Reactive Preferential Structures and Nonmonotonic Consequence

2012

Ann Math Artif Intell

This paper introduces the idea of reactive semantics and reactive Beth tableaux for modal logic and quotes some of its applications. The reactive idea is very simple. Given a system with states and the possibility of transitions moving from one state to another, we can naturally imagine a path beginning at an initial state and moving along the path following allowed transitions. If our starting point is s 0 , and the path is s 0 , s 1 , . . . , s n , then the system is ordinary non-reactive system if the options available at s n (i.e., which states t we can go to from s n ) do not depend on the path s 0 , . . . , s n (i.e., do not depend on how we got to s n ). Otherwise if there is such dependence then the system is reactive. It seems that the simple idea of taking existing systems and turning them reactive in certain ways, has many new applications. The purpose of this paper is to introduce reactive tableaux in particular and illustrate and present some of the applications of reactivity in general. Mathematically one can take a reactive system and turn it into an ordinary system by taking the paths as our new states. This is true but from the point of view of applications there is serious loss of information here as the applicability of the reactive system comes from the way the change occurs along the path. In any specific application, the states have meaning, the transitions have meaning and the paths have

“…For general references on tableaux and on modal tableaux, see [1,[3][4][5]20]. For general papers on Reactivity and its use see [2,6,7,10,11,[13][14][15][16][17][18][19].…”

Section: Setting the Scenementioning

confidence: 99%

Introducing reactive modal tableaux

2012

Ann Math Artif Intell

“…So we do not gain in expressive power by going reactive (as we do in the case of reactive context free grammars) but we do gain simplicity. [9,12,13]. Figure 9 is an example of a network.…”

Section: Theorem 15mentioning

confidence: 99%

“…This also means that the classical modal reading of 13 We cannot fold t A → B as t A implies t B is too poor to bring out the reachness of reactive Kripke models and we need something A ⇒ B reading first evaluate A and then evaluate B.…”

Section: Ternary Relationsmentioning

confidence: 99%

Completeness theorems for reactive modal logics

2012

Ann Math Artif Intell

Mathematics Subject Classifications (2010) 03B44 · 03B45 OverviewThis paper gives completeness theorems for some basic reactive Kripke models and semantics. This section will 1. Introduce reactivity 2. Discuss and compare the kind of Kripke semantics we get with reactivity 3. Explain the challenges in obtaining completeness theorems Fibring and reactivityOur starting point is an ordinary Kripke model for modal logic. This has the form m = (S, R, a) where S is a nonempty set of worlds, a ∈ S is the initial (actual) world and R ⊆ S × S is the accessibility relation. The model also has an assignment h, giving for each atomic q a subset h(q) ⊆ S.We shall focus on R.

“…Indeed, there are applications of the reactive ideas in such diverse areas as modal logic, preferential non-monotonic logic, inheritance systems, context-free grammars, automata theory, deontic logic and contrary to duty, argumentation and other networks, see papers [5,20,12,16,23,21,22,17,28]. For example when one adds these kind of double arrows to the structure of an automata, one allows it to modify its transition relation while reading a sentence, that is, one makes it reactive.…”

Section: Introductionmentioning

confidence: 99%

Global View on Reactivity: Switch Graphs and their Logics

Reactive Kripke Semantics

2013

Self Cite

The notion of reactive graph generalises the one of graph by allowing the base accessibility relation to change when its edges are traversed. Can we represent these more general structures using points and arrows? We prove this can be done by introducing higher order arrows: the switches.The possibility of expressing the dependency of the future states of the accessibility relation on individual transitions by the use of higher-order relations, that is, coding meta-relational concepts by means of relations, strongly suggests the use of modal languages to reason directly about these structures. We introduce a hybrid modal logic for this purpose and prove its completeness.