Infinite Strings Generated by Insertions

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1 Programming and Computer Software, Vol. 30, No. 2, 2004, pp Translated from Programmirovanie, Vol. 30, No. 2, Original Russian Text Copyright 2004 by Golubitsky, Falconer. Infinite Strings Generated by Insertions O. D. Golubitsky* and S. Falconer** *Department of Mechanics and Mathematics, Moscow State University, Vorob evy gory, Moscow, Russia **Faculty of Computer Science, University of New Brunswick, 540 Windsor St., Fredericton, E3B5A3, Canada Received June 5, 2003 Abstract The process of generation of strings over a finite alphabet by inserting characters at an arbitrary place in the string is considered. A topology on the set of strings is introduced, in which closed sets are defined to be sets of strings that are invariant with respect to the insertion of characters. An infinite insertion string is defined to be a set of infinite sequences of insertions ending in the same open sets. The number of infinite insertion strings is proved to be countable. It is proved also that there is a relationship between the countability of the completion of partially well-ordered sets by infinite elements and the fulfillment of an analogue of Dickson s and Higman s lemmas for them. 1. INTRODUCTION This work emerged as a result of the authors research on the development of a model of structural data representation known as the Evolving Transformation Systems (ETS) model [1, 2]. Strings over finite alphabets are a particular case (which, however, plays an important role in mathematics and applications) of structural representations described by this model. The applications of strings are so numerous that several terms exist for example, words or sequences of characters that denote the same obects. In our opinion, the latter fact is not occasional; it bears evidence of the existence of several very different interpretations of one obect, namely, a string over a finite alphabet. In particular, there may be defined various operations on strings, such as concatenation, insertion, replacement, deletion, symmetric reflection, and so on. We believe that definitions of the operations are closely related to the process of string construction (or generation), which is often assumed only implicitly. Moreover, the choice of the generation process affects the extension of the notion of a finite string to the infinite case. In this paper, we show that, if characters can be inserted into any place of the string, the set of the corresponding infinite strings is countable, whereas, if the strings are generated by adding characters only to the end of the string, the set obtained is a set of infinite sequences and, thus, has continuum cardinality. Examples of the generation processes for sets of positive integers, monomials, strings, and trees are presented, and the question of countability of the completion of these sets by infinite obects is studied. The generation process naturally defines a partial order relation on the obects being generated: an obect A is less than an obect B if B can be generated from A. It turns out that the cardinality of a set of infinite obects is related to the so-called well quasi orderedness property of this relation [3]. A partially ordered set is said to be well quasi ordered if any of its subsets has only a finite number of minimal elements. In particular, according to the well-known Dickson lemma [4], this condition holds for the set of monomials in a finite number of variables ordered by the divisibility relation; according to Higman s lemma [3, 5], it holds also for strings under the assumption that a string u is less than a string v if u is a subsequence (not necessarily contiguous) of v. The main result of this work is a theorem stating that, for a countable partially ordered set, the properties of being well quasi ordered and having a countable completion by infinite obects are equivalent. 2. TOPOLOGY ON THE SET OF STRINGS Let Σ be a finite alphabet and Σ* be a set of strings over it. The partial order relation defined below corresponds to the process of string generation by inserting characters at any place in a string. Definition 1. If v = a 1 a m, u = a i1 a ik Σ*, where 1 i 1 < < i k m, then the string u is called an ancestor of the string v, and the string v is called a descendant of u, which is denoted as u v. One of the ways to complete a set by infinite elements is to introduce a topology on the set. We introduce the topology on the set of strings that corresponds to the partial order relation defined above. Definition 2. A subset X Σ* is said to be closed if, for each string u X, it contains also all its descendants. A subset u X is said to be open if its complement X is closed. Lemma 1 (properties of open and closed sets). 1. A set Y is open if and only if, for any string u Y, all its ancestors also belong to Y /04/ åäiä Nauka /Interperiodica

