Formalisms For Nonmonotonic Reasoning Research Articles

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.

A Unifying Approach for Nonmonotonic S4F, (Reflexive) Autoepistemic Logic, and Answer Set Programming

This paper presents a general strategy, bringing together some major types of nonmonotonic reasoning under a monotonic bimodal setting. Such formalisms are also of interest to the fields of knowledge representation and declarative programming. We exemplify the methodology, capturing minimal model reasoning that underlies nonmonotonicity over S4F first, but then we also show how to apply the technique to other nonmonotonic logics respectively based on the modal logics KD45 and SW5. We naturally succeed it, by modifying only the axioms of the underlying modal logic and show that it successfully works. The last two formalisms are also known as autoepistemic logic (AEL) and its reflexive extension (RAEL) in the given order: AEL is an important form of nonmonotonic reasoning, introduced by Robert C. Moore in order to allow an agent to reason about his own knowledge. Equilibrium logic (EL) is a general-purpose nonmonotonic reasoning formalism, proposed more recently by David Pearce as a semantical framework for answer set programming (ASP). The latter is an efficient declarative problem solving approach with lots of applications to science and technology. Fariñas et al. have embedded EL (and so ASP) into a monotonic bimodal logic. We take this work as an initiative and successfully apply a similar methodology to closely aligned nonmonotonic modal logics. We finally discuss the potential capability to subsume the epistemic extensions of ASP within our unified paradigm.

Sequent-Type Calculi for Three-Valued and Disjunctive Default Logic

Default logic is one of the basic formalisms for nonmonotonic reasoning, a well-established area from logic-based artificial intelligence dealing with the representation of rational conclusions, which are characterised by the feature that the inference process may require to retract prior conclusions given additional premisses. This nonmonotonic aspect is in contrast to valid inference relations, which are monotonic. Although nonmonotonic reasoning has been extensively studied in the literature, only few works exist dealing with a proper proof theory for specific logics. In this paper, we introduce sequent-type calculi for two variants of default logic, viz., on the one hand, for three-valued default logic due to Radzikowska, and on the other hand, for disjunctive default logic, due to Gelfond, Lifschitz, Przymusinska, and Truszczyński. The first variant of default logic employs Łukasiewicz’s three-valued logic as the underlying base logic and the second variant generalises defaults by allowing a selection of consequents in defaults. Both versions have been introduced to address certain representational shortcomings of standard default logic. The calculi we introduce axiomatise brave reasoning for these versions of default logic, which is the task of determining whether a given formula is contained in some extension of a given default theory. Our approach follows the sequent method first introduced in the context of nonmonotonic reasoning by Bonatti, which employs a rejection calculus for axiomatising invalid formulas, taking care of expressing the consistency condition of defaults.

On the Computation of Paracoherent Answer Sets

Answer Set Programming (ASP) is a well-established formalism for nonmonotonic reasoning. An ASP program can have no answer set due to cyclic default negation. In this case, it is not possible to draw any conclusion, even if this is not intended. Recently, several paracoherent semantics have been proposed that address this issue,and several potential applications for these semantics have been identified. However, paracoherent semantics have essentially been inapplicable in practice, due to the lack of efficient algorithms and implementations. In this paper, this lack is addressed, and several different algorithms to compute semi-stable and semi-equilibrium models are proposed and implemented into an answer set solving framework. An empirical performance comparison among the new algorithms on benchmarks from ASP competitions is given as well.

On the relationship between fuzzy autoepistemic logic and fuzzy modal logics of belief

Autoepistemic logic is an important formalism for nonmonotonic reasoning originally intended to model an ideal rational agent reflecting upon his own beliefs. Fuzzy autoepistemic logic is a generalization of autoepistemic logic that allows to represent an agent's rational beliefs on gradable propositions. It has recently been shown that, in the same way as autoepistemic logic generalizes answer set programming, fuzzy autoepistemic logic generalizes fuzzy answer set programming as well. Besides being related to answer set programming, autoepistemic logic is also closely related to several modal logics. To investigate whether a similar relationship holds in a fuzzy logical setting, we firstly generalize the main modal logics for belief to the setting of finitely-valued Łukasiewicz logic with truth constants Łkc, and secondly we relate them with fuzzy autoepistemic logics. Moreover, we show that the problem of satisfiability checking in these logics is NP-complete. Finally, we generalize Levesque's results on stable expansions, belief sets, and “only knowing” operators to our setting, and provide a complete axiomatization for a logic of “only knowing” in the Łkc framework.

