28707658
Bounds on Certain Multiplications of Affine Combinations
Let A and B be n × n matrices the entries of which are affine combinations of the variables a1,…,am, b1,…,bm over GF(2). Suppose that, for each i, 1 ⩽ i ⩽ m, the term aibi is an element of the product matrix C = A · B. What is the maximum value that m can have as a function of n? This question arises from a recent technique for improving the communication complexity of zero-knowledge proofs. 
 
The obvious upper bound of n2 is improved to n233+ O(n). Tighter bounds are obtained for smaller values of n. The bounds for n = 2, n = 3, and n = 4 are tight.
34233329
Search Trees with Relaxed Balance and Near-Optimal Height
We introduce the relaxed k-tree, a search tree with relaxed balance and a height bound, when in balance, of (1 + Ɛ) log2 n + 1, for any Ɛ > 0. The rebalancing work is amortized O(1/Ɛ) per update. This is the first binary search tree with relaxed balance having a height bound better than c ċ log2 n for a fixed constant c. In all previous proposals, the constant is at least 1/ log2 Φ > 1.44, where Φ is the golden ratio. 
 
As a consequence, we can also define a standard (non-relaxed) k-tree with amortized constant rebalancing per update, which is an improvement over the original definition. 
 
Search engines based on main-memory databases with strongly fluctuating workloads are possible applications for this line of work.
80794
Efficient Rebalancing of Chromatic Search Trees
Chromatic trees were defined by Nurmi and Soisalon-Soininen, as a new type of binary search tree for databases. The aim is to improve runtime performance by allowing a greater degree of concurrency, which, in turn, is obtained by uncoupling updating from rebalancing. This also allows rebalancing to be postponed completely or partially until after peak working hours. The advantages of the proposal of Nurmi and Soisalon-Soininen are quite significant, but there are definite problems with it. First, they give no explicit upper bound on the complexity of their algorithm. Second, some of their rebalancing operations can be applied many more times than necessary. Third, some of their operations, when removing one problem, create another. We define a new set of rebalancing operations which we prove give rise to at most @?log"2(N+1)@?-1 rebalancing operations per insertion and at most @?log"2(N+1)@?-2 rebalancing operations per deletion, where N is the maximum size the tree could ever have, given its initial size and the number of insertions performed. Most of these rebalancing operations, in fact, do no restructuring; they simply move weights around. The number of operations which actually change the structure of the tree is at most one per update.
27033090
Exponentially Decreasing Number of Operations in Balanced Trees
While many tree-like structures have been proven to support amortized constant number of operations after updates, considerably fewer structures have been proven to support the more general exponentially decreasing number of operations with respect to distance from the update. In addition, all existing proofs of exponentially decreasing operations are tailor-made for specific structures. We provide the first formalization of conditions under which amortized constant number of operations imply exponentially decreasing number of operations. Since our proof is constructive, we obtain the constants involved immediately. Moreover, we develop a number of techniques to improve these constants.
30882911
Complexity of Layered Binary Search Trees with Relaxed Balance
When search trees are made relaxed, balance constraints are weakened such that updates can be made without immediate rebalancing. This can lead to a speed-up in some circumstances. However, the weakened balance constraints also make it more challenging to prove complexity results for relaxed structures.In our opinion, one of the simplest and most intuitive presentations of balanced search trees has been given via layered trees. We show that relaxed layered trees are among the best of the relaxed structures. More precisely, rebalancing is worst-case logarithmic and amortized constant per update, and restructuring is worst-case constant per update.
6986183
Efficient Simplification of Bisimulation Formulas
The problem of checking or optimally simplifying bisimulation formulas is likely to be computationally very hard. We take a different view at the problem: we set out to define a very fast algorithm, and then see what we can obtain. Sometimes our algorithm can simplify a formula perfectly, sometimes it cannot. However, the algorithm is extremely fast and can, therefore, be added to formula-based bisimulation model checkers at practically no cost. When the formula can be simplified by our algorithm, this can have a dramatic positive effect on the better, but also more time consuming, theorem provers which will finish the job.
44597345
Verification of Hierarchical State/Event Systems Using Reusability and Compositionality
We investigate techniques for verifying hierarchical systems, i.e., finite state systems with a nesting capability. The straightforward way of analysing a hierarchical system is to first flatten it into an equivalent non-hierarchical system and then apply existing finite state system verification techniques. Though conceptually simple, flattening is severely punished by the hierarchical depth of a system. To alleviate this problem, we develop a technique that exploits the hierarchical structure to reuse earlier reachability checks of superstates to conclude reachability of substates. We combine the reusability technique with the successful compositional technique of [13] and investigate the combination experimentally on industrial systems and hierarchical systems generated according to our expectations to real systems. The experimental results are very encouraging: whereas a flattening approach degrades in performance with an increase in the hierarchical depth (even when applying the technique of [13]), the new approach proves not only insensitive to the hierarchical depth, but even leads to improved performance as the depth increases.
2304363
Time-Optimal Real-Time Test Case Generation Using Uppaal
Testing is the primary software validation technique used by industry today, but remains ad hoc, error prone, and very expensive. A promising improvement is to automatically generate test cases from formal models of the system under test.
30176003
Lower and Upper Bounds in Zone Based Abstractions of Timed Automata
Timed automata have an infinite semantics. For verification purposes, one usually uses zone based abstractions w.r.t. the maximal constants to which clocks of the timed automaton are compared. We show that by distinguishing maximal lower and upper bounds, significantly coarser abstractions can be obtained. We show soundness and completeness of the new abstractions w.r.t. reachability. We demonstrate how information about lower and upper bounds can be used to optimise the algorithm for bringing a difference bound matrix into normal form. Finally, we experimentally demonstrate that the new techniques dramatically increases the scalability of the real-time model checker Uppaal.
18794799
Model-Checking for Real-Time Systems
Efficient automatic model-checking algorithms for real-time systems have been obtained in recent years based on the state-region graph technique of Alur, Courcoubetis and Dill. However, these algorithms are faced with two potential types of explosion arising from parallel composition: explosion in the space of control nodes, and explosion in the region space over clock-variables.
8766873
Synthesis of Optimal Strategies Using HyTech
Priced timed (game) automata extend timed (game) automata with costs on both locations and transitions. The problem of synthesizing an optimal winning strategy for a priced timed game under some hypotheses has been shown decidable in P. Bouyer, F. Cassez, E. Fleury, and K.G. Larsen. Optimal strategies in priced timed game automata. Research Report BRICS RS-04-4, Denmark, Feb. 2004. Available at http://www.brics.dk/RS/04/4/]. In this paper, we present an algorithm for computing the optimal cost and for synthesizing an optimal strategy in case there exists one. We also describe the implementation of this algorithm with the tool HyTech and present an example.