In search of a “true” logic of knowledge: the nonmonotonic perspective

AbstractModal logics are currently widely accepted as a suitable tool of knowledge representation, and the question what logics are better suited for representing knowledge is of particular importance. Usually, some axiom list is given, and arguments are presented justifying that suggested axioms agree with intuition. The question why the suggested axioms describe all the desired properties of knowledge remains answered only partially, by showing that the most obvious and popular additional axioms would violate the intuition.We suggest the general paradigm of maximal logics and demonstrate how it can work for nonmonotonic modal logics. Technically, we prove that each of the modal logics KD45, SW5, S4F and S4.2 is the strongest modal logic among the logics generating the same nonmonotonic logic. These logics have already found important applications in knowledge representation, and the obtained results contribute to the explanation of this fact

Reasoning about Minimal Belief and Negation as Failure

We investigate the problem of reasoning in the propositional fragment of MBNF, the logic of minimal belief and negation as failure introduced by Lifschitz, which can be considered as a unifying framework for several nonmonotonic formalisms, including default logic, autoepistemic logic, circumscription, epistemic queries, and logic programming. We characterize the complexity and provide algorithms for reasoning in propositional MBNF. In particular, we show that entailment in propositional MBNF lies at the third level of the polynomial hierarchy, hence it is harder than reasoning in all the above mentioned propositional formalisms for nonmonotonic reasoning. We also prove the exact correspondence between negation as failure in MBNF and negative introspection in Moore's autoepistemic logic

The Gödel and the Splitting Translations

When the new research area of logic programming and non-monotonic reasoning emerged at the end of the 1980s, it focused notably on the study of mathematical relations between different non-monotonic formalisms, especially between the semantics of stable models and various non-monotonic modal logics. Given the many and varied embeddings of stable models into systems of modal logic, the modal interpretation of logic programming connectives and rules became the dominant view until well into the new century. Recently, modal interpretations are once again receiving attention in the context of hybrid theories that combine reasoning with non-monotonic rules and ontologies or external knowledge bases. In this talk I explain how familiar embeddings of stable models into modal logics can be seen as special cases of two translations that are very well-known in non-classical logic. They are, first, the translation used by Godel in 1933 to em- ¨ bed Heyting’s intuitionistic logic H into a modal provability logic equivalent to Lewis’s S4; second, the splitting translation, known since the mid-1970s, that allows one to embed extensions of S4 into extensions of the non-reflexive logic, K4. By composing the two translations one can obtain (Goldblatt, 1978) an adequate provability interpretation of H within the Goedel-Loeb logic GL, the system shown by Solovay (1976) to capture precisely the provability predicate of Peano Arithmetic. These two translations and their composition not only apply to monotonic logics extending H and S4, they also apply in several relevant cases to non-monotonic logics built upon such extensions, including equilibrium logic, non-monotonic S4F and autoepistemic logic. The embeddings obtained are not merely faithful and modular, they are based on fully recursive translations applicable to arbitrary logical formulas. Besides providing a uniform picture of some older results in LPNMR, the translations yield a perspective from which some new logics of belief emerge in a natural wa

Notes on Some Ideas in Lloyd Humberstone’s Philosophical Applications of Modal Logic

Lloyd Humberstone’s recently published Philosophical Applications of Modal Logic presents a number of new ideas in modal logic as well explication and critique of recent work of many others. We extend some of these ideas and answer some questions that are left open in the book

On embedding default logic into Moore's autoepistemic logic

AbstractRecently Gottlob proved [2] that there does not exist a faithful modular translation of default logic into autoepistemic logic, and presented a non-modular translation. Gottlob's translation, however, is indirect (it uses “nonmonotonic logic N” as an intermediate point), quite complex and exploits sophisticated encoding of proof theory in autoepistemic formulas. We provide a simpler and more intuitive (non-modular) direct translation. In addition, our argument is purely model-theoretic

Notes on Some Ideas in Lloyd Humberstone’s Philosophical Applications of Modal Logic

Lloyd Humberstone’s recently published Philosophical Applications of Modal Logic presents a number of new ideas in modal logic as well explication and critique of recent work of many others. We extend some of these ideas and answer some questions that are left open in the book

Current and Future Challenges in Knowledge Representation and Reasoning

Knowledge Representation and Reasoning is a central, longstanding, and active area of Artificial Intelligence. Over the years it has evolved significantly; more recently it has been challenged and complemented by research in areas such as machine learning and reasoning under uncertainty. In July 2022 a Dagstuhl Perspectives workshop was held on Knowledge Representation and Reasoning. The goal of the workshop was to describe the state of the art in the field, including its relation with other areas, its shortcomings and strengths, together with recommendations for future progress. We developed this manifesto based on the presentations, panels, working groups, and discussions that took place at the Dagstuhl Workshop. It is a declaration of our views on Knowledge Representation: its origins, goals, milestones, and current foci; its relation to other disciplines, especially to Artificial Intelligence; and on its challenges, along with key priorities for the next decade

Modular Logic Programming: Full Compositionality and Conflict Handling for Practical Reasoning