2 INFINITE STRINGS Intersection and union of any number of open (closed) sets are open (closed). Proof. 1. Let Y be open and u Y. Suppose that v u and v Y. Then, v belongs to the set Σ*\Y, which is closed by definition. Hence, all descendants of v also belong to Σ*\Y, which contradicts the condition u Y. Conversely, let all descendants of u belong to the set Y for all u Y. Let us show that the set Σ*\Y is closed. Indeed, if v Σ*\Y, then none of the descendants of v can belong to Y, since, in this case, Y would have to contain not only this descendant but also its ancestor v, which is impossible. 2. Let {X α } α A be a family of open (closed) sets. Let u X, then u X α for all α A. By the definition of a closed set, it follows from the first assertion α A α that each descendant (ancestor) of u also belongs to X α for all α A; hence, each descendant (ancestor) of u belongs to X. The case of the union operation α A α is proved similarly or by means of the formula X α α A c =, c X α α A where the superscript c stands for the complement. To describe closed sets, we introduce the following definition. Definition 3. A closure of a subset X Σ* is the intersection of all closed sets containing X and is denoted as X. Lemma 2. Any closed set is the closure of some finite set. Proof. In accordance with Higman s lemma [5], the set of minimal elements (with respect to ) of any nonempty set X is finite. We denote this set as min(x) and prove that min( X) = X. Let us assume the contrary, i.e., Y = X\ min( X). Since Y Σ* and Y, Y contains, at least, one minimal element y Y. Then, y is minimal in X (otherwise, there exists z X\Y = min( X) such that z y, and, hence, y min( X), which contradicts the assumption). However, y min(x), which contradicts the definition of the set min(x). Corollary 1. Closed and open sets are regular sets. This corollary allows us to specify closed and open sets in a compact form. It should be noted, however, that these sets do not exhaust all possible regular sets. Example 1. The set Σ*\ { ab} is open and is determined by the regular expression b*a*. 3. PATHS The concept of a path, which is defined below, is similar to that of a monotone sequence of numbers. In what follows, infinite insertion strings are defined as equivalence classes of paths, similarly to the definition of real numbers as equivalence classes of monotonically increasing sequences of rational numbers. Definition 4. A path is a sequence of strings monotonically increasing with respect to. If a path p is finite, its last element is denoted by term(p). Definition 5. A path p = {u 1, u 2, } ends in a set X if there exists N such that u i X for all i N. Lemma If a set X is closed and up X, then p ends in X. 2. If a set Y is open and p ends in Y, then p Y. Proof. 1. Let X be closed, p = {u 1, u 2, }, and u i X for some i. Then, by the definition of a path, for > i, u s u i ; hence, u X. 2. Let Y be open and p Y. Then, there exists an element p belonging to the complement of Y, which is a closed set. As proved above, this implies that p ends in the complement of Y, which is impossible. Definition 6. A path p = {u 1, u 2, } is called an ancestor of a path q = {v 1, v 2, } if, for any u i, there exists v such that u i v. In this case, we write p q. Lemma The relation on the set of paths is reflexive and transitive. 2. For finite paths p and q, p q term(p) term(q). Proof. 1. The reflexivity is evident. Let p, q, and r be paths, and let p q and q r. Then, for any element u i from p, there exists a descendant v from q, for which, in turn, there exists a descendant w k from r. Since the relation on strings is a partial order relation, we have u i w k and, hence, p r. 2. Let p q. Then, for any element p, in particular, for term(p), there exists a descendant v in q. However, v term(q); therefore, term(p) term(q). Conversely, if term(p) term(q), then, for any element u i from p, we have u i term(p) term(q), which means that p q. Example A path consisting of an empty string λ is an ancestor of any other path. 2. The path (ab) = {λ, a, ab, aba, abab, ababa, ababab, } is a descendant of any other path.

