On the Shadow Simplex Method for Curved Polyhedra

We study the simplex method over polyhedra satisfying certain “discrete curvature” lower bounds, which enforce that the boundary always meets vertices at sharp angles. Motivated by linear programs with totally unimodular constraint matrices, recent results of Bonifas et al (SOCG 2012), Brunsch and Röglin (ICALP 2013), and Eisenbrand and Vempala (2014) have improved our understanding of such polyhedra. We develop a new type of dual analysis of the shadow simplex method which provides a clean and powerful tool for improving all previously mentioned results. Our methods are inspired by the recent work of Bonifas and the first named author [4], who analyzed a remarkably similar process as part of an algorithm for the Closest Vector Problem with Preprocessing. For our first result, we obtain a constructive diameter bound of O( n2 ln n ) for n-dimensional polyhedra with curvature parameter 2 [0, 1]. For the class of polyhedra arising from totally unimodular constraint matrices, this implies a bound of O(n3 ln n). For linear optimization, given an initial feasible vertex, we show that an optimal vertex can be found using an expected O( n3 ln n ) simplex pivots, each requiring O(mn) time to compute. An initial feasible solutioncan be found using O(mn3 ln n ) pivot steps

Soundness and completeness proofs by coinductive methods

We show how codatatypes can be employed to produce compact, high-level proofs of key results in logic: the soundness and completeness of proof systems for variations of first-order logic. For the classical completeness result, we first establish an abstract property of possibly infinite derivation trees. The abstract proof can be instantiated for a wide range of Gentzen and tableau systems for various flavors of first-order logic. Soundness becomes interesting as soon as one allows infinite proofs of first-order formulas. This forms the subject of several cyclic proof systems for first-order logic augmented with inductive predicate definitions studied in the literature. All the discussed results are formalized using Isabelle/HOL’s recently introduced support for codatatypes and corecursion. The development illustrates some unique features of Isabelle/HOL’s new coinductive specification language such as nesting through non-free types and mixed recursion–corecursion

From fuzzy to annotated semantic web languages

The aim of this chapter is to present a detailed, selfcontained and comprehensive account of the state of the art in representing and reasoning with fuzzy knowledge in Semantic Web Languages such as triple languages RDF/RDFS, conceptual languages of the OWL 2 family and rule languages. We further show how one may generalise them to so-called annotation domains, that cover also e.g. temporal and provenance extensions

Finite Satisfiability in Infinite-Valued Łukasiewicz Logic

Although it is well-known that every satisfiable formula in Łukasiewicz’ infinite-valued logic L∞ can be satisfied in some finite-valued logic, practical methods for finding an appropriate number of truth degrees do currently not exist. As a first step towards efficient reasoning in L∞ , we propose a method to find a tight upper bound on this number which, in practice, often significantly improves the worst-case upper bound of Aguzzoli et al