## The Complexity of Finding SUBSEQ(A)

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@MISC{Fenner_thecomplexity,

author = {Stephen Fenner and Brian Postow and et al.},

title = {The Complexity of Finding SUBSEQ(A)},

year = {}

}

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### Abstract

Higman showed that if A is any language then SUBSEQ(A) is regular. His proof wasnonconstructive. We show that the result cannot be made constructive. In particular we show that if f takes as input an index e of a total Turing Machine Me, and outputs a DFA forSUBSEQ(L(M e)), then;00 ^T f (f is \Sigma 2-hard). We also study the complexity of going from Ato SUBSEQ(A) for several representations of A and SUBSEQ(A).

### Citations

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Citation Context ...ill be oracle Turing machines, and the oracle can be considered to be the input; the Turing machine will output answers from time to time. This is similar to the Inductive Inference model of learning =-=[6, 7, 15]-=-, and we study the task of learning SUBSEQ(A) more fully in another paper [10]. Definition 6.1 Let M () be an oracle Turing Machine and A an oracle. 1. M A ↓= e means that M A , when run, will run for... |

503 |
Recursively enumerable sets and degrees
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Citation Context ... {REG, NROCA, coNROCA, CFL, coCFL, P, CE}, then F C,CE is computable. 2. If D ∈ {REG, NROCA, coNROCA, CFL, coCFL, P, DEC, CE}, then F REG,D is computable. The notation below is standard. reference is =-=[26]-=-. For the notation that relates to computability theory, our Notation 2.19 Let e, s ∈ N and x ∈ Σ ∗ . 1. The empty string is denoted by λ. 2. Me,s(x) is the result of running Me on x for s steps. 3. W... |

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Citation Context ...ee. 3. Show that if A is c.e. 1 then SUBSEQ(A) is c.e. What happens if A is decidable? Clearly if A is decidable then SUBSEQ(A) is c.e. But is SUBSEQ(A) decidable? A corollary of a theorem of Higman (=-=[19]-=- but also see the appendix for his proof and a new proof) supplies far more information: If A is any language whatsoever, then SUBSEQ(A) is regular. ∗ University of South Carolina, Department of Compu... |

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Citation Context ...ill be oracle Turing machines, and the oracle can be considered to be the input; the Turing machine will output answers from time to time. This is similar to the Inductive Inference model of learning =-=[6, 7, 15]-=-, and we study the task of learning SUBSEQ(A) more fully in another paper [10]. Definition 6.1 Let M () be an oracle Turing Machine and A an oracle. 1. M A ↓= e means that M A , when run, will run for... |

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Citation Context ... let f(e) be the index of a CFG or DFA recognizing Σ ∗ . 9 Relation to Recursive and Reverse Mathematics Nerode’s Recursive Mathematics Program [20, 8] and Simpson and Friedman’s Reverse Math Program =-=[23, 24]-=- both attempt to pin down what it means for a proof to be noneffective or nonconstructive. Corollary 3.4 yields results in both of these programs. Recall Higman’s result, which we denote by H: If A ⊆ ... |

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Citation Context ...ill be oracle Turing machines, and the oracle can be considered to be the input; the Turing machine will output answers from time to time. This is similar to the Inductive Inference model of learning =-=[6, 7, 15]-=-, and we study the task of learning SUBSEQ(A) more fully in another paper [10]. Definition 6.1 Let M () be an oracle Turing Machine and A an oracle. 1. M A ↓= e means that M A , when run, will run for... |

54 | An almost optimal algorithm for unbounded searching
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Citation Context ... be a function of the output. The final upshot will be that this problem has the exact same results as the unbounded search problem. 5.1 Unbounded Search The material in this subsection is taken from =-=[5, 2]-=-. The base of the log function is 2 throughout. Definition 5.1 The Unbounded Search Problem is as follows. Alice has a natural number n (there are no bounds on n). Bob is trying to determine what n is... |