3 112 GOLUBITSKY, FALCONER (ab) aa ab ba bb a... λ Fig. 1. A diagram for the set of insertion strings. 4. EQUIVALENCE OF PATHS Definition 7. Paths p and q are said to be equivalent if one of the following equivalent conditions is fulfilled: 1. For any closed set X, (p ends in X) (q ends in X). 2. For any open set Y, (p ends in Y) (q ends in Y). 3. p q p. The equivalence of the paths is written as p ~ q. Lemma 5. The conditions specified in Definition 7 are equivalent. Proof. (1 2): Let, for any closed set X (p ends in X) (q ends in X). Let Y be open and p end in Y. Then, according to Lemma 3, p Y, and, hence, p does not end in Y c. In this case, q does not end in Y c either. Hence, by virtue of Lemma 3, q Y; i.e., q ends in Y. (2 3): Let, for any open set Y (p ends in Y) (q ends in Y). Suppose that there exists an element u i from p such that, for all v from q, u i v. Consider the open set Y = { u i } c. We have p Y; hence, by Lemma 4, p does not end in Y, and, thus, q does not end in Y, and q Y. Then, there exists an element v from q that belongs to the complement of Y, i.e., to the closed set { u i }. This implies that u i v, and we arrive at the contradiction. Hence, p q and, similarly, q p. (3 1): Let p q p. Let p end in the closed set X, and let u i be an element of p belonging to X. Since p q, there exists an element v from q such that u i v. Then, v X, and all subsequent elements of q also belong to X; i.e., q ends in X. The converse implication is proved similarly. Example 3. The following paths are mutually equivalent: ( ab) = { λ, a, ab, aba, abab, ababa, ababab, }, b ( aab) = { λ, a, aa, aab, aaba, aabaa, aabaab, }, ( a b ) = { λ, ab, aabbab, aaabbbaabbab, aaaabbbbaaabbbaabbab }. Lemma 6 (ustification of the definition). For finite paths, p ~ q term(p) = term(q). Proof. Let p ~ q. Then, p q p, which, by Lemma 4, implies that term(p) term(q) term(p). Hence, it follows that term(p) = term(q). Conversely, let term(p) = term(q). Then, term(p) term(q) term(p), which, by Lemma 4, implies that p q p; i.e., p ~ q. 5. INSERTION STRINGS Definition 8. An insertion string [p] is a path equivalence class with respect to ~. This definition is suitable both for the case of finite paths (in this case, we obtain an ordinary finite string) and the case of infinite paths. Theorem 1. The set of insertion strings is countable. Proof. We will prove the theorem for the case of an alphabet consisting of two characters. In the case of an arbitrary finite alphabet, the proof is similar. Let u be an insertion string. There exist two following possibilities: 1. (ab) u. Then, u = [(ab) ]. 2. (ab) u. In this case, there exists a maximal number n such that (ab) n is an ancestor of a certain string from u, and, for any string v from any path p u, Let v b k 1 = K i a k 2 b k 3 a k 2n b k 2n + 1 a k 2n + 2, k i 0. ( p) = max( k i ) { 0, 1, } { }. Then, p q K 1 ( p) = K 1 ( q),, K 2n + 2 ( p) = K 2n + 2 ( q), and the number of all sets of the form (K 1,, K 2n + 2 ) is countable. In the next section, we will prove a more general theorem. In the framework of this theorem, according to Higman s lemma, the insertion strings are a particular case. The above proof makes it possible to prove Higman s lemma independently for the particular case of a finite alphabet. A partially ordered set of insertion strings can be represented by means of the diagram shown in Fig. 1.

4 INFINITE STRINGS THE PROPERTY OF BEING WELL QUASI ORDERED Lemma 7 (Dickson). For any infinite sequence of monomials {u 1, u 2, } over a finite set of variables, there exist two indices i < such that u i u. Lemma 8 (Higman). For any infinite sequence of strings {u 1, u 2, } over a finite alphabet, there exist two indices i < such that u i u. The notions of an open and closed subset, closure, path, partial order and equivalence relations on the set of paths are easily extended from the case of strings over finite alphabets to the case of an arbitrary countable partially well-ordered set (S, ) (under a partially well-ordered set, we mean a partially ordered set in which any nonempty subset has a minimal element). It is not difficult to see that Lemmas 1, 3 6 remain valid in this case. Lemma 2 is valid if and only if the following condition holds. Definition 9. A countable partially well-ordered set (S, ) is a well quasi ordered set if, for any infinite sequence {s 1, s 2, }, there exist two indices i < such that u i u. Theorem 2. If (S, ) is a well quasi ordered set, then the number of equivalence classes of paths in S is countable. Proof. The property of being well quasi ordered implies that any closed subset in S is a closure of a certain finite set. Therefore, the number of all closed subsets is countable. For each path p, we consider the closed set U p that is the union of all closed sets X in which p does not end. Then, p q U p = U q. Indeed, let p ~ q. Then, p and q, by definition, end in the same closed sets; hence, U p = U q. Conversely, let U p = U q. If p does not end in a closed set X, then, by Lemma 3, p X =, and, therefore, p U p =. Similarly, q U q =. Since U p = U q, we have