The scientific legacy of Marco Cadoli in Artificial Intelligence

It was in 1988 when Marco Cadoli asked me (Maurizio Lenzerini) if I could propose a master thesis work to him. I had just finished my first course as a professor. The course was about Software Engineering and Programming Languages, and during the lectures I had noticed Marco’s outstanding performance in the course, both in the part related to Software Design, and in the part devoted to object-oriented, functional and logic programming. I thought it was great that such a brilliant student was willing to work with me. So, we started discussing about possible topics, and I suggested Artificial Intelligence and in particular Knowledge Representation. More specifically, I talked to him about the research program I was pursuing, centered around the idea of understanding the exact computational complexity of various KR formalisms, in particular the formalisms for nonmonotonic reasoning. Although we now take it for granted that analyzing the computational complexity of a logical formalism is an essential tool for characterizing the properties of the formalism itself, at that time this kind of investigation was not so common. Only recently, people like Hector Levesque, Ron Brachman, Ray Reiter and others had started pointing out the importance of understanding the inherent complexity of reasoning tasks, and relating such complexity with the expressive power of the representation language. Marco had studied Theoretical Computer Science in

Fuzzy autoepistemic logic and its relation to fuzzy answer set programming

Autoepistemic logic is an important formalism for nonmonotonic reasoning. It extends propositional logic by offering the ability to reason about an agent's (lack of) beliefs. Moreover, it is well known to generalize the stable model semantics of answer set programming. Fuzzy logics on the other hand are multi-valued logics, which allow to model the intensity to which properties are satisfied. We combine these ideas to a fuzzy autoepistemic logic which can be used to reason about one's beliefs in the degrees to which properties are satisfied. We show that many properties from classical autoepistemic logic, e.g. the equivalence between autoepistemic models and stable expansions, remain valid under this generalization. In this paper, we consider a version of fuzzy answer set programming and show that its answer sets can be equivalently described as models in fuzzy autoepistemic logic. We also define a fuzzy logic of minimal belief and negation-as-failure and use this as a tool to show that fuzzy autoepistemic logic generalizes fuzzy answer set programming.

A defeasible reasoning model of inductive concept learning from examples and communication

This paper introduces a logical model of inductive generalization, and specifically of the machine learning task of inductive concept learning (ICL). We argue that some inductive processes, like ICL, can be seen as a form of defeasible reasoning. We define a consequence relation characterizing which hypotheses can be induced from given sets of examples, and study its properties, showing they correspond to a rather well-behaved non-monotonic logic. We will also show that with the addition of a preference relation on inductive theories we can characterize the inductive bias of ICL algorithms. The second part of the paper shows how this logical characterization of inductive generalization can be integrated with another form of non-monotonic reasoning (argumentation), to define a model of multiagent ICL. This integration allows two or more agents to learn, in a consistent way, both from induction and from arguments used in the communication between them. We show that the inductive theories achieved by multiagent induction plus argumentation are sound, i.e. they are precisely the same as the inductive theories built by a single agent with all data.

Abnormality and randomness

With respect to any inference we might make about an individual having a certain property, Kyburg’s theory of epistemological probability requires us to assume that the individual under discussion is randomly drawn from some reference class. The probability that the individual has the property is equal to the proportion of individuals in the reference class having that property. When this proportion, or its lower bound, is sufficiently high, the individual is said to be practically certain to hold the property.The ideas of epistemological randomness and practical certainty address the same inferential problems addressed by the manipulation of abnormality predicates in circumscription and other nonmonotonic reasoning formalisms. This article revisits the goals and problems of nonmonotonic reasoning through the lens of epistemological probability.

