# Applicable geometry: global and local convexity by Heinrich W Guggenheimer

By Heinrich W Guggenheimer

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Qualitative Spatial Representation and Reasoning ledge and as such marked a starting point for qualitative physics [54, 165]. A further milestone towards establishing qualitative physics (or qualitative reasoning as it is now called) as an important sub-discipline of Artiﬁcial Intelligence was the Naive Physics Manifesto [82, 83, 84] which, among other things, proposes to represent space-time with four-dimensional “histories”. Based on Hayes’s histories, Forbus [52] presented a system which reasoned about motion through free space by using again both qualitative and quantitative information.

So far, any algorithm for an NP-complete problem has an exponential or at least superpolynomial running time. 2 Phase Transitions Having proved a problem to be NP-complete is not the end of the computational analysis of a problem, rather its beginning. NP-completeness is just a worst-case measure of a problem. It means that for any algorithm there is at least one instance of an NP-complete problem which cannot be solved in polynomial time. It is possible that only one in a million instances is very hard and that the other instances can be solved eﬃciently.

Whether a region consists of one-piece or of multiple pieces, or the existence of holes. The diﬀerent approaches are compared in [29, 30]. All of these approaches have in common that the relations are axiomatized and deﬁned in ﬁrst-order logic which provides them with formal semantics. The formal properties of ﬁrst-order theories based on a connectedness relation were studied by Grzegorczyk [78], Dornheim [39] and Pratt and Schoop [132, 148]. It turns out that all of them are incomplete and, moreover, reasoning about these theories is undecidable.