Nonmonotonic Reasoning Research Articles

Logics for at most countable first-order structures

Abstract In this paper we present two extensions of $\omega $-logic with infinitary inference rules, denoted $Arch$-$\omega $-logic and $non$-$Arch$-$\omega $-logic. We provide the corresponding Hilbert-style axiomatizations and prove their strong completeness with respect to countable Archimedean and non-Archimedean fields, respectively. Through several examples we illustrate a natural representation of various weight functions within the proposed framework and applications to non-monotonic reasoning and neuro-symbolic computing.

Exploring the Definition of Non-Monotonicity – Logical and Psychological Considerations

When humans reason, they are able to revise their beliefs in light of new information and abandon obsolete conclusions. Logicians argued, that in some cases, such reasonings appear to be non-monotonic. Thus, many different, seemingly non-monotonic systems were created to formally model such cases. The purpose of this article is to re-examine the definition of non-monotonicity and its implementation in non-monotonic logics and in examples of everyday human reasoning. We will argue that many non-monotonic logics employ some weakened versions of the definitions of non-monotonicity, since in-between different steps of reasoning they either: a) allow previously accepted premises to be removed, or b) change the rules of inference. Of the two strategies, the second one seems downright absurd, since changing the rules of a given logic is a mere replacement of that logic with the rules of another. As a consequence we obtain two logics, whereas the definition of a non-monotonic logic is supposed to define one. The definition of non-monotonicity does not permit either of these cases, which means that such logics are monotonic.

Diverging Approaches to Skeptical Inference in Non-monotonic Reasoning

Our paper addresses the problem of a two-fold approach to skeptical inferences in the context non-monotonic logics. We tackle the problem through the analysis of ambiguous theories, such as the Nixon Diamond, as instantiated in non-monotonic inheritance networks, and the notion of an extension. Our paper presents a detailed description of the inner mechanisms underlying both approaches to skeptical inference, i.e. direct and indirect skepticism, and how each information processing policy is applied to ambiguous networks like the Nixon Diamond. Finally, we discuss the extent and limitation of each approach, and we propose an alternative stance towards the existence of diverging implementations to skeptical inferences in non-monotonic reasoning.

Dynamic superiorities in Defeasible Logic

Abstract In many logical rule-based systems, it often happens that opposite statements can be concluded, hence superiorities (or priorities) among rules are needed to solve such conflicts. Since the specific circumstances describing the scenario at hand determine which rules may apply, we thus need a mechanism able to compute such superiorities dynamically, case-by-case. We formalise a non-monotonic logic able to dynamically compute superiorities over rules. Finally, we advance efficient algorithms to compute the (positive and negative) conclusions of such theories, and prove their computational properties.

Partial MaxSAT Approach to Nonmonotonic Reasoning with System W

The only recently introduced System W is a nonmonotonicinductive inference operator exhibiting some notable proper-ties like extending rational closure and satisfying syntax split-ting postulates for inference from conditional belief bases.A semantic model of system W is given by its underlyingpreferred structure of worlds, a strict partial order on the setof propositional interpretations, also called possible worlds,over the signature of the belief base. Existing implementa-tions of system W are severely limited by the number ofpropositional variables that occur in a belief base becauseof the exponentially growing number of possible worlds. Inthis paper, we present an approach to realizing nonmono-tonic reasoning with system W by using partial maximumsatisfiability (PMaxSAT) problems and exploiting the powerof current PMaxSAT solvers. An evaluation of our approachdemonstrates that it outperforms previous implementations ofsystem W and scales reasoning with system W up to a newdimension.

CLKR: Conditional Logic and Knowledge Representation

CLKR (Conditional Logic and Knowledge Representation) is an online repository of conditional logic resources for knowledge representation and reasoning. The question which entailments should follow from a conditional knowledge base consisting of a set of conditionals “If A then usually B“ is central in logic-based AI. In order to support the practical side of this question, CLKR provides various collections of conditional knowledge bases and related resources. All knowledge bases available in CLKR can be processed directly with a corresponding reasoning system like InfOCF-Web. The sets of knowledge bases include examples as they are used in the literature for illustration, application knowledge bases from different domains, and systematically generated knowledge bases for evaluating implementations of nonmonotonic reasoning. A main emphasis of the current version of CLKR is on providing collections of knowledge bases in various normal forms that have been proposed for conditional knowledge bases, e.g., conditional normal form, antecedent normal form, and renaming normal form.

