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A New Order Theory of Set Systems and Better Quasi-Orderings

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A New Order Theory of Set Systems and Better Quasi-Orderings 03 Progress in Informatics, No. 9, pp.9–18, (2012) 9 Special issue: Theoretical computer science and discrete mathematics Research Paper A new order theory of set systems and better quasi-orderings Yohji AKAMA1 1Mathematical Institute, Tohoku University ABSTRACT By reformulating a learning process of a set system L as a game between Teacher (presenter of data) and Learner (updater of abstract independent set), we define the order type dim L of L to be the order type of the game tree. The theory of this new order type and continuous, monotone function between set systems corresponds to the theory of well quasi-orderings (WQOs). As Nash-Williams developed the theory of WQOs to the theory of better quasi- orderings (BQOs), we introduce a set system that has order type and corresponds to a BQO. We prove that the class of set systems corresponding to BQOs is closed by any monotone function. In (Shinohara and Arimura. “Inductive inference of unbounded unions of pattern languages from positive data.” Theoretical Computer Science, pp. 191–209, 2000), for any set system L, they considered the class of arbitrary (finite) unions of members of L.Fromview- point of WQOs and BQOs, we characterize the set systems L such that the class of arbitrary (finite) unions of members of L has order type. The characterization shows that the order structure of the set system L with respect to the set inclusion is not important for the result- ing set system having order type. We point out continuous, monotone function of set systems is similar to positive reduction to Jockusch-Owings’ weakly semirecursive sets. KEYWORDS Better elasticity, continuous deformation, powerset orderings, linearization, unbounded unions, wqo 1 Introduction ory. In [22], [29], [39], sufficient conditions for set sys- A set system L over a set T, a subfamily of the pow- tems being learnable is studied with K¨onig’s lemma erset P(T), is a topic of (extremal) combinatorics[5], and Ramsey’s theorem, In [38], for a set system L, [21], as well as a target of an algorithm to learn in com- Shinohara-Arimura considered the unbounded unions <ω putational learning theory [26]. of L, that is, the class L of nonempty finite unions of A well quasi-ordering [16] (WQO for short) is, by members of L, and then they used Higman’s theorem to definition, a quasi-ordering (X, ) which has nei- study a sufficient condition for it being learnable. In [9], ther an infinite antichain nor an infinite descending de Brecht employed WQOs to calibrate mind change chain. WQOs are employed in algebra [16], combina- complexity of unbounded unions of restricted pattern torics [23], [35], formal language theory [6], [7], [14], languages. Motivated by [22], [29], [39], a somehow [32], and so on. systematic study on the relation between WQOsanda WQOs and related theorems such as Higman’s the- class of learnable set systems is done in [3], as follows: orem [16], K¨onig’s lemma and Ramsey’s theorem [5] are sometimes employed in computational learning the- (i) By reformulating a learning process of a set sys- tem L as a game between Teacher (presenter of data) Received October 31, 2011; Revised January 10, 2012; Accepted January 10, 2012. and Learner (updater of abstract independent set), we 1) [email protected] define the order type dim L of L to be the order type DOI: 10.2201/NiiPi.2012.9.3 c 2012 National Institute of Informatics 03 10 Progress in Informatics, No. 9, pp.9–18, (2012) of the game tree, if the tree is well-founded. According The class of WQOs is closed under finitary operations to computational learning theory, if an indexed family such as Higman embedding [16] and topological minor L of recursive languages has well-defined dim L then relation [13 Sect. 1.7] between finite trees [13 Ch. 12], L is learnable by an algorithm from positive data. If [23]. The class of finite graphs is a WQO under the mi- a set system has the well-defined order type, then we nor relation. Robertson-Seymour’s proof of it is given callitafinitely elastic set system (FESS for short). See in the numbers IV-VII, IX-XII and XIV-XX of their se- Definition 12. ries of over 20 papers under the common title of Graph Minors, which has been appearing in the Journal of (ii) For each quasi-ordering X = (X, ), we consider Combinatorial Theory, Series B, since 1983. For a the set system ss (X) consisting of upper-closed subsets shorter proof, see recent papers by Kawarabayashi and of X. The set system has the order type equal to the his coauthors. maximal order type [12] of X. Furthermore, the con- The class of FESSs enjoys a useful closure condition: struction ss (•) has a left-inverse qo (·). Hereforaset L M L (L) L, L Proposition 1 ( [3]) For any set systems and and system ,qo is a quasi-ordering ( )such M that any continuous deformation O : {0, 1} →L,ifM is an FESS, so is the image O [M] of M by O. x L y ⇐⇒ ∀ L ∈L, (x ∈ L =⇒ y ∈ L) . By it, we prove that for various (nondeterminis- The maximal order type otp(X)ofX is defined if and tic) language operators (e.g. Kleene-closure, shuffle- only if X is a WQO. For any quasi-ordering X, if one of product [34], [37], shuffle-closure [17], (iterated) literal otp(X) and dim ss (X) is defined then the other side is shuffle [4], union, product, intersection), the element- defined with the same ordinal number. So FESSs corre- wise application of such operator to (an) FESS(s) in- spond to WQOs. duces an FESS. Roughly speaking, a deformation transforms any ∀ (iii) For every nonempty set U, the product topologi- quasi-ordering to the powerset ordering [28] ∃. cal space {0, 1}U of the discrete topology {0, 1} is called It is through our correspondence (ss (•) , qo (·))be- a Cantor space. A subspace of a Cantor space is repre- tween quasi-orderings to set systems (Section 3). Al- sented by L, M,....WesayafunctionfromM to L is though the powerset ordering of Rado’s WQO [33] is continuous, if it is continuous with respect to the sub- not a WQO [28 Corollary 12], the class of better quasi- spaces M, L of the Cantor spaces. We identify {0, 1}U orderings [31] (BQOs for short) is closed with respect with the powerset P(U), and a function from {0, 1}U to to the powerset ordering. There are infinitary opera- {0, 1}U with a function from P(U)toP(U). We say tions under which the class of WQOs is not closed but O : M→Lis a deformation, if it is monotone (i.e. the class of BQOs is. So we introduce a better elastic M ⊆ M implies O(M) ⊆ O(M ).) If a deformation is set system (BESS for short), as a set system correspond- continuous, then it has following finiteness condition: ing to a BQO. We show that the class of BESSsisclosed under the image of any deformation (Section 3), where Lemma 1 Let O : {0, 1} M →L. any deformations are infinitary in a sense of Lemma 1. 1. O is a deformation, if and only if there is a binary By this and (i), we can develop the computational learn- relation R ⊆ ( L) × P ( M) such that ing theory of ω-languages [36]. The notion of BESS is useful in investigating a fol- ∀M ∈M∀x ∈ L lowing set system O ⇔∃ ⊆ . , . (M) x v M R(x v) (1) L<ω := { M ; ∅ M⊆L, #M < ∞}. (2) 2. O is a continuous deformation, if and only if there It is studied for the learnability of language classes such exists R ⊆ ( L) × P ( M) such that (1) holds, but as a class of regular pattern languages [38]. We charac- visafinite set whenever R(x, v) holds, and there are <ω terize the set systems L such that L are again FESSs, only finitely many such v’s for each x. ( [3]) <ω and then we prove that for every BESS L, L is an For each binary relation R ⊆ ( L) × P( M),the FESS. function O satisfying (1) is unique. So we write it by M We remark that another importance of set system OR. Conversely, every deformation O : {0, 1} →L <ω L . We conjecture that the order type of an FESS is the is written as O by a binary relation R supremum, actually the maximum, of the order types of the “linearizations” of the FESS. This conjecture corre- R := (x, v) ∈ L × P M ; O(v)(x) = 1 . sponds to a proposition useful in investigating WQOs: 03 A new order theory of set systems and better quasi-orderings 11 Proposition 2 ( [12]) The order type of a WQO is the of U such that the cardinality of X is less than α (equal <α α maximum order type of the linearizations of the WQO. to α, resp.) is denoted by [U] ([U] , resp.). A set X ⊆ ω is often identified with the sequence enumerating it L A “linearization” of an FESS seems to be a subfamily in a strictly increasing order. of L<ω. <ω We hope that our study on FESSsandBESSsareuse- Definition 1 1. We say B ⊆ [ω] abarrier,if(1)aset ful in solving problems (e.g. decision problem of timed B is infinite; (2) for all σ ∈ [ B]ω there exists Petri-nets [1], [2], the multiplicative exponential linear s ∈ B such that s is a prefix of σ; and (3) for all logic [11]) which are related to WQOsandBQOsbut s, t ∈ B, s t.
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