Nonmonotonic Modal Logic Research Articles

Representation of gene regulation networks by hypothesis logic-based Boolean systems

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.

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.

A nonmonotonic modal authorization logic for pervasive computing

AbstractModal logics have proven invaluable for authorization in distributed systems. The logics devised so far, however, are inadequate to meet the requirements of pervasive environments. Such environments are, in general, characterized as open systems in which computing and communication facilities are provided to human users in a dynamic manner. These features suggest the need for the modification of existing logics in two directions. First, users’ capabilities being intrinsic to pervasive computing should be incorporated into the underlying modal logic. Second, the logic should be equipped with appropriate machinery so that it can deal with the imperfection in the information required for authorization. This paper has contributions in both directions. We present a logic that reflects how the capabilities of users change in different contexts. Nonmonotonicity is then added to the logic so that earlier decisions based on imperfect information can be retracted. The usefulness of our formulation is demonstrated through the added capacity it provides for specifying and enforcing access control policies in real‐life environments. We also present a minimal model semantics that reflects nonmonotonicity through the way it gives meaning to the formulas of the logic. Finally, we propose a sound and complete decision procedure based on semantic tableaux. Copyright © 2014 John Wiley & Sons, Ltd.

Algebraic semantics for modal and superintuitionistic non-monotonic logics

The paper provides a preliminary study of algebraic semantics for modal and superintuitionistic non-monotonic logics. The main question answered is: how can non-monotonic inference be understood algebraically?

From answer set logic programming to circumscription via logic of GK

We first embed Pearce's equilibrium logic and Ferraris's propositional general logic programs in Lin and Shoham's logic of GK, a nonmonotonic modal logic that has been shown to include as special cases both Reiter's default logic in the propositional case and Moore's autoepistemic logic. From this embedding, we obtain a mapping from Ferraris's propositional general logic programs to circumscription, and show that this mapping can be used to check the strong equivalence between two propositional logic programs in classical logic. We also show that Ferraris's propositional general logic programs can be extended to the first-order case, and our mapping from Ferraris's propositional general logic programs to circumscription can be extended to the first-order case as well to provide a semantics for these first-order general logic programs.

On a Modal Epistemic Axiom Emerging from McDermott-Doyle Logics

An important question in modal nonmonotonic logics concerns the limits of propositional definability for logics of the McDermott-Doyle family. Inspired by this technical question we define a varian...

On a Modal Epistemic Axiom Emerging from McDermott-Doyle Logics

An important question in modal nonmonotonic logics concerns the limits of propositional definability for logics of the McDermott-Doyle family. Inspired by this technical question we define a variant of autoepistemic logic which provably corresponds to the logic of the McDermott-Doyle family that is based on themodal axiom p5: ◊p⊃ (¬sp⊃s¬sp). This axiomis a naturalweakening of classical negative introspection restricting its scope to possible facts. It closely resembles the axiom w5: p⊃ (¬sp⊃s¬sp) which restricts the effect of negative introspection to true facts. We examine p5 in the context of classical possible-worlds Kripke models, providing results for correspondence, completeness and the finite model property. We also identify the corresponding condition for p5 in the context of neighbourhood semantics. Although rather natural epistemically, this axiom has not been investigated in classical modal epistemic reasoning, probably because its addition to S4 gives the well-known strong modal system S5.

First-Order Ground Non-Monotonic Modal Logic

We study the extension of propositional ground non-monotonic modal logic to the firstorder case. We show that first-order ground non-monotonic modal logic well complies with first-order default log...

A non-preferential semantics of non-monotonic modal logic

We present a new semantical description of non-monotonic modal logic whose underlying (monotonic) modal logic S is characterized by a class of cluster-decomposable Kripke interpretations: it is shown that S-expansions coincide with the theories of isolated clusters of the canonical model of S.

WITHDRAWN: Non-monotonic modal logic of belief

WITHDRAWN: Non-monotonic modal logic of belief

Ground Nonmonotonic Modal Logic S5: New Results

We study logic programs under Gelfond's translation in the context of modal logic S5. We show that for arbitrary logic programs (propositional theories where logic negation is associated with default negation) ground nonmonotonic modal logics between T and S5 are equivalent. Furthermore, we also show that these logics are equivalent to a nonmonotonic logic that we construct using the well known F O U R bilattice. We will call this semantic GNM-S5 as a reminder of its origin in the logic S5. Finally we show that, for normal programs, our approach is closely related to theWell-Founded-by-Cases Semantics introduced by Schlipf and the WFS+ proposed by Dix. We prove that GNM-S5 has the properties of classicality and extended cut. While WFS+ also supports classicality it fails to satisfy the extended cut principle, an important property available in other semantics such as stable models. Hence, we claim that GNM-S5 is a good candidate for defining a nonmonotonic semantics closer to the direction of classical logic.

More on Bounding Introspection in Modal Nonmonotonic Logics

More on Bounding Introspection in Modal Nonmonotonic Logics

More on Bounding Introspection in Modal Nonmonotonic Logics

By $$ {\user1{\mathcal{L}}} $$ we denote the set of all propositional formulas. Let $$ {\user1{\mathcal{C}}} $$ be the set of all clauses. Define $$ {\user1{\mathcal{C}}}_{0} = {\user1{\mathcal{C}}} \cup {\left\{ {\neg L\eta :\eta \in {\user1{\mathcal{C}}}} \right\}} $$ . In Sec. 2 of this paper we prove that for normal modal logics $$ {\user1{\mathcal{S}}} $$ , the notions of $$ {\left( {{\user1{\mathcal{S}}},{\user1{\mathcal{C}}}_{0} } \right)} $$ -expansions and $$ {\user1{\mathcal{S}}} $$ -expansions coincide. In Sec. 3, we prove that if I consists of default clauses then the notions of $$ {\user1{\mathcal{S}}} $$ -expansions for I and $$ {\left( {{\user1{\mathcal{S}}},{\user1{\mathcal{C}}}} \right)} $$ -expansions for I coincide. To this end, we first show, in Sec. 3, that the notion of $$ {\user1{\mathcal{S}}} $$ -expansions for I is the same as that of $$ {\left( {{\user1{\mathcal{S}}},{\user1{\mathcal{L}}}} \right)} $$ -expansions for I.