Preface to the Special Issue on the 2022 Conference on Logic Programming and Nonmonotonic Reasoning

Preface to the Special Issue on the 2022 Conference on Logic Programming and Nonmonotonic Reasoning

"EVERYDAY THINKING" AND LOGICAL RATIONALITY

Knowing or not knowing, people find themselves in various unpleasant situations due to incorrect reasoning. Is it because they don't know logic and don't think logically? And those who know logic always think correctly and do not make mistakes? What is the relationship between correct reasoning and logical rationality? The purpose of the article is to determine the answers to these questions on the basis of psychologism in logic, non-monotonic logic, phenomenology of intersubjectivity.

A Unified View on Forgetting and Strong Equivalence Notions in Answer Set Programming

Answer Set Programming (ASP) is a prominent rule-based language for knowledge representation and reasoning with roots in logic programming and non-monotonic reasoning. The aim to capture the essence of removing (ir)relevant details in ASP programs led to the investigation of different notions, from strong persistence (SP) forgetting, to faithful abstractions, and, recently, strong simplifications, where the latter two can be seen as relaxed and strengthened notions of forgetting, respectively. Although it was observed that these notions are related, especially given that they have characterizations through the semantics for strong equivalence, it remained unclear whether they can be brought together. In this work, we bridge this gap by introducing a novel relativized equivalence notion, which is a relaxation of the recent simplification notion, that is able to capture all related notions from the literature. We provide the necessary and sufficient conditions for relativized simplifiability, which shows that the challenging part is for when the context programs do not contain all the atoms to remove. We then introduce an operator that combines projection and a relaxation of SP-forgetting to obtain the relativized simplifications. We furthermore provide complexity results that complete the overall picture.

Defeasible Normative Reasoning: A Proof-Theoretic Integration of Logical Argumentation

We present a novel computational approach to resolving conflicts among norms by nonmonotonic normative reasoning (in constrained I/O logics). Our approach extends standard sequent-based proof systems and makes them more adequate to nonmonotonic reasoning by adding to the sequents annotations that keep track of what is known about the defeasible status of the derived sequents. This makes transparent the reasons according to which norms should be applicable or inapplicable, and accordingly the sequents that make use of such norms are accepted or retracted. We also show that this proof theoretic method has tight links to the semantics of formal argumentation frameworks. The outcome of this paper is thus a threefold characterization result that relates, in the context of nonmonotonic normative reasoning, three traditional ingredients of AI-based reasoning methods: maximally consistent sets of premises (in constrained I/O logics), derived sequents (which are accepted in corresponding annotated sequent calculi), and logical arguments (that belong to the grounded extensions of the induced logical argumentation frameworks).

Embedding justification theory in approximation fixpoint theory

Approximation Fixpoint Theory (AFT) and Justification Theory (JT) are two frameworks to unify logical formalisms. AFT studies semantics in terms of fixpoints of lattice operators, and JT in terms of so-called justifications, which are explanations of why certain facts do or do not hold in a model. While the approaches differ, the frameworks were designed with similar goals in mind, namely to study the different semantics that arise in (mainly) non-monotonic logics. The first contribution of our current paper is to provide a formal link between the two frameworks. To be precise, we show that every justification frame induces an approximator and that this mapping from JT to AFT preserves all major semantics. The second contribution exploits this correspondence to extend JT with a novel class of semantics, namely ultimate semantics: we formally show that ultimate semantics can be obtained in JT by a syntactic transformation on the justification frame, essentially performing a sort of resolution on the rules.

Non-deterministic approximation fixpoint theory and its application in disjunctive logic programming

Approximation fixpoint theory (AFT) is an abstract and general algebraic framework for studying the semantics of nonmonotonic logics. It provides a unifying study of the semantics of different formalisms for nonmonotonic reasoning, such as logic programming, default logic and autoepistemic logic. In this paper, we extend AFT to dealing with non-deterministic constructs that allow to handle indefinite information, represented e.g. by disjunctive formulas. This is done by generalizing the main constructions and corresponding results of AFT to non-deterministic operators, whose ranges are sets of elements rather than single elements. The applicability and usefulness of this generalization is illustrated in the context of disjunctive logic programming.