With the recent development of a new ubiquitous nature of data and the profusity of available knowledge, there is nowadays the need to reason from multiple sources of often incomplete and uncertain knowledge. Our goal was to provide a way to combine declarative knowledge bases – represented as logic programming modules under the answer set semantics – as well as the individual results one already inferred from them, without having to recalculate the results for their composition and without having to explicitly know the original logic programming encodings that produced such results. This posed us many challenges such as how to deal with fundamental problems of modular frameworks for logic programming, namely how to define a general compositional semantics that allows us to compose unrestricted modules. Building upon existing logic programming approaches, we devised a framework capable of composing generic logic programming modules while preserving the crucial property of compositionality, which informally means that the combination of models of individual modules are the models of the union of modules. We are also still able to reason in the presence of knowledge containing incoherencies, which is informally characterised by a logic program that does not have an answer set due to cyclic dependencies of an atom from its default negation. In this thesis we also discuss how the same approach can be extended to deal with probabilistic knowledge in a modular and compositional way. We depart from the Modular Logic Programming approach in Oikarinen & Janhunen (2008); Janhunen et al. (2009) which achieved a restricted form of compositionality of answer set programming modules. We aim at generalising this framework of modular logic programming and start by lifting restrictive conditions that were originally imposed, and use alternative ways of combining these (so called by us) Generalised Modular Logic Programs. We then deal with conflicts arising in generalised modular logic programming and provide modular justifications and debugging for the generalised modular logic programming setting, where justification models answer the question: Why is a given interpretation indeed an Answer Set? and Debugging models answer the question: Why is a given interpretation not an Answer Set? In summary, our research deals with the problematic of formally devising a generic modular logic programming framework, providing: operators for combining arbitrary modular logic programs together with a compositional semantics; We characterise conflicts that occur when composing access control policies, which are generalisable to our context of generalised modular logic programming, and ways of dealing with them syntactically: provided a unification for justification and debugging of logic programs; and semantically: provide a new semantics capable of dealing with incoherences. We also provide an extension of modular logic programming to a probabilistic setting. These goals are already covered with published work. A prototypical tool implementing the unification of justifications and debugging is available for download from http://cptkirk.sourceforge.net

Logical considerations on default semantics

We consider a reinterpretation of the rules of default logic. We make Reiter’s default rules into a constructive method of building models, not theories. To allow reasoning in first‐order systems, we equip standard first‐order logic with a (new) Kleene 3‐valued partial model semantics. Then, using our methodology, we add defaults to this semantic system. The result is that our logic is an ordinary monotonic one, but its semantics is now nonmonotonic. Reiter’s extensions now appear in the semantics, not in the syntax. As an application, we show that this semantics gives a partial solution to the conceptual problems with open defaults pointed out by Lifschitz [V. Lifschitz, On open defaults, in: Proceedings of the Symposium on Computational Logics (1990)], and Baader and Hollunder [F. Baader and B. Hollunder, Embedding defaults into terminological knowledge representation formalisms, in: Proceedings of Third Annual Conference on Knowledge Representation (Morgan‐Kaufmann, 1992)]. The solution is not complete, chiefly because in making the defaults model‐theoretic, we can only add conjunctive information to our models. This is in contrast to default theories, where extensions can contain disjunctive formulas, and therefore disjunctive information. Our proposal to treat the problem of open defaults uses a semantic notion of nonmonotonic entailment for our logic, related to the idea of “only knowing”. Our notion is “only having information” given by a formula. We discuss the differences between this and “minimal‐knowledge” ideas. Finally, we consider the Kraus–Lehmann–Magidor [S. Kraus, D. Lehmann and M. Magidor, Nonmonotonic reasoning, preferential models, and cumulative logics, Artificial Intelligence 44 (1990) 167–207] axioms for preferential consequence relations. We find that our consequence relation satisfies the most basic of the laws, and the Or law, but it does not satisfy the law of Cut, nor the law of Cautious Monotony. We give intuitive examples using our system, on the other hand, which on the surface seem to violate these two laws. We make some comparisons, using our examples, to probabilistic interpretations for which these laws are true, and we compare our models to the cumulative models of Kraus, Lehmann, and Magidor. We also show sufficient conditions for the laws to hold. These involve limiting the use of disjunction in our formulas in one way or another. We show how to make use of the theory of complete partially ordered sets, or domain theory. We can augment any Scott domain with a default set. We state a version of Reiter’s extension operator on arbitrary domains as well. This version makes clear the basic order‐theoretic nature of Reiter’s definitions. A three‐variable function is involved. Finding extensions corresponds to taking fixed points twice, with respect to two of these variables. In the special case of precondition‐free defaults, a general relation on Scott domains induced from the set of defaults is shown to characterize extensions. We show how a general notion of domain theory, the logic induced from the Scott topology on a domain, guides us to a correct notion of “affirmable sentence” in a specific case such as our first‐order systems. We also prove our consequence laws in such a way that they hold not only in first‐order systems, but in any logic derived from the Scott topology on an arbitrary domain.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/41771/1/10472_2004_Article_325429.pd