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Citation Context ...lem can be solved with h(n) queries iff h satisfies Kraft’s inequality. 165.2 Definitions, Notation, and Lemmas from Bounded Queries The Definitions and Notations in this section are originally from =-=[1, 4]-=-, but are also in [14]. We only touch on the parts of bounded queries that we need; there are many variants on these definitions. Definition 5.7 Let A be a set, and let n ≥ 1. 1. C A n : N n → {0, 1} ... |

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Citation Context ...m.org. 1 The notation “c.e.” means “computably enumerable” and is used in this paper to denote what are otherwise called r.e. (recursively enumerable) sets. Our notation follows a suggestion of Soare =-=[27]-=-. 2 Two proofs of Higman’s theorem—his proof and a new proof—are included as appendices to an on-line draft of this paper, available from the first author’s homepage. 11. Show that if A is decidable ... |

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Citation Context ...dex of a CFG or DFA recognizing SUBSEQ(D); otherwise, we let f(e) be the index of a CFG or DFA recognizing Σ ∗ . 9 Relation to Recursive and Reverse Mathematics Nerode’s Recursive Mathematics Program =-=[20, 8]-=- and Simpson and Friedman’s Reverse Math Program [23, 24] both attempt to pin down what it means for a proof to be noneffective or nonconstructive. Corollary 3.4 yields results in both of these progra... |

26 |
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(Show Context)
Citation Context ...unctions that take an index i for a machine in C and produce an index j for a machine in D such that L(Dj) = SUBSEQ(L(Ci)). We use the notion of Muchnik reducibility (also known as weak reducibility) =-=[22]-=- to measure the complexity of the various F C,D . Definition 2.10 (after Muchnik [22]) Let F and G be classes of functions N → N. We say that F Muchnik reduces to G (denoted F ≤w G) 2 if (∀g ∈ G)(∃f ∈... |

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Citation Context ... {REG, NROCA, coNROCA, CFL, coCFL, P, CE}, then F C,CE is computable. 2. If D ∈ {REG, NROCA, coNROCA, CFL, coCFL, P, DEC, CE}, then F REG,D is computable. The notation below is standard. reference is =-=[26]-=-. For the notation that relates to computability theory, our Notation 2.19 Let e, s ∈ N and x ∈ Σ ∗ . 1. The empty string is denoted by λ. 2. Me,s(x) is the result of running Me on x for s steps. 3. W... |

17 | On the complexity of finding the chromatic number of a recursive graph I: The bounded case, Ann. Pure Appl. Logic 45
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Citation Context ...ed for certain problems. They will provide problems to reduce to in order to get lower bounds. Lemma 5.15 ([4]) # K n−1 ̸∈ EN(n − 1) The following lemma is a modification of a similar lemma from both =-=[3]-=- and [13]. It will enable us to prove lower bounds on the number of queries certain functions require to compute. Lemma 5.16 Let X ⊆ N, and h : N → N be any function. Let γ : N → N. If γ ∈ FQ(h(γ(x)),... |

16 |
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Citation Context ...counter machines were shown to be universal by Minsky [21] (improved by Fischer [11]). Deterministic one-counter automata (DOCAs) were first defined and their properties studied by Valiant & Paterson =-=[28]-=-. Real-time counter machines were studied by Fischer, Meyer, & Rosenberg [12]. ROCAs were introduced in the context of machine learning by Fahmy & Roos [9]. 2.3 Rectangular Traces The concepts in this... |

15 |
verbose sets
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Citation Context ...lem can be solved with h(n) queries iff h satisfies Kraft’s inequality. 165.2 Definitions, Notation, and Lemmas from Bounded Queries The Definitions and Notations in this section are originally from =-=[1, 4]-=-, but are also in [14]. We only touch on the parts of bounded queries that we need; there are many variants on these definitions. Definition 5.7 Let A be a set, and let n ≥ 1. 1. C A n : N n → {0, 1} ... |