Abwägung, „Belief Revision“ und Rangtheorie

An approach to analyse the structure of balancing is the idea of defeasible reasoning. This is often explained by using a non-monotonic logic, admitting that new information can invalidate formerly valid inferences. A more fruitful alternative seems to be the the- ory of belief revision, which allows not only invalidating inferences, but to define how beliefs should change facing new information. A sophisticated version of belief revi- sion theory is developed by Wolfgang Spohn in his “ranking theory”. The issue of this paper is to analyse how this “ranking theory” fits into the structure of normative argu- mentation, distinguishing four types of balancing: epistemic, normative, relative, and intersubjective balancing.

A Supraclassical Probabilistic Entailment Relation

The paper presents an original supraclassical nontrivial plausible entailment relation $\vapprox$ that employs Kolmogorov's probability theory. Its crucial feature is the primitiveness of a conditional probability, which one calculates with the help of the method of truth tables for classical propositional logic. I study the properties of the entailment relation in question. In particular, I show that while being supraclassical, i.e., all classical entailments and valid formulas are $\vapprox$-valid, but not vice versa, it is not trivial and enjoys the same form of inconsistency as classical entailment ⊧ does. I specify the place of the proposed probability entailment relation in certain classifications of nonclassical entailment relations. In particular, I use Douven's analysis of some probabilistic entailment relations that contains dozens of properties that are crucial for any probabilistic entailment relation, as well as Hlobil's choosing your nonmonotonic logic: shopper’s guide, due to the fact that $\vapprox$ is not monotonic, and Cobreros, Egré, Ripley, van Rooij's entailment relations for tolerant reasoning. At last, I perform a comparative analysis of classical, the proposed, and some other entailment relations closely related to the latter: those introduced by Bocharov, Markin, Voishvillo, Degtyarev, Ivlev, where the last two entailment relations are based on the so-called principle of reverse deduction, which is an intuitively acceptable way to connect classical and probabilistic entailment relations.

Assumption-based argumentation for extended disjunctive logic programming and its relation to nonmonotonic reasoning

The motivation of this study is that Reiter’s default theory as well as assumption-based argumentation frameworks corresponding to default theories have difficulties in handling disjunctive information, while a disjunctive default theory (ddt) avoids them. This paper presents the semantic correspondence between generalized assumption-based argumentation (ABA) and extended disjunctive logic programming as well as the correspondence between ABA and nonmonotonic reasoning approaches such as disjunctive default logic and prioritized circumscription. To overcome the above-mentioned difficulties of ABA frameworks corresponding to default theories, we propose an assumption-based framework (ABF) translated from an extended disjunctive logic program (EDLP) since an EDLP can be translated into a ddt. Our ABF incorporates explicit negation and the connective of disjunction “|” to Heyninck and Arieli’s ABF induced by a disjunctive logic program. In this paper, first, we show how arguments are constructed from disjunctive rules in our proposed ABF. Then, we show the correspondence between answer sets of an EDLP P and stable extensions of the ABF translated from P with trivialization rules. After defining rationality postulates, we show answer sets of a consistent EDLP are captured by consistent stable extensions of the translated ABF with no trivialization rules. Finally, we show the correspondence between ABA and disjunctive default logic (resp. prioritized circumscription). The relation between ABA and possible model semantics of EDLPs is also discussed.

The use of logic and argumentation in therapy of sex offenders

Abstract This paper is intended first for the formal argumentation community (see https://comma.csc.liv.ac.uk/). This community develops logics and systems modelling argumentation and dialogues. The community is in search of major applications areas for their models. One such application area e.g. is Law. The message of this paper is that there is another major application area for formal argumentation. There is an international community of sex offender therapist that is well established and well funded, and their therapy methods use (methods that can be modelled by) formal argumentation and logic. This community presents a natural application area for formal argumentation. We thus describe in this paper how the sex offender therapists work, to give the formal argumentation researcher a view of this application area. What is especially important about this application area is that in order to model it and learn from it, the formal argumentation community have to evolve their formal methods and adapt to this new application. Part of this enhancement is to modify and import certain methods from other areas of Logic e.g. from Non-Monotonic logic. The members of the formal argumentation community are not familiar, on average, with other areas of logic, and so we also describe in this paper, what we need from neighbouring logics. This makes this paper of interest also to sex offender therapist as well. They may be already familiar with their own practices, but the additional logics described will be of interest to them.

