Three-element Codes Research Articles

On the proportion of prefix codes in the set of three-element codes

Let L be a finite sequence of natural numbers. In Woryna (2017, 2018), we derived some interesting properties for the ratio ρn,L=|PRn(L)|∕|UDn(L)|, where UDn(L) denotes the set of all codes over an n-letter alphabet and with length distribution L, and PRn(L)⊆UDn(L) is the corresponding subset of prefix codes. In the present paper, we study the case when the length distributions are three-element sequences. We show in this case that the ratio ρn,L is always greater than αn, where αn=(n−2)∕n for n>2 and α2=1∕6. Moreover, the number αn is the best possible lower bound for this ratio, as the length distributions of the form L=(1,1,c) and L=(1,2,c) assure that the ratios asymptotically approach αn. Namely, if L=(1,1,c), then ρn,L tends to (n−2)∕n with c→∞, and, if L=(1,2,c), then ρ2,L tends to 1∕6 with c→∞.

RATIONAL CHOICE OF MACHINING TOOLS USING PREDICTION PROCEDURES

Introducing the methods and procedures for predictive analysis into the design process contours of a variety of machining tools (MT) of metal cutting machines is the main aim of this article. A sequence of realization of prediction object (PO) choice as an initial stage of search of perspective designs is offered. Effective in this regard is the "Tree of objectives" apparatus, on the basis of which many ways of improving MT are formed, selecting progressive (reducing the dimension of the problem) at each level of the hierarchy of the constructed graph-tree. The procedure for selecting the prediction method (PM) as a means of generating the forecast data is developed. The task of choosing a method is structured in detail and uses "Information supply"as the main criterion. To this end, assessment scales of choice criteria have been formed, on the basis of which it is possible to evaluate their effectiveness for the PM selection process. The rules forPOcoding are introduced by a three-element information code, including information source classes – static data, expert estimates and patent data. The process of forecasting the MT components by the method of engineering forecasting on the basis of a representative patent fund is realized. The General Definition Table has been built (GDT "Machining tools") and estimates of the prospects of design solutions have been obtained. A fragment of the database of 3D models of promising MT designs in the integrated computer-aided design KOMPAS-3D is proposed.

A first step in characterizing three-element codes

It is always an interesting subject to investigate whether a three-element language is a code or not. In this paper, we consider a special class of three-element languages, where two words have the same length which is less than the length of the third word. We give a necessary and sufficient condition to state whether a three-element language in this class is a code. This result partially resolves the problem proposed by Professor H. J. Shyr in 1990s.

Three-element codes with one d-primitive word

In the field of combinatorics on words, it is known that {u, v}, the language with two words is a code if and only if $uv\neq vu$ . But up to now general characterization for codes consisting of any three words has not been found. In 1998, Fan, Shyr and Yu provided a characterization for codes consisting of three d-primitive words. In this paper, we give characterizations for a code with three words in which one of the words is d-primitive.

Conway's problem for three-word sets

We prove two results on commutation of languages. First, we show that the maximal language commuting with a three-element language, i.e. its centralizer, is rational, thus giving an affirmative answer to a special case of a problem proposed by Conway in 1971. Second, we characterize all languages commuting with a three-element code. The characterization is similar to the one proved by Bergman for polynomials over noncommuting variables (see Trans. Am. Math. Soc. 137 (1969) 327 and Algebraic Combinatorics on Words, Cambridge University Press, Cambridge, 2000): A language commutes with a three-element code X if and only if it is a union of powers of X.

On codes, ω-codes and ω-generators

Abstract Up to now, the question “given a rational language R, how to decide if there exists an ω-code C such that Cω = Rω?” remains with no general solution. We give an answer assuming that the greatest ω-generator of Rω exists and is a free submonoid. In the case where the greatest generator is generated by a three-element code the question was recently solved (Julia, 1996). We extend the result to the whole class of rational codes.

On three-element codes

We show that three-element codes have some special properties which do not hold even for four-element codes. Firstly, for each three-element code A, if u and v are words in pref( xA ω ) ∩ pref( yA ω ), with x, y ϵ A , x ≠ y, then one of them is a prefix of the other, i.e., among the words which can be covered in two different ways from left to right there exists a unique maximal (possibly infinite) element. Secondly, each three-element code has a bounded delay in at least one direction.

A property of three-element codes

Let w be a word and A a language of a finitely generated free monoid Σ ∗ . We say that w is ambiguously covered by A if there exist words α and β in A, with α ≠ β, such that w ϵ pref( αA +) ∩ pref( βA +). We show that if A is a three-element code, then any two words which are ambiguously covered by A are comparable, i.e., one of them is a prefix of the other. This property is characteristic for three-element codes.

On three-element codes

On three-element codes