Phase transitions are familiar phenomena in physical systems. But they also occur in many probabilistic and combinatorial models, including random versions of some classic problems in theoretical computer science.

In this talk, I will discuss phase transitions in several systems, including the random graph — a simple probabilistic model which undergoes a critical transition from a disordered to an ordered phase; k-satisfiability — a canonical model in theoretical computer science which undergoes a transition from solvability to insolvability; and optimum partitioning — a fundamental problem in combinatorial optimization, which undergoes a transition from existence to non-existence of a perfect optimizer.

Technically, phase transitions only occur in infinite systems. However, real systems and the systems we simulate are large, but finite. Hence the question of finite-size scaling: Namely, how does the transition behavior emerge from the behavior of large, finite systems? Results on the random graph, satisfiability and optimum partitioning locate the critical windows of these transitions and establish interesting features within these windows.

No knowledge of phase transitions will be assumed in this talk

The Internet and the worldwide web, unlike all other computational artifacts, were not deliberately designed by a single entity, but emerged from the complex interactions of many. As a result, they must be approached very much the same way that cells, galaxies or markets are studied in other sciences: By speculative (and falsifiable) theories trying to explain how selfish algorithmic actions could have led to what we observe. I present several instances of recent work on this theme, with several collaborators

The Probabilistic Method is a lasting legacy of the late Paul Erdős. We give two examples — both problems first formulated by Erdős in the 1960s with new results in the last few years and both with substantial open questions. Further in both examples we take a Computer Science vantagepoint, creating a probabilistic algorithm to create the object (coloring, packing respectively) and showing that with positive probability the created object has the desired properties.

Given \(m\) sets each of size \(n\) (with an arbitrary intersection pattern) we want to color the underlying vertices Red and Blue so that no set is monochromatic. Erdős showed this may always be done if \(m< 2(n-1)\), we give a recent argument of Srinivasan and Radhukrishnan that extends this to \(m < c 2n\sqrt{n/\ln n}\). One first colors randomly and then recolors the blemishes with a clever random sequential algorithm.

In a universe of size \(N\) we have a family of sets, each of size \(k\), such that each vertex is in \(D\) sets and any two vertices have only \(o(D)\) common sets. Asymptotics are for fixed \(k\) with \(N,D\). We want an asymptotic packing, a subfamily of \(N/k\) disjoint sets. Erdős and Hanani conjectured such a packing exists (in an important special case of asymptotic designs) and this conjecture was shown by Rödl. We give a simple proof of the speaker that analyzes the random greedy algorithm.

In this paper we present a collection of problems which have defied solution for some time. We hope that this paper will stimulate renewed interest in these problems, leading to solutions to at least some of them.

Quantum computers are the only model of computing to credibly violate the modified Church-Turing thesis, which states that any reasonable model of com-putation can be simulated by a probabilistic Turing Machine with at most poly-nomial factor simulation overhead. This is dramatically demonstrated by Shor’s polynomial time algorithms for factorization and discrete logarithms [13]. Shor’s algorithm, as well as the earlier algorithm due to Simon [12] can both be cast into the general framework of the hidden subgroup problem (see for example [10]). Two recent papers [11],  [9] study how well this framework extends to solving the hidden subgroup problem for non-abelian groups (which includes the graph iso-morphism problem).

Grigni, M., Schulman, S., Vazirani, M., Vazirani, U.. Quantum Mechnical Algorithms for the Non beli n Hidden Subgroup Problem. In: Proceedings of the Thirty-third Annual ACM Symposium on the Theory of Computing Crete, Greece, 2001.

L. Hales and S. Hallgren. Quantum Fourier S mpling Simplified. In: Proceedings of the Thirty-first Annual ACM Symposium on the Theory of Computing pages 330–338,Atlanta,Georgia, 1–4 May 1999.

Hallgren, S., Russell, A., T-Shma, A.. Normal subgroup reconstruction and quantum computation using group representations. In: Proceedings of the 32nd Annual ACM Symposium on Theory of Computing 627–635, 2000.

D. Simon. On the power of quantum computation. In: Proc. 35th Symposium on Foundations of Computer Science (FOCS), 1994.

Shor P. W.. Polynomial-time algorithms for prime factorization and discrete alogarithms on a quantum computer. SIAM J. Comp., 26 No.5, pp 1484–1509, October 1997. Mihalis Yannakakis (Avaya Laboratories), Testing and Checking of Finite State Systems

Finite state machines have been used to model a wide variety of systems, including sequential circuits, communication protocols, and other types of reactive systems, i.e., systems that interact with their environment. In testing problems we are given a system, which we may test by providing inputs and observing the outputs produced.

The goal is to design test sequences so that we can deduce desired information about the given system under test, such as whether it implements correctly a given specification machine (conformance testing), or whether it satisfies given requirement properties (black-box checking).

In this talk we will review some of the theory and algorithms on basic testing problems for systems modeled by different types of finite state machines. Conformance testing of deterministic machines has been investigated for a long time; we will discuss various efficient methods. Testing of nondeterministic and probabilistic machines is related to games with incomplete information and to partially observable Markov decisions processes.

The verification of properties for finite state systems with a known structure (i.e., “white box” checking) is known as the model-checking problem, and has been thoroughly studied for many years, yielding a rich theory, algorithms and tools. Black-box checking, i.e., the verification of properties for systems with an unknown structure, combines elements from model checking, conformance testing and learning.