A theory of nonmonotonic rule systems I

Full classical S5 in natural deduction with weak normalization

This is the second in a series of workshops that are bringing together researchers from the theoretical end of both the logic programming and artificial intelligence communities to discuss their mutual interests. This workshop emphasizes the relationship between logic programming and non-monotonic reasoning.Luis' Moniz Pereira is Professor in the Department of Computer Science at the Universidade Nova Lisboa, Portugal. Anil Nerode is Professor and Director of the Mathematical Sciences Institute at Cornell University.Topics include: Stable Semantics. Autoepistemic Logic. Abduction. Implementation Issues. Well-founded Semantics. Truth Maintenance. Probabilistic Theories. Applications. Default Logic. Diagnosis. Complexity and Theory. Handling Inconsistency.

Compiling specificity into approaches to nonmonotonic reasoning

Reiter’s pioneering work on default reasoning aimed to formalize the notion of quickly “jumping to conclusions”. Unfortunately, his resulting formalism, default logic, proved more computationally complex than classical logics. The same has been true for the other major formalisms for nonmonotonic reasoning, circumscription [20], nonmonotonic logic [22], [21], and autoepistemic logic [25]. This has dramatically limited the applicability of formal methods vis a vis real problems involving defaults. The complexity of consistency checking is one of the two problems that must be addressed to reduce the complexity of default reasoning. We propose an approach to default reasoning based on approximate consistency checking, using a novel synthesis of limited contexts and fast incomplete checks, and argue that this combination overcomes the limitations of its component parts. Our approach trades correctness for speed, but we argue that the nature of default reasoning makes this trade relatively inexpensive and intuitively plausible. This approach not only accords well with Reiter’s original motivations, but converges to Default Logic in the computational limit. We describe a prototype implementation of a default reasoner based on these ideas, and a preliminary empirical evaluation.KeywordsConsistency CheckDefense Advance Research Project AgencyDefault LogicNonmonotonic ReasoningDefault ReasoningThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Intuitionistic logic encompasses the principles of reasoning which were used by L. Brouwer in developing his intuitionistic mathematics, beginning in [Bro07]. Intuitionistic logic can be succinctly described as classical logic without the law ϕ ∨ ¬ϕ of excluded middle. Brouwer observed that this law was abstracted from finite situations and its application to statements about infinite collections is not justified. One of the consequences of the rejection of the law of excluded middle is that every intuitionistic proof of an existential sentence can be effectively transformed into an intuitionistic proof of an instance of that sentence. More precisely, if a formula of the form ∃xϕ(x) without free variables is provable in the intuitionistic predicate logic, then there is a term t without free variables such that ϕ(t) is provable. In particular, if ϕ ∨ ψ is provable, then either ϕ or ψ is provable. In that sense intuitionistic logic may provide a logical basis for constructive reasoning. A formal system of intuitionistic logic was proposed in [Hey30]. A relationship between the classical propositional logic PC and intuitionistic propositional logic INT was proved by Glivenko in [Gli29], namely a PC-formula ϕ is PC-provable if and only if ¬ ¬ϕ is INT-provable. Kripke semantics for intuitionistic logic was developed in [Kri65].

The default logic (Rei 80) and the autoepistemic logic (Mo 85) are non-monotonic logics. In default logic a default theory consists of a set of propositions sentences W and a set of defaults D. A sentence is defined to be derivable from a given default theory if it belongs to an extension of the default theory. A central question is: Given a default theory Δ = (D, W) and a sentence β, is there an extension of Δ which contains β?

We consider a general format for sequent rules for not necessarily normal modal logics based on classical or intuitionistic propositional logic and provide relatively simple local conditions ensuring cut elimination for such rule sets. The rule format encompasses e.g. rules for the boolean connectives and transitive modal logics such as S4 or its constructive version. We also adapt the method of constructing suitable rule sets by saturation to the intuitionistic setting and provide a criterium for translating axioms for intuitionistic modal logics into sequent rules. Examples include constructive modal logics and conditional logic $\mathbb{VA}$ .

Among non-monotonic systems of reasoning, non-monotonic modal logics, and autoepistemic logic in particular, have had considerable success. The presence of explicit modal operators allows flexibility in the embedding of other approaches. Also several theoretical results of interest have been established concerning these logics. In this paper we introduce non-monotonic modal logics based on many-valued logics, rather than on classical logic. This extends earlier work of ours on many-valued modal logics. Intended applications are to situations involving several reasoners, not just one as in the standard development.

