds of assessments: an external one (what are the alternatives) and an internal one (which logic should govern the set of beliefs). We exploit this scheme to capture in novel ways basic nonmonotonic systems such as autoepistemic logic, default logic, one of the best known interpretations of negation-as-failure, as well as some conditional logics of normality. Among the aims of this paper is to explain why there are substantial overlaps between these systems and also to settle some problematic features that are of concern in logical investigations on common sense reasoning.

An Alternative Definition of Stable Models Via Łukasiewicz Logic

Extensions of G3 (3-valued Godel logic) were studied as tools for knowledge representation and nonmonotonic reasoning. One of these extensions was studied and baptized as G′ 3 by Osorio et al. as a tool to define semantics of logic programming. In this work, we will explore the possibility to use another fuzzy logic for knowledge representation. In particular, we show that Łukasiewicz 3-valued logic (for short, Ł3) can be used for knowledge representation based on logic programming. Firstly, we prove that the definition of stable model for Ł3 is equivalent to the obtained through G3 for augmented programs, but when we consider more general programs we obtain more answers for stable models on Ł3 than G3. Finally, we present and explore a new definition of stable model based on Lukasiewicz n-valued logic via the use of Monteiro-Baaz ∆ operator.

A comprehensive account of the burden of persuasion in abstract argumentation

AbstractIn this paper, we provide a formal framework for modeling the burden of persuasion in legal reasoning. The framework is based on abstract argumentation, a frequently studied method of non-monotonic reasoning, and can be applied to different argumentation semantics; it supports burdens of persuasion with arbitrary many levels, and allows for the placement of a burden of persuasion on any subset of an argumentation framework’s arguments. Our framework can be considered an extension of related works that raise questions on how burdens of persuasion should be handled in some conflict scenarios that can be modeled with abstract argumentation. An open source software implementation of the introduced formal notions is available as an extension of an argumentation reasoning library. A theoretical analysis shows that our approach can be generalized to a novel method for the preference-based selection of extensions from argumentation frameworks.

Preface: Special Issue on Logic and Argumentation

This special issue of the Journal of Logic and Computation presents 11 carefully selected articles in the field of Logic and Argumentation. Authors of the strongest contributions to 4th International Conference on Logic and Argumentation (CLAR 2021) were invited to submit extended versions of their papers for inclusion in this special issue. All articles in this volume were carefully reviewed by two or three expert reviewers before acceptance was finally granted. CLAR 2021 was held in Hangzhou, 20–22 October 2021 at the Zhejiang University City College in China. This special issue was made possible by the numerous strong submissions (58 in total) we received for the conference, from which 30 presentations and 4 additional invited talks were selected for presentation at the conference and for publication in the CLAR 2021 proceedings published as volume 13040 in the Springer series Lecture Notes in Computer Science. The contributions to this special issue, and more generally to CLAR 2021, cover a wide range of topics that are the focus of the CLAR conference series, including in particular abstract and structured argumentation, logic and argumentation, argumentation with qualitative and quantitative uncertainty, knowledge representation and reasoning, modal and non-classical logics and non-monotonic logics, with applications in a variety of fields like law, ethics, game theory, linguistics and medical reasoning.

Detecting Malware Activities With MalpMiner: A Dynamic Analysis Approach

Day by day, malware as a service becomes more popular and easy to acquire, thus allowing anyone to start an attack without any technical background. In addition, advanced malware attacks get more sophisticated such as Advanced Persistent Threats (APTs). Such reasons introduce challenges for detecting such attacks. One of those challenges is the detection of malware activities early in order to prevent harm as much as possible. Therefore, this paper presents a trusted dynamic analysis approach based on Answer Set Programming (ASP), a logic engine inference named Malware-Logic-Miner (MalpMiner). ASP is a nonmonotonic reasoning engine built on an open-world assumption. Thus, it allows MalpMiner to adopt commonsense reasoning when capturing malware activities of any given binary. Furthermore, MalpMiner requires no prior training; therefore, it can scale up quickly to include more malware-attack attributes. Besides that, it considers the invoked application programming interfaces’ values, resulting in correct malware behaviour modelling. The baseline experiments prove the correctness of MalpMiner related to recognizing malware activities. Moreover, MalpMiner achieved a detection ratio of 99% with a false-positive rate of less than 1% while maintaining low computational costs and explaining the detection decision.

Defeasible RDFS via rational closure

In the field of non-monotonic logics, the notion of Rational Closure (RC) is acknowledged as a notable approach. In recent years, RC has gained popularity in the context of Description Logics (DLs), the logic underpinning the standard semantic Web Ontology Language OWL 2, whose main ingredients are classes, the relationship among classes and roles, which are used to describe the properties of classes.In this work, we show instead how to integrate RC within the triple language RDFS (Resource Description Framework Schema), which together with OWL 2 is a major standard semantic web ontology language.To do so, we start from ρdf, a minimal, but significant RDFS fragment that covers the essential features of RDFS, and then extend it to ρdf⊥, allowing to state that two entities are incompatible/disjoint with each other. Eventually, we propose defeasible ρdf⊥ via a typical RC construction allowing to state default class/property inclusions.Furthermore, to overcome the main weaknesses of RC in our context, i.e., the “drowning problem” (viz. the “inheritance blocking problem”), we further extend our construction by leveraging Defeasible Inheritance Networks (DIN) defining a new non-monotonic inference relation that combines the advantages of both (RC and DIN). To the best of our knowledge this is the first time of such an attempt.In summary, the main features of our approach are: (i) the defeasible ρdf⊥ we propose here remains syntactically a triple language by extending it with new predicate symbols with specific semantics; (ii) the logic is defined in such a way that any RDFS reasoner/store may handle the new predicates as ordinary terms if it does not want to take account of the extra non-monotonic capabilities; (iii) the defeasible entailment decision procedure is built on top of the ρdf⊥ entailment decision procedure, which in turn is an extension of the one for ρdf via some additional inference rules favouring a potential implementation; (iv) the computational complexity of deciding entailment in ρdf and ρdf⊥ are the same; and (v) defeasible entailment can be decided via a polynomial number of calls to an oracle deciding ground triple entailment in ρdf⊥ and, in particular, deciding defeasible entailment can be done in polynomial time.

