Micha A. Perles

The Theory of Definite Automata

A definite automaton is, roughly speaking, an automaton (sequential circuit) with the property that for some fixed integer k its action depends only on the last k inputs. The notion of a definite event introduced by Kleene, as well as the related concepts of definite automata and tables, are studied here in detail. Basic results relating to the minimum number of states required for synthesizing an automaton of a given degree of definiteness are proved. We give a characterization of all k-definite events definable by k+1 state automata. Various decision problems pertaining to definite automata are effectively solved. We also solve effectively the problem of synthesizing a minimal automaton defining a given definite event. The solutions of decision and synthesis problems given here are practical in the sense that if the problem is presented by n units of information, then the algorithm in question requires about n3 steps of a very elementary nature (rather than requiring about 2n steps as some algorithms for automata do, which puts them beyond the capacity of the largest computers even for relatively small values of n). A notion of equivalence of definite events is introduced and the uniqueness of the minimal automaton defining an event in an equivalence class is proved.

The rooted tree embedding problem into points in the plane

In this paper we show that any rooted tree ofn vertices can be straight-line embedded into any setS ofn points in the plane in general position so that the image of the root is arbitrarily specified.

A proof of dilworth’s decomposition theorem for partially ordered sets

A short proof of the following theorem is given: LetP be a finite partially ordered set. If the maximal number of elements in an independent subset ofP isk, thenP is the union ofk chains.

On the Number of Maximal Regular Simplices Determined by n Points in Rd

A set V = {x 1,…, x n } of n distinct points in Euclidean d-space ℝ d determines 2 n distances ∥x j − x i ∥ (1 ≤ i < j ≤ n). Some of these distances may be equal. Many questions concerning the distribution of these distances have been asked (and, at least partially, answered). E.g., what is the smallest possible number of distinct distances, as a function of d and n? How often can a particular distance (say, one) occur and, in particular, how often can the largest (resp., the smallest) distance occur?

Quotient polytopes of cyclic polytopes part I: Structure and characterization

We investigate the quotient polytopesC/F, whereC is a cyclic polytope andF is a face ofC. We describe the combinatorial structure of such quotients, and show that under suitable restrictions the pair (C, F) is determined by the combinatorial type ofC/F. We describe alternative constructions of these quotients by “splitting vertices” of lower-dimensional cyclic polytopes. Using Gale diagrams, we show that every simpliciald-polytope withd+3 vertices is isomorphic to a quotient of a cyclic polytope.

Extremal theory for convex matchings in convex geometric graphs

AbstractA convex geometric graphG of ordern consists of the set of vertices of a plane convexn-gonP together with some edges, and/or diagonals ofP as edges. CallG 1-free ifG does not havel disjoint edges in convex position.We answer the following questions:(a)What is the maximum possible number of edges ofG ifG isl-free (as a function ofn andl)?(b)What is the minimum possible number of edges ofG ifG isl-free and saturated, i.e., ifG∪{e} is notl-free for any edge or diagonale ofP that is not, already inG.. We also fully describe the graphsG where the maximum (in (a)) or the minimum (in (b)) is attained. Then we remove the word “disjoint” from the definition of “l-free” and do the same over again. The results obtained are quite similar and closely related to the corresponding results (Turáns theorem, etc) in extremal abstract graph theory.

A closed (n + 1)-convex set inR 2 is a union ofn 6 convex sets

It is shown that if a closed setS in the plane is (n+1)-convex, then it has no more thann4 holes. As a consequence, it can be covered by≤n6 convex subsets. This is an improvement on the known bound of 2n·n3.

Staircase Connected Sets

A compact set

Blockers for Noncrossing Spanning Trees in Complete Geometric Graphs

S \subset {\Bbb R}^2

On the intersection of edges of a geometric graph by straight lines

is staircase connected if every two points