Semantics and complexity of abduction from default theories

1 Modal logic: a semantic perspective

Knowledge formalization and reasoning automation are central within Artificial Intelligence. Classical logic has been traditionally used for such purposes. However, it is better suited to deal with complete knowledge in ideal circumstances. In real situations, in which the knowledge is partial, classical logic is not sufficient since it is monotonic. No monotonic logics have been proposed to better cope with practical reasoning. A successful formalization of no monotonic reasoning is Reiter's default logic which extends classical logic with default rules. Unfortunately, default logic is undividable. One reason for that is the use of classical logic as its monotonic basis. In this work, we change the default logic monotonic basis and propose a new default logic based on the description logic ALC. This new default logic is decidable and useful to formalize practical reasoning on hierarchical ontology's with exceptions, specially the ones that deals with legal knowledge and reasoning. On the default counterpart, we add some restrictions to the application of defaults in order to obtain nice properties such as coherence and elimination of anomalous extensions. We present the main algorithms used to build the extensions for this logic with its complexity analysis.

The purpose of the research presented in this paper is to extend the author's results on sequent forms of Herbrand theorems for classical and intuitionistic logics onto classical and intuitionistic modal sequent logics. It was found that the technique reported at the SYNASC 2008 symposium and based on the original notions of admissibility and compatibility can satisfactorily be applied for proving Herbrand theorems for the logics under consideration in the sequent form.

We identify a pervasive contrast between implicit and explicit stances in logical analysis and system design. Implicit systems change received meanings of logical constants and sometimes also the notion of consequence, while explicit systems conservatively extend classical systems with new vocabulary. We illustrate the contrast for intuitionistic and epistemic logic, then take it further to information dynamics, default reasoning, and other areas, to show its wide scope. This gives a working understanding of the contrast, though we stop short of a formal definition, and acknowledge limitations and borderline cases. Throughout we show how awareness of the two stances suggests new logical systems and new issues about translations between implicit and explicit systems, linking up with foundational concerns about identity of logical systems. But we also show how a practical facility with these complementary working styles has philosophical consequences, as it throws doubt on strong philosophical claims made by just taking one design stance and ignoring alternative ones. We will illustrate the latter benefit for the case of logical pluralism and hyper-intensional semantics.

Default logic is the most popular non monotonic formalism. We present a general framework, defining the classical default extensions as “justified pre-extensions” having a particular property, “stability”. These pre-extensions have a constructive definition, contrary to classical extensions, and they give rise to a notion of “formula provable by default” which has appeared yet under several formulations in the literature. Also, we show how a less known proposal of this kind gives a very interesting “semi-constructive” definition of default extensions. Finally our framework applies to some modal translations of defaults, making precise the relations between the stability in default logic and in autoepistemic modal logic.

The present paper studies two approaches to the expressiveness of propositional modal logics based on first-degree entailment logic, FDE. We first consider the basic FDE-based modal logic BK and certain systems in its vicinity, and then turn to some FDE-based modal logics in a richer vocabulary, including modal bilattice logic, MBL. On the one hand, model-theoretic proofs of the definability of connectives along the lines of [McCullough, “Logical connectives for intuitionistic propositional logic”, Journal of Symbolic Logic 36, 1 (1971): 15–20. DOI: 10.2307/2271511] and [[17] Wansing, “Logical connectives for constructive modal logic”, Synthese 150, 3 (2006): 459–482. DOI: 10.1007/s11229-005-5518-5] are given for various FDE-based modal logics. On the other hand, building on [Odintsov and Wansing, “Disentangling FDE-based paraconsistent modal logics, Studia Logica 105, 6 (2017): 1221–1254. DOI: 10.1007/s11225-017-9753-9], expressibility is considered in terms of mutual faithful embeddability of one logic into another logic. A distinction is drawn between definitional equivalence, which is defined with respect to a pair of structural translations between two languages, and weak definitional equivalence, which is defined with respect to a weaker notion of translations. Moreover, the definitional equivalence of some FDE-based modal logics is proven, especially the definitional equivalence of MBL and a conservative extension of the logic BK□×BK□, which underlines the central role played by BK among FDE-based modal logics.