Recently, Gargani and Refalo (G&R) presented an elegant model for the Steel Mill Slab Design Problem (Problem 38 in the CSP LIB). Contrary to earlier approaches, their model does not use 0/1 variables but exploits the traditional expressiveness of constraint programming. G&R indicated that static symmetry-breaking constraints proposed earlier are not effective on this model, as these interact w...

An important point is that, although this is not usually stressed, the standardNP -completeness reductions also reduce the solution or witness from one problem to the other. That is, along with the function reduce(·) we also have a function red′ that maps a number w such that w2 = x to a 3-coloring c : V → {R,G,B} of the graph G = reduce(x, n). This can be used for the prover to convert their p...

Given a graph G with a speciied root node r. A spanning tree in G where each node has distance at most 2 from r is called a 2-hop spanning tree. For given edge weights the 2-hop spanning tree problem is to nd a minimum weight 2-hop spanning tree. The problem is NP-hard and has some interesting applications. We study a polytope associated with a directed model of the problem give a completeness ...

Godel's proofs of the incompleteness theorems are frequently discussed in the context of constructive or nitary viewpoint. Godel showed a concrete independent statement of arithmetic and the proof does not contain any transcendental argument. On the other hand, the completeness theorem asserts that a sentence is not derivable from a theory if and only if there exists a model of the theory whi...

and so geomxi is the geometric mean of xi. The goal of this note is to sketch the proof of the following theorem, concentrating on the iteration scheme from [4] in the two dimensional case. This justifies (19) of [4] without an appeal to [1]. We also include some routine applications of Hölder, Minkowski, etc. that are omitted from [4] for completeness. We will prove Theorem 1. Let 0 < δ ≤ 1. F...

Alternative proofs for Guaspari and Solovay's completeness theorems for R are presented. R is an extension of the provability logic L and was developed in order to study the formal properties of the provability predicate of PA occurring in sentences that may contain connectives for witness comparison. (The primary example of sentences involving witness comparison is the Rosser sentence.) In thi...

Given a graph G with a speciied root node r. A spanning tree in G where each node has distance at most 2 from r is called a 2-hop spanning tree. For given edge weights the 2-hop spanning tree problem is to nd a minimum weight 2-hop spanning tree. The problem is NP-hard and has some interesting applications. We study a polytope associated with a directed model of the problem give a completeness ...

Abstract Let $$G \subsetneq \mathbb {C}$$ G ⊊ C be a bounded, simply connected domain in $$\mathbb , and denote by $$\begin{aligned} Q(G):= \left\{ f: G \longrightarrow \ |\ f \text { is quasiconformal mapping of } onto \right\} \end{aligned}$$ <mml:mta...

The Boolean ring $B$ of measurable subsets of the unit interval, modulo sets of measure zero, has proper radical ideals (for example, ${0})$ that are closed under the natural metric, but has no prime ideal closed under that metric; hence closed radical ideals are not, in general, intersections of closed prime ideals. Moreover, $B$ is known to be complete in its metric. Togethe...

In this paper we discuss two hybrid languages , L(∀) and L(↓), and provide them with complete axiomatizations. Both languages combine features of modal and classical logic. Like modal languages, they contain modal operators and have a Kripke semantics. Unlike modal languages, in these systems it is possible to ‘label’ states by using ∀ and ↓ to bind special state variables . This paper explores...