First-order Non-monotonic Modal Logics

In this paper we introduce first-order non-monotonic modal logic and study its properties. Our definition is semantical and is based on the intuition similar to that lying behind the definition of first-order default logic. Thus, our definition of first-order non-monotonic modal logic well complies with that of first-order default logic and circumscription, providing a substantial evidence for its acceptance. In addition, our definition implies all properties of propositional non-monotonic modal logic, which supplies an additional support for its adequacy.

A logic of universal causation

For many commonsense reasoning tasks associated with action domains, only a relatively simple kind of causal knowledge is required—knowledge of the conditions under which facts are caused. This note introduces a modal nonmonotonic logic for representing causal knowledge of this kind, relates it to other nonmonotonic formalisms, and shows that a variety of causal theories of action can be expressed in it, including the recently proposed causal action theories of Lin. The new logic extends the causal theories formalism of McCain and Turner, and provides a more adequate semantic account of it. A useful subset of the logic has a concise translation into classical propositional logic, and so can be used for automated planning and reasoning about action. A larger subset is closely related to logic programming under the answer set semantics, yielding another approach to automated reasoning.

Reasoning about Minimal Knowledge in Nonmonotonic Modal Logics

We study the problem of embedding Halpern and Mosess modal logic of minimal knowledge states into two families of modal formalism for nonmonotonic reasoning, McDermott and Doyles nonmonotonic modal...

Many-Valued Modal Non-Monotonic Reasoning: Sequential Stable Sets and Logics with Linear Truth Spaces

A family of many-valued modal logics which correspond to possible-worlds models with many-valued accessibility relations, has been recently proposed by M. Fitting [7, 8]. Non-monotonic extensions of these logics are introduced with a fixpoint construction a la McDermott & Doyle and employ sequential belief sets as epistemic states [9]. In this paper we take a logical investigation of many-valued modal non-monotonic reasoning in Fitting's formal framework. We examine the notion of MV-stable sets which emerges as a sequential many-valued analog of Stalnaker-Moore stable sets and prove that several attractive epistemic properties are essentially retained in the many-valued setting, esp. when focusing on a syntactically simple epistemic fragment of MV-stable sets. We show that MV-stable sets are always closed under S4 consequence and identify three sufficient conditions for capturing axioms of negative introspection. Also, the relation of MV-stable sets to many-valued analogs of classical S5 models and to many-valued extensions of universal models is discussed. Finally, we pay special attention to the subclass of logics built on linear Heyting algebras and show that inside this subclass, the situation is very similar - in many respects - to the machinery devised by W. Marek, G. Schwarz and M. Truszczynski. In particular, the normal fragments of the two important classical ranges of modal non-monotonic logics remain intact: many-valued autoepistemic logic is captured by any non-monotonic logic in K5 - KD45 and many-valued reflexive autoepistemic logic corresponds to KTw5 - Sw5.

An epistemic operator for description logics

Description logics (also called terminological logics, or concept languages) are fragments of first-order logic that provide a formal account of the basic features of frame-based systems. However, there are aspects of frame-based systems—such as nonmonotonic reasoning and procedural rules—that cannot be characterized in a standard first-order framework. Such features are needed for real applications, and a clear understanding of the logic underlying them is necessary for principled implementations. We show how description logics enriched with an epistemic operator can formalize such aspects. The logic obtained is a fragment of a first-order nonmonotonic modal logic. We show that the epistemic operator formalizes procedural rules, as provided in many knowledge representation systems, and enables sophisticated query formulation, including various forms of closed-world reasoning. We provide an effective procedure for answering epistemic queries posed to a knowledge base expressed in a description logic and extend this procedure in order to deal with rules. We also address the computational complexity of reasoning with the epistemic operator, identifying cases in which an appropriate use of the epistemic operator can help in decreasing the complexity of reasoning.

An abstract, argumentation-theoretic approach to default reasoning

We present an abstract framework for default reasoning, which includes Theorist, default logic, logic programming, autoepistemic logic, non-monotonic modal logics, and certain instances of circumscription as special cases. The framework can be understood as a generalisation of Theorist. The generalisation allows any theory formulated in a monotonic logic to be extended by a defeasible set of assumptions. An assumption can be defeated (or “attacked”) if its “contrary” can be proved, possibly with the aid of other conflicting assumptions. We show that, given such a framework, the standard semantics of most logics for default reasoning can be understood as sanctioning a set of assumptions, as an extension of a given theory, if and only if the set of assumptions is conflict-free (in the sense that it does not attack itself) and it attacks every assumption not in the set. We propose a more liberal, argumentation-theoretic semantics, based upon the notion of admissible extension in logic programming. We regard a set of assumptions, in general, as admissible if and only if it is conflict-free and defends itself (by attacking) every set of assumptions which attacks it. We identify conditions for the existence of extensions and for the equivalence of different semantics.

Intuitionistic autoepistemic logic

In this paper we address the problem of combining a logic Λ with nonmonotonic modal logic. In particular we study the intuitionistic case. We start from a formal analysis of the notion of intuitionistic consistency via the sequent calculus. The epistemic operator M is interpreted as the consistency operator of intuitionistic logic by introducing intuitionistic stable sets. On the basis of a bimodal structure we also provide a semantics for intuitionistic stable sets.