15 |
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Citation Context ...er [11]). Deterministic one-counter automata (DOCAs) were first defined and their properties studied by Valiant & Paterson [28]. Real-time counter machines were studied by Fischer, Meyer, & Rosenberg =-=[12]-=-. ROCAs were introduced in the context of machine learning by Fahmy & Roos [9]. 2.3 Rectangular Traces The concepts in this section will be used for the proofs of Lemma 4.4 and Theorem 4.5. We’ll use ... |

13 |
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Citation Context ...mpty marker), and at most one symbol pushed or popped at a time. Counter machines have been studied by many people. Two-counter machines were shown to be universal by Minsky [21] (improved by Fischer =-=[11]-=-). Deterministic one-counter automata (DOCAs) were first defined and their properties studied by Valiant & Paterson [28]. Real-time counter machines were studied by Fischer, Meyer, & Rosenberg [12]. R... |

13 |
Introduction to the Theory of Computation, 2nd ed
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Citation Context ...ctangular Traces The concepts in this section will be used for the proofs of Lemma 4.4 and Theorem 4.5. We’ll use a standard notion of a Turing machine with a single one-way infinite tape. See Sipser =-=[25]-=- for details. We do not need to assume that M1, M2, M3, . . . all share the same input alphabet, but we will assume WLOG that 0 belongs to the input alphabets of all the Me. Fix a deterministic Turing... |

10 |
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Citation Context ...es to nondeterministic real-time onecounter automata (NROCAs—see Section 2.2). The proof technique (also used in the proof of Theorem 4.5) is a routine adaptation of a standard technique of Hartmanis =-=[16]-=-, who showed that the set of invalid Turing machine computations is context-free, yielding the undecidability of EMPTYcoCFL. Lemma 4.4 improves this to one-counter machines. Although a result of this ... |

10 |
Effective constructions in well-partially-ordered free monoids
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Citation Context ...is total for all oracles, and hence A ≤tt X via M () i . 2. If there is an e such that N (X⊕A)′ = e, then L(Fe) ̸= SUBSEQ(A). 7 The Complexity of F CFL,REG The following theorem is due to van Leeuwen =-=[29]-=-. For completeness, we present it here with an altered proof. Theorem 7.1 (van Leeuwen [29]) F CFL,REG is computable. Proof: In this proof, as is customary, we will often identify regular expressions ... |

9 | Unbounded searching algorithms
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- 1990
(Show Context)
Citation Context ... be a function of the output. The final upshot will be that this problem has the exact same results as the unbounded search problem. 5.1 Unbounded Search The material in this subsection is taken from =-=[5, 2]-=-. The base of the log function is 2 throughout. Definition 5.1 The Unbounded Search Problem is as follows. Alice has a natural number n (there are no bounds on n). Bob is trying to determine what n is... |

8 |
Handbook of Recursive Mathematics
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- 1998
(Show Context)
Citation Context ...dex of a CFG or DFA recognizing SUBSEQ(D); otherwise, we let f(e) be the index of a CFG or DFA recognizing Σ ∗ . 9 Relation to Recursive and Reverse Mathematics Nerode’s Recursive Mathematics Program =-=[20, 8]-=- and Simpson and Friedman’s Reverse Math Program [23, 24] both attempt to pin down what it means for a proof to be noneffective or nonconstructive. Corollary 3.4 yields results in both of these progra... |

8 |
Bounded Queries in Recursion Theory
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(Show Context)
Citation Context ...(n) queries iff h satisfies Kraft’s inequality. 165.2 Definitions, Notation, and Lemmas from Bounded Queries The Definitions and Notations in this section are originally from [1, 4], but are also in =-=[14]-=-. We only touch on the parts of bounded queries that we need; there are many variants on these definitions. Definition 5.7 Let A be a set, and let n ≥ 1. 1. C A n : N n → {0, 1} n is defined by C A n ... |

8 |
On the succinctness of different representations of languