Many-valued non-monotonic modal logics

It has been well recognized that common-sense reasoning is nonmonotonic. Nonmonotonic reasoning is prototypically described by the reasoning about normality and exception. In this paper, we firstly propose a modal logic about normality and exception. We then present a nonmonotonic modal logic which formalizes the intuitions of nonmonotonic reasoning directly, rather than go around to indirect criterion. The results can be seen by comparing our logic with major nonmonotonic logics in recent literature. In particular, we provide a first-order extension of default logic and autoepistemic logic, and the capability of circumscription to infer universal statement. As a result, the relations among various existing nonmonotonic logics coincide in our logic. >

Chapter 20 - Automated Deduction for Many-Valued Logics

We generalize Moore's autoepistemic logic to multi-valued autoepistemic logic, where the set of truth-values can be any complete lattice. Multi-valued autoepistemic extensions can be characterized by admissible belief interpretations which are the conrete approximations of extensions and are appropriate to be computed and manipulated. We prove that multi-valued autoepistemic extensions are exactly the theories of maximal multi-valued Kripke models. The class of stratified theories is investigated and it is shown that stratified theories have exactly one multi-valued autoepistemic extension. Finally we present a sequent calculus for multi-valued logic which serves as a tool for a decision procedure for multi-valued autoepistemic logic.

The ability to “define” propositions using default assumptions about the same propositions is identified as a significant source of computational complexity in nonmonotonic reasoning. If such constructs are not allowed, i.e. the knowledge base is stratified, a notable computational advantage is obtained. This is demonstrated by developing an iterative algorithm for propositional stratified autoepistemic theories the complexity of which is dominated by required classical reasoning. Thus efficient subclasses of stratified nonmonotonic reasoning can be obtained by further restricting the form of sentences in the knowledge base. As an example quadratic and linear time algorithms are derived for specific subclasses of stratified autoepistemic theories. The results are shown to imply efficient reasoning methods for stratified cases of default logic, logic programs, truth maintenance systems, and nonmonotonic modal logics.

The goal of this note is to provide a background and references for the invited lecture presented at Computer Science Logic 2006. We briefly discuss motivations that led to the emergence of nonmonotonic logics and introduce two major nonmonotonic formalisms, default and autoepistemic logics. We then point out to algebraic principles behind the two logics and present an abstract algebraic theory that unifies them and provides an effective framework to study properties of nonmonotonic reasoning. We conclude with comments on other major research directions in nonmonotonic logics.

Many-valued logic based interval reasoning plays an increasingly important role in fuzzy and other many-valued extensions of two-valued (crisp) logic. It is often overlooked that many-valued logic (MVL) reasoning has a richer inference rule base than the crisp (i.e. classical two-valued) logic. Indeed some rules that are not possible in the crisp logic (which is point-based) do come to existence when we accept intervals as the basic elements of the semantic space ([0, 1] or a more general lattice) into which the logic expressions are valuated [11, 8, 37].

We present a semantic tableaux calculus for propositional nonmonotonic modal logics, based on possible-worlds characterisations for nonmonotonic modal logics. This method is parametric with respect to both the modal logic and the preference semantics, since it handles in a uniform way the entailment problem for a wide class of nonmonotonic modal logics: McDermott and Doyle's logics and ground logics. It also achieves the computational complexity lower bounds.KeywordsModal LogicKripke ModelGround LogicDefault LogicNonmonotonic 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.

Multi-valued logic and Gröner bases with applications to modal logic

Sören Halldén. On the logic of ‘better.’ Library of Theoria, no. 2. C. W. K. Gleerup, Lund, and Ejnar Munksgaard, Copenhagen, 1957, 112 pp. - Lennart Åqvist. Deontic logic based on a logic of ‘better.’Proceedings of a Colloquium on Modal and Many-valued Logics, Helsinki, 23–26 August, 1962, Acta philosophica Fennica, no. 16, Helsinki1963, pp. 285–290. - Mark Fisher. A three-valued calculus for deontic logic. Theoria (Lund), vol. 27 (1961), pp. 107–118. - Lennart Åqvist. Postulate sets and decision procedures for some systems of deontic logic. Theoria (Lund), vol. 29 (1963), pp. 154–175. - Volume 31 Issue 2

Some find it plausible that a sufficiently long duration of torture is worse than any duration of mild headaches. Similarly, it has been claimed that a million humans living great lives is better than any number of worm-like creatures feeling a few seconds of pleasure each. Some have related bad things to good things along the same lines. For example, one may hold that a future in which a sufficient number of beings experience a lifetime of torture is bad, regardless of what else that future contains, while minor bad things, such as slight unpleasantness, can always be counterbalanced by enough good things. Among the most common objections to such ideas are sequence arguments. But sequence arguments are usually formulated in classical logic. One might therefore wonder if they work if we instead adopt many-valued logic. I show that, in a common many-valued logical framework, the answer depends on which versions of transitivity are used as premises. We get valid sequence arguments if we grant any of several strong forms of transitivity of ‘is at least as bad as’ and a notion of completeness. Other, weaker forms of transitivity lead to invalid sequence arguments. The plausibility of the premises is largely set aside here, but I tentatively note that almost all of the forms of transitivity that lead to valid sequence arguments seem intuitively problematic. Still, a few moderately strong forms of transitivity that might be acceptable lead to valid sequence arguments, although weaker statements of the initial value claims avoid these arguments at least to some extent.

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.

1 Modal logic: a semantic perspective

Analogic: Inference beyond logic

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.

We introduce here the study of generalnonmonotonic rule systems. These deal with situations where a conclusion is drawn from a system of beliefsS (and seen to be inS), basedboth on some being inS and on some restraints not being inS. In the monotone systems of traditional logic there are no restraints, conclusions are drawn solely based on premises being inS. Nonmonotonic rule systems capture the essential syntactic, semantic, and algorithmic features of many nonmonotone systems such as default logic, negation as failure, truth maintenance, autoepistemic logic, and also important combinatorial questions from mathematics such as the marriage problem. This reveals semantics and syntax and proof procedures and algorithms for computing belief sets in many cases where none were previously available and entirely uniformly. In particular, we introduce and study deductively closed sets, extensions and weak extensions. Semantics of nonmonotonic rule systems is studied in part II of this paper and extensions to predicate classical, intuitionistic, and modal logics are left to a later paper.