Efficient Timed Reachability Analysis using Clock Difference Diagrams

One of the major problems in applying automatic verication tools to industrial-size systems is the excessive amount of memory required during the state-space exploration of amodel. In the setting of real-time, this problem of state-explosion requires extra attention as information must be kept not only on the discrete control structure but also on the values of continuous clock variables. In this paper, we present Clock Dierence Diagrams, CDD's, a BDD-like data-structure forrepresenting and eectively manipulating certain non-convex subsets of the Euclidean space, notably those encountered during verication of timed automata. A version of the real-time verication tool Uppaal using CDD's as a compact datastructurefor storing explored symbolic states has been implemented. Our experimental results demonstrate signicant space-savings: for 8 industrial examples, the savings are between 46%and 99% with moderate increase in runtime. We further report on how the symbolic state-space exploration itself may be carried out using CDD's

Heegaard-Floer homology and string links

We extend knot Floer homology to string links in D^{2} \times I and to d-based links in arbitrary three manifolds, without any hypothesis on the null-homology of the components. As for knot Floer homology we obtain a description of the Euler characteristic of the resulting homology groups (in D^{2} \times I) in terms of the torsion of the string link. Additionally, a state summation approach is described using the equivalent of Kauffman states. Furthermore, we examine the situtation for braids, prove that for alternating string links the Euler characteristic determines the homology, and develop similar composition formulas and long exact sequences as in knot Floer homology.Comment: 57 page

Odd q-State Clock Spin-Glass Models in Three Dimensions, Asymmetric Phase Diagrams, and Multiple Algebraically Ordered Phases

Distinctive orderings and phase diagram structures are found, from renormalization-group theory, for odd q-state clock spin-glass models in d=3 dimensions. These models exhibit asymmetric phase diagrams, as is also the case for quantum Heisenberg spin-glass models. No finite-temperature spin-glass phase occurs. For all odd $q\geqslant 5$, algebraically ordered antiferromagnetic phases occur. One such phase is dominant and occurs for all $q\geqslant 5$. Other such phases occupy small low-temperature portions of the phase diagrams and occur for $5 \leqslant q \leqslant 15$. All algebraically ordered phases have the same structure, determined by an attractive finite-temperature sink fixed point where a dominant and a subdominant pair states have the only non-zero Boltzmann weights. The phase transition critical exponents quickly saturate to the high q value.Comment: Published version, 9 pages, 10 phase diagrams, 5 figures, 1 tabl