Proof complexity of propositional default logic

Default logic is one of the most popular and successful formalisms for non-monotonic reasoning. In 2002, Bonatti and Olivetti introduced several sequent calculi for credulous and skeptical reasoning in propositional default logic. In this paper we examine these calculi from a proof-complexity perspective. In particular, we show that the calculus for credulous reasoning obeys almost the same bounds on the proof size as Gentzen’s system LK. Hence proving lower bounds for credulous reasoning will be as hard as proving lower bounds for LK. On the other hand, we show an exponential lower bound to the proof size in Bonatti and Olivetti’s enhanced calculus for skeptical default reasoning.

Logical foundations and complexity of 4QL, a query language with unrestricted negation

The paper discusses properties of 4QL, a DATALOG⌉⌉-like query language, originally outlined by Maluszyński and Szalas (Maluszyński & Szalas, 2011). 4QL allows one to use rules with negation in heads and bodies of rules. It is based on a simple and intuitive semantics and provides uniform tools for “lightweight” versions of known forms of nonmonotonic reasoning. Negated literals in heads of rules may naturally lead to inconsistencies. On the other hand, rules do not have to attach meaning to some literals. Therefore 4QL is founded on a four-valued semantics, employing the logic introduced in (Maluszyński et al., 2008; Vitória et al., 2009) with truth values: ‘true’, ‘false’, ‘inconsistent’ and ‘unknown’. In addition, 4QL is tractable w.r.t. data complexity and captures PTIME queries. Even though DATALOG⌉⌉ is known as a concept for the last 30 years, to our best knowledge no existing approach enjoys these properties. In the current paper we: investigate properties of well-supported models of 4QL prove the correctness of the algorithm for computing well-supported models show that 4QL has PTIME data complexity and captures PTIME.

Well-founded semantics for defeasible logic

Fixpoint semantics are provided for ambiguity blocking and propagating variants of Nute’s defeasible logic. The semantics are based upon the well-founded semantics for logic programs. It is shown that the logics are sound with respect to their counterpart semantics and complete for locally finite theories. Unlike some other nonmonotonic reasoning formalisms such as Reiter’s default logic, the two defeasible logics are directly skeptical and so reject floating conclusions. For defeasible theories with transitive priorities on defeasible rules, the logics are shown to satisfy versions of Cut and Cautious Monotony. For theories with either conflict sets closed under strict rules or strict rules closed under transposition, a form of Consistency Preservation is shown to hold. The differences between the two logics and other variants of defeasible logic—specifically those presented by Billington, Antoniou, Governatori, and Maher—are discussed.

Partial equilibrium logic

Partial equilibrium logic (PEL) is a new nonmonotonic reasoning formalism closely aligned with logic programming under well-founded and partial stable model semantics. In particular it provides a logical foundation for these semantics as well as an extension of the basic syntax of logic programs. In this paper we describe PEL, study some of its logical properties and examine its behaviour on disjunctive and nested logic programs. In addition we consider computational features of PEL and study different approaches to its computation.

Equilibrium logic

Equilibrium logic is a general purpose nonmonotonic reasoning formalism closely aligned with answer set programming (ASP). In particular it provides a logical foundation for ASP as well as an extension of the basic syntax of answer set programs. We present an overview of equilibrium logic and its main properties and uses.

A compositional reasoning system for executing nonmonotonic theories of reasoning

In this paper the framework DESIRE for the design of compositional reasoning systems and multi-agent systems was applied to build a generic nonmonotonic reasoning system. The outcome is a general reasoning system that can be used to model different nonmonotonic reasoning formalisms and that can be executed by a generic execution mechanism. The main advantages of using DESIRE (for example, compared to a direct implementation in a programming language such as PROLOG) are that the design is generic and has a transparent compositional structure, and the explicit declarative specification of both the static and dynamic aspects of the nonmonotonic reasoning processes, including their control. © 2003 Wiley Periodicals, Inc.