q U p =. Let X be a closed set in which p does not end. Then X U p ; hence, q X =. This, in particular, implies that q does not end in X. Since the number of all closed subsets is countable, the number of all sets of the form U p is also countable (since these sets are unions of closed subsets and, thus, are closed). Therefore, the number of equivalence classes of paths is countable. In accordance with Lemma 6, elements of the countable partially well-ordered set (S, ) can be identified with equivalence classes of finite paths. The union of S and the set of all equivalence classes of infinite paths in S is referred to as the completion of S, and the equivalence classes of infinite paths are called infinite elements of the completion. In this case, Theorem 2 can be stated as follows: the completion of a countable well quasi ordered set is countable. 7. INVERSE THEOREM Denote by Fin(S) the set of finite subsets of the set S. If the set S is partially ordered by the relation, then this relation can naturally be extended to Fin(S): for S 1, S 2 S, we assume that S 1 S 2 if there exists an inection f: S 1 S 2 such that s f(s) for all s S 1. Theorem 3. Let (S, ) be a partially well-ordered set, and let the completion of the set Fin(S) be countable. Then, (S, ) is a well quasi ordered set. Proof. Suppose that the assertion of the theorem is not true. Let us select an infinite subset X S all elements of which are minimal. Then, all possible subsets of X correspond to various elements of the completion of the set Fin(S), and the set of all subsets of X has continuum cardinality. Theorem 4 (criterion). A countable partially wellordered set (S, ) is well quasi ordered if and only if the completion of the set Fin(S) is countable. Proof. Let (S, ) be a Dickson set. Let us show that Fin(S) is also well quasi ordered. Indeed, according to the general Higman s lemma [3, 5, 6], if (S, ) is well quasi ordered, then the set S* of finite ordered subsets of S is also well quasi ordered. Note that the partial order on S* is defined as follows: s 1 s m t 1 t n if an only if there exists a monotonically increasing function f: {1,, m} {1,, n} such that s i t f(i) for all i {1,, m}. It is easy to see that s 1 s m t 1 t n implies {s 1,, s m } {t 1,, t n }. Let A 1, A 2, be an infinite sequence in Fin(S). Let us fix an arbitrary linear order on each set A i ; thus, we obtain an infinite sequence A 1 ', A 2 ', in S*. Since S* is well quasi ordered, there exist indices i < such that A i ' A'. Then, it follows that A i A, and, hence, Fin(S) is well quasi ordered. The converse assertion of the criterion is valid by virtue of Theorem EXAMPLES OF COMPLETIONS 1. Positive integers: (a) =, one infinite element + ;

5 114 GOLUBITSKY, FALCONER,,,,... Fig. 2. The infinite set of trees that violates the property of being well quasi ordered. (b) =, continuum of infinite elements; the property of being well quasi ordered is violated for the set of prime numbers. 2. Monomials over a finite set of variables: =, a countable number of infinite monomials; Dickson s lemma is valid. 3. Strings over a finite alphabet: (a) u v w Σ* v = uw; continuum of infinite elements; the property of being well quasi ordered is violated for the set b, ab, aab, aaab, ; (b) u v x, y Σ* v = xuy; continuum of infinite elements; the property of being Dickson is violated for the set aa, aba, abba, abbba, ; (c) u v u is a subsequence of v; a countable number of infinite strings; Higman s lemma is valid. 4. Unlabelled forests: T 1 T 2 T 1 is a subgraph of T 2. The set of infinite forests has continuum power, since the set of trees depicted in Fig. 2 violates the property of being well quasi ordered. 9. UNSOLVED PROBLEMS AND DIRECTIONS OF FUTURE RESEARCH To conclude the paper, we list some problems that are, in our opinion, of interest and continue the study initiated in this paper: 1. Classification of countable topological spaces. 2. Study of continuous mappings between them. 3. Computations in countable topological spaces. 4. Relationship between the process of obect construction and admissible operations on them (for example, systems of rewriting rules). 5. Investigation of random processes in countable topological spaces. REFERENCES 1. Goldfarb, L., Golubitsky, O., and Korkin, D., What Is a Structural Representation? Techn. Report TR00-137, Faculty Comput. Sci., U.N.B., Golubitsky, O., On the Generating Process and the Class Typicality Measure, Techn. Report TR02-151, Faculty Comput. Sci., U.N.B., Dershowitz, N. and Jouannaud, J.-P., Rewrite Systems, Handbook of Theoretical Computer Science, van Leeuwen, J., Ed., Amsterdam: North-Holland, 1990, vol. B, ch. 6, pp Cox, D., Little, J., and O Shea, D., Ideals, Varieties, and Algorithms. An Introduction to Computational Algebraic Geometry and Commutative Algebra, New York: Springer, Translated under the title Idealy, mnogoobraziya i algoritmy, Moscow: Mir, Higman, G., Ordering by Divisibility in Abstract Algebras, Proc. London Math. Soc., 1952, vol. 2, no. 7, pp Murthy, C. and Russell, J.R., A Constructive Proof of Higman s Lemma, LICS, 1990, pp