György Turán

Threshold circuits of bounded depth

Abstract We examine a powerful model of parallel computation: polynomial size threshold circuits of bounded depth (the gates compute threshold functions with polynomial weights). Lower bounds are given to separate polynomial size threshold circuits of depth 2 from polynomial size threshold circuits of depth 3 and from probabilistic polynomial size circuits of depth 2. With regard to the unreliability of bounded depth circuits, it is shown that the class of functions computed reliably with bounded depth circuits of unreliable ∨, ∧, ¬ gates is narrow. On the other hand, functions computable by bounded depth, polynomial-size threshold circuits can also be computed by such circuits of unreliable threshold gates. Furthermore we examine to what extent imprecise threshold gates (which behave unpredictably near the threshold value) can compute nontrivial functions in bounded depth and a bound is given for the permissible amount of imprecision. We also discuss threshold quantifiers and prove an undefinability result for graph connectivity.

On the complexity of cutting-plane proofs

Abstract As introduced by Chvatal, cutting planes provide a canonical way of proving that every integral solution of a given system of linear inequalities satisfies another specified inequality. In this note we make several observations on the complexity of such proofs in general and when restricted to proving the unsatisfiability of formulae in the propositional calculus.

On the succinct representation of graphs

Abstract It is shown that unlabeled planar graphs can be encoded using 12 n bits, and an asymptotically optimal representation is given for labeled planar graphs.

Lower Bound Methods and Separation Results for On-Line Learning Models

We consider the complexity of concept learning in various common models for on-line learning, focusing on methods for proving lower bounds to the learning complexity of a concept class. Among others, we consider the model for learning with equivalence and membership queries. For this model we give lower bounds on the number of queries that are needed to learn a concept classC in terms of the Vapnik-Chervonenkis dimension ofC, and in terms of the complexity of learningC with arbitrary equivalence queries. Furthermore, we survey other known lower bound methods and we exhibit all known relationships between learning complexities in the models considered and some relevant combinatorial parameters. As it turns out, the picture is almost complete. This paper has been written so that it can be read without previous knowledge of Computational Learning Theory.

On the communication complexity of graph properties

We prove <italic>&thgr;</italic>(<italic>n</italic> log <italic>n</italic>) bounds for the deterministic 2-way communication complexity of the graph properties CONNECTIVITY, <italic>s</italic>-<italic>t</italic>-CONNECTIVITY and BIPARTITENESS (for arbitrary partitions of the variables into two sets of equal size). The proofs are based on combinatorial results of Dowling-Wilson and Lovász-Saks about partition matrices using the Möbius function, and the Regularity Lemma of Szemerédi. The bounds imply improved lower bounds for the VLSI complexity of these decision problems and sharp bounds for a generalized decision tree model which is related to the notion of evasiveness.

Algorithms and Lower Bounds for On-Line Learning of Geometrical Concepts

The complexity of on-line learning is investigated for the basic classes of geometrical objects over a discrete (“digitized”) domain. In particular, upper and lower bounds are derived for the complexity of learning algorithms for axis-parallel rectangles, rectangles in general position, balls, half-spaces, intersections of half-spaces, and semi-algebraic sets. The learning model considered is the standard model for on-line learning from counterexamples.

Searching in trees, series-parallel and interval orders

Linial and Saks [2] have shown that

On the definability of properties of finite graphs

O(\log N)

On frequent sets of Boolean matrices

evaluations of an order preserving map

On linear decision trees computing Boolean functions

f:p \to \mathbb{R}