Non-Monotonic Reasoning in Medieval Theology: Problems and Assumptions

Abstract Some interesting cases of non-monotonic reasoning have already been identified in medieval theological texts. Jacob Archambault proved in 2015 that the argumentation presented by St Anselm of Canterbury in his Proslogion has non-monotonic “embeddings”. My own contribution from 2011 indicated that we can argue that a non-monotonic logic underlies some discussions provided by St Thomas Aquinas in his Summa theologiae, and showed that Boethius of Dacia used non-monotonic reasoning in his De aeternitate mundi. In this article, I would like to briefly present these examples and verify whether we can speak about similar cases in medieval Biblical exegesis. My aim is to outline particular problems connected with the identification of non- monotonicity which are specific to theology, as well as assumptions that should be adopted to successfully discuss this issue.

Reasoning with Expectations About Causal Relations

Abstract Reasoning is not just following logical rules, but a large part of human reasoning depends on our expectations about the world. To some extent, non-monotonic logic has been developed to account for the role of expectations. In this article, the focus is on expectations based on actions and their consequences. The analysis is based on a two-vector model of events where an event is represented in terms of two main components – the force of an action that drives the event, and the result of its application. Actions are modelled in terms of the force domain and the results are modelled with the aid of different domains for locations or properties of objects. As a consequence, the assumption that reasoning about causal relations should be made in terms of propositional structures becomes very unnatural. Instead, the reasoning will be based on the geometric and topological properties of causes and effects modelled in conceptual spaces.

Paraconsistent inference relations induced from inconsistency measures

This article presents a family of paraconsistent inference relations which aim at reasoning with inconsistent knowledge bases. These inference relations are defined from inconsistency measures. We check under which conditions these inference relations satisfy a series of properties for nonmonotonic reasoning. We also show that they are paraconsistent when the corresponding inconsistency measures satisfy some rationality postulates, and even strictly paraconsistent in most cases. In addition, some dependencies and incompatibilities among some rationality postulates are shown.

Representation of gene regulation networks by hypothesis logic-based Boolean systems

Boolean Dynamical Systems (BDSs) are networks described by Boolean variables. A new representation of BDSs is presented in this article by using modal non-monotonic logic (\({\mathcal {H}}\)). This approach allows Boolean Networks to be represented by a set of modal formulas and therefore can be used to describe and learn their properties. The study of a BDS focuses in particular on the search of stable configurations, limit cycles and unstable cycles, which help to characterize a large type of Gene Networks. In this article is presented the identification of such asymptotic properties by introduction of a new concept, ghost extensions. Using ghost extensions, it is possible to translate BDSs in propositional calculus and consequently to use SAT algorithms.

Comparing and extending the use of defeasible argumentation with quantitative data in real-world contexts

Dealing with uncertain, contradicting, and ambiguous information is still a central issue in Artificial Intelligence (AI). As a result, many formalisms have been proposed or adapted so as to consider non-monotonicity. A non-monotonic formalism is one that allows the retraction of previous conclusions or claims, from premises, in light of new evidence, offering some desirable flexibility when dealing with uncertainty. Among possible options, knowledge-base, non-monotonic reasoning approaches have seen their use being increased in practice. Nonetheless, only a limited number of works and researchers have performed any sort of comparison among them. This research article focuses on evaluating the inferential capacity of defeasible argumentation, a formalism particularly envisioned for modelling non-monotonic reasoning. In addition to this, fuzzy reasoning and expert systems, extended for handling non-monotonicity of reasoning, are selected and employed as baselines, due to their vast and accepted use within the AI community. Computational trust was selected as the domain of application of such models. Trust is an ill-defined construct, hence, reasoning applied to the inference of trust can be seen as non-monotonic. Inference models were designed to assign trust scalars to editors of the Wikipedia project. Scalars assigned to recognised trustworthy editors provided the basis for the analysis of the models’ inferential capacity according to evaluation metrics from the domain of computational trust. In particular, argument-based models demonstrated more robustness than those built upon the baselines despite the knowledge bases or datasets employed. This study contributes to the body of knowledge through the exploitation of defeasible argumentation and its comparison to similar approaches. It provides publicly implementations for the designed models of inference, which might be a useful aid to scholars interested in performing non-monotonic reasoning activities. It adds to previous works, empirically enhancing the generalisability of defeasible argumentation as a compelling approach to reason with quantitative data and uncertain knowledge.

Tree-Like Justification Systems are Consistent

Justification theory is an abstract unifying formalism that captures semantics of various non-monotonic logics. One intriguing problem that has received significant attention is the consistency problem: under which conditions are justifications for a fact and justifications for its negation suitably related. Two variants of justification theory exist: one in which justifications are trees and one in which they are graphs. In this work we resolve the consistency problem once and for all for the tree-like setting by showing that all reasonable tree-like justification systems are consistent.