Modal correspondence theory is a powerful and effective way to guarantee that adding specific syntactic axioms to a modal logic is mirrored by requiring 'corresponding' properties of the underlying Kripke models. However, such axioms not only  quantify over all formulas, but they are also global  in the sense that the corresponding semantic property is assumed to hold for all states. However, in  for instance epistemic logic one would like to have  the flexibility to say that certain properties (like "agent b knows at least what agent a knows") are  true locally in a specific state, but not necessarily  globally, in all states. This would enable one to  say "currently, b knows at least what a knows, but  this is not common knowledge," or ". . . but this is  not always true," or ". . . but this could be changed  by action α." We offer a logic for "knowing at least  as," where the (global) axiom scheme Ka ϕ → Kb ϕ  is replaced by a (local) inference rule. We give  a complete modal system, and discuss some consequences of the axiom in an epistemic setting.  Our completeness proof also suggests how achieving such local properties can be generalized to other  axioms schemes and modal logics. 

In this article we discuss two alternative proposals for neighbourhood semantics (which we call strict and loose neighbourhood semantics, NSS and NSL respectively) that have been previously introduced in the literature. Our main tools are suitable notions of bisimulation. While an elegant notion of bisimulation exists for NSL, the required bisimulation for NSS is rather involved. We propose a simple extension of NSS with a universal modality that we call NSS(E), which comes together with a natural notion of bisimulation. We also investigate the complexity of the satisfiability problem for NSL and NSS(E).

In a seminal paper, Lin and Reiter introduced the notion of progression for basic action theories in the situation calculus. Unfortunately, progression is not first-order definable in general. Recently, Vassos, Lakemeyer, and Levesque showed that in case actions have only local effects, progression is first-order representable. However, they could show computability of the first-order representation only for a restricted class. Also, their proofs were quite involved. In this paper, we present a result stronger than theirs that for local-effect actions, progression is always first-order definable and computable. We give a very simple proof for this via the concept of forgetting. We also show first-order definability and computability results for a class of knowledge bases and actions with non-local effects. Moreover, for a certain class of local-effect actions and knowledge bases for representing disjunctive information, we show that progression is not only first-order definable but also efficiently computable.

Splitting a logic program allows us to reduce the task of computing its stable models to similar tasks for smaller programs.  This idea is extended here to the general theory of stable models that replaces traditional logic programs by arbitrary first-order sentences and distinguishes between intensional and extensional predicates.  We discuss two kinds of splitting: a set of intensional predicates can be split into subsets, and a formula can be split into its conjunctive terms.

Sensors provide computer systems with a window to the outside world. Activity recognition "sees" what is in the window to predict the locations, trajectories, actions, goals and plans of humans and objects. Building an activity recognition system requires a full range of interaction from statistical inference on lower level sensor data to symbolic AI at higher levels, where prediction results and acquired knowledge are passed up each level to form a knowledge food chain. In this article, I will give an overview of some of the current activity recognition research works and explore a life-cycle of learning and inference that allows the lowest-level radio-frequency signals to be transformed into symbolic logical representations for AI planning, which in turn controls the robots or guides human users through a sensor network, thus completing a full life-cycle of knowledge.

On the other hand, the Recently, Ferraris, Lee and Lifschitz presented a solution provided by answer set programming (ASP), that is general definition of a stable model that is similar carried over to high level action language A [Gelfond and to the definition of circumscription, and can even Lifschitz, 1998] and many of its descendants that are based be characterized in terms of circumscription. In on ASP, uses both default negation (not) and strong negation this paper, we show the opposite direction, which ()--the idea of which is closely related to Reiter's default is, how to turn circumscription into the general stable logic solution [Reiter, 1980]. Interestingly, the development model semantics, and based on this, how to turn of the event calculus has spanned over both classical circumscriptive event calculus into answer set programs.

We propose a new framework for reasoning about knowledge, action and time for domains that include actions with non-deterministic and context-dependent effects. The axiomatization is based on the Event Calculus and combines the expressiveness of possible worlds semantics with the efficiency of approaches that dispense the use of the accessibility relation. The framework is proved logically sound and, when restricted to deterministic domains, is also logically complete. To prove correctness of the approach, we construct a knowledge theory based on a branching version of the Event Calculus and study their correlation.

A general framework is proposed for integration of rules and external first order theories. It is based on the well-founded semantics of normal logic programs and inspired by ideas of Constraint Logic Programming (CLP) and constructive negation for logic programs. Hybrid rules are normal clauses extended with constraints in the bodies; constraints are certain formulae in the language of the external theory. A hybrid program is a pair of a set of hybrid rules and an external theory. Instances of the framework are obtained by specifying the class of external theories, and the class of constraints. An example instance is integration of (non-disjunctive) Datalog with ontologies formalized as description logics. The paper defines a declarative semantics of hybrid programs and a goal-driven formal operational semantics. The latter can be seen as a generalization of SLS-resolution. It provides a basis for hybrid implementations combining Prolog with constraint solvers. Soundness of the operational semantics is proven. Sufficient conditions for decidability of the declarative semantics, and for completeness of the operational semantics are given.

Equilibrium logic is an approach to nonmonotonic reasoning that extends the stable-model and answer-set semantics for logic programs. In particular, it includes the general case of nested logic programs, where arbitrary Boolean combinations are permitted in heads and bodies of rules, as special kinds of theories. In this paper, we present polynomial reductions of the main reasoning tasks associated with equilibrium logic and nested logic programs into quantified propositional logic, an extension of classical propositional logic where quantifications over atomic formulas are permitted. We provide reductions not only for decision problems, but also for the central semantical concepts of equilibrium logic and nested logic programs. In particular, our encodings map a given decision problem into some formula such that the latter is valid precisely in case the former holds. The basic tasks we deal with here are the consistency problem, brave reasoning, and skeptical reasoning. Additionally, we also provide encodings for testing equivalence of theories or programs under different notions of equivalence, viz. ordinary, strong, and uniform equivalence. For all considered reasoning tasks, we analyse their computational complexity and give strict complexity bounds.

We present the CIFF proof procedure for abductive logic programming with constraints, and we prove its correctness. CIFF is an extension of the IFF proof procedure for abductive logic programming, relaxing the original restrictions over variable quantification (allowedness conditions) and incorporating a constraint solver to deal with numerical constraints as in constraint logic programming. Finally, we describe the CIFF system, comparing it with state of the art abductive systems and answer set solvers and showing how to use it to program some applications. (To appear in Theory and Practice of Logic Programming - TPLP).