Deontic meta-rules

Abstract The use of meta-rules in logic, i.e., rules whose content includes other rules, has recently gained attention in the setting of non-monotonic reasoning: a first logical formalisation and efficient algorithms to compute the (meta)-extensions of such theories were proposed in Olivieri et al. (2021, Computing defeasible meta-logic. In JELIA 2021, LNCS, vol. 12678, pp. 69–84. Springer.). This work extends such a logical framework by considering the deontic aspect. The resulting logic will not just be able to model policies but also tackle well-known aspects that occur in numerous legal systems. The use of Defeasible Logic to model meta-rules in the application area we just alluded to has been investigated. Within this line of research, the study mentioned above was not focusing on the general computational properties of meta-rules. This study fills this gap with two major contributions. First, we introduce and formalise two variants of Defeasible Deontic Logic (DDL) with meta-rules to represent (i) defeasible meta-theories with deontic modalities and (ii) two different types of conflicts among rules: Simple Conflict DDL and Cautious Conflict DDL. Second, we advance efficient algorithms to compute the extensions for both variants.

Modelling supra-classical logic in a Boltzmann neural network: III adaptation

Abstract The field of belief revision in logic is still in evolution and holds a variety of disparate approaches; a consequence of theoretical conjecture. As a probabilistic model of supra-classical, non-monotonic (SCNM) logic, the Boltzmann machine, offers an experimental gateway into the field. How does the Boltzmann network adapt to new information? Catastrophic forgetting is the default response to retraining in any neural network. We have moderated this irrational non-monotonicity by alterations in the Boltzmann learning algorithm. The spectrum of experimental belief change is limited by the availability of ‘new’ information, a pragmatic realization co-related to the property of Rational Monotonicity in the domain of SCNM logic. Recognizing this upper boundary of defeasible belief simplifies the task of experimentally exploring machine adaptation. A minority of belief revisions involve new, but unsurprising information, that is at least partially consistent with the previous learned beliefs. In these circumstances, the Boltzmann network incrementally adjusts the priority of model state exemplars in accordance with preference; the traditional approach in SCNM logic. However, in the majority of situations the new information will be surprisingly inconsistent with the previous beliefs. In these circumstances, the pre-order on model states stratified by preference, will not have sufficient granularity to represent the conflicting requirements of ranking based on compositional atomic typicality. This novel experimental finding has not previously been considered in the logical conjecture on Belief Revision.

Theoretical analysis and implementation of abstract argumentation frameworks with domain assignments

A representational limitation of current argumentation frameworks is their inability to deal with sets of entities and their properties, for example to express that an argument is applicable for a specific set of entities that have a certain property and not applicable for all the others. In order to address this limitation, we recently introduced Abstract Argumentation Frameworks with Domain Assignments (AAFDs), which extend Abstract Argumentation Frameworks (AAFs) by assigning to each argument a domain of application, i.e., a set of entities for which the argument is believed to apply. We provided formal definitions of AAFDs and their semantics, showed with examples how this model can support various features of commonsense and non-monotonic reasoning, and studied its relation to AAFs. In this paper, aiming to provide a deeper insight into this new model, we present more results on the relation between AAFDs and AAFs and the properties of the AAFD semantics, and we introduce an alternative, more expressive way to define the domains of arguments using logical predicates. We also offer an implementation of AAFDs based on Answer Set Programming (ASP) and evaluate it using a range of experiments with synthetic datasets.

Quantum projections on conceptual subspaces

One of the main challenges of cognitive science is to explain the representation of conceptual knowledge and the mechanisms involved in evaluating the similarities between these representations. Theories that attempt to explain this phenomenon should account for the fact that conceptual knowledge is not static. In line with this thinking, many studies suggest that the representation of a concept changes depending on context. Traditionally, concepts have been studied as vectors within a geometric space, sometimes called Semantic-Vector Space Models (S-VSMs). However, S-VSMs have certain limitations in emulating human biases or context effects when the similarity of concepts is judged. Such limitations are related to the use of a classical geometric approach that represents a concept as a point in space. Recently, some theories have proposed the use of sequential projections of subspaces based on Quantum Probability Theory (Busemeyer and Bruza, 2012; Pothos et al., 2013). They argue that this theoretical approach may facilitate accounting for human similarity biases and context effects in a more natural way. More specifically, Pothos and Busemeyer (2011) proposed the Quantum Similarity Model (QSM) to determine expectation in conceptual spaces in a non-monotonic logic frame. To the best of our knowledge, previous data-driven studies have used the QSM subspaces in a unidimensional way. In this paper, we present a data-driven method to generate these conceptual subspaces in a multidimensional manner using a traditional S-VSM. We present an illustration of the method taking Tversky’s classical examples to explain the effects of Asymmetry, Triangular Inequality, and the Diagnosticity by means of sequential projections of those conceptual subspaces.