Translating Multi-Agent Autoepistemic Logic into Logic Program

In order to develop a proof procedure of multi-agent autoepistemic Logic (MAEL), a natural framework to formalize belief and reasoning including inheritance, persistence, and causality, we introduce a method that translates a MAEL theory into a logic program with integrity constraints. It is proved that there exists one-to-one correspondence between extensions of a MAEL theory and stable models of a logic program translated from it. Our approach has the following advantages: (1) We can obtain all extensions of a MAEL theory if we compute all stable models of the translated logic program. (2) We can fully use efficient techniques or systems for computing stable models of a logic program. We also investigate the properties of reasoning in MAEL through this translation. The fact that the extension computing problem can be reduced to the stable model computing problem implies that there are close relationships between MAEL and other formalizations of nonmonotonic reasoning.

Translating Multi-Agent Autoepistemic Logic into Logic Program

In this paper, we develop a proof procedure for multi-agent autoepistemic logic (MAEL) by translating it into a logic program with stable model semantics. We introduce a method that translates a MAEL theory in normal form into a logic program which includes integrity constrains, and prove some theorems that guarantee soundness and completeness of the translation. In fact, there is a one-to-one correspondence between MAEL extensions of a theory and stable models of a logic program which is translated from the theory. Our approach has the following advantages compared with the former ones whose decision procedures are based on tableaux and resolution technique: (1) We can get all extensions (all inference results) of a MAEL theory if we compute all stable models of the logic program. (2) We can fully use efficient techniques or systems for computing stable models of a logic program on the process of MAEL theorem proving. Furthermore, we investigate properties of inference on MAEL through this translation. The fact that the extension computing problem of MAEL can be reduced to a stable model computing problem of logic programs implies that there are close relationships between MAEL and other formalizations of nonmonotonic reasoning.

Logic programming revisited

Logic programming has been introduced as programming in the Horn clause subset of first-order logic. This view breaks down for the negation as failure inference rule. To overcome the problem, one line of research has been to view a logic program as a set of iff-definitions. A second approach was to identify a unique canonical, preferred , or intended model among the models of the program and to appeal to common sense to validate the choice of such model. Another line of research developed the view of logic programming as a nonmonotonic reasoning formalism strongly related to Default Logic and Autoepistemic Logic. These competing approaches have resulted in some confusion about the declarative meaning of logic programming. This paper investigates the problem and proposes an alternative epistemological foundation for the canonical model approach, which is not based on common sense but on a solid mathematical information principle. The thesis is developed that logic programming can be understood as a natural and general logic of inductive definitions . In particular, logic programs with negation represent nonmonotone inductive definitions . It is argued that this thesis results in an alternative justification of the well-founded model as the unique intended model of the logic program. In addition, it equips logic programs with an easy-to-comprehend meaning that corresponds very well with the intuitions of programmers.

Representation results for defeasible logic

The importance of transformations and normal forms in logic programming, and generally in computer science, is well documented. This paper investigates transformations and normal forms in the context of Defeasible Logic, a simple but efficient formalism for nonmonotonic reasoning based on rules and priorities. The transformations described in this paper have two main benefits: on one hand they can be used as a theoretical tool that leads to a deeper understanding of the formalism, and on the other hand they have been used in the development of an efficient implementation of defeasible logic.

Nonmonotonic reasoning as prioritized argumentation

This paper proposes a formalism for nonmonotonic reasoning based on prioritized argumentation. We argue that nonmonotonic reasoning in general can be viewed as selecting monotonic inferences by a simple notion of priority among inference rules. More importantly, these types of constrained inferences can be specified in a knowledge representation language where a theory consists of a collection of rules of first order formulas and a priority among these rules. We recast default reasoning as a form of prioritized argumentation and illustrate how the parameterized formulation of priority may be used to allow various extensions and modifications to default reasoning. We also show that it is possible, but more difficult, to express prioritized argumentation by default logic: Even some particular forms of prioritized argumentation cannot be represented modularly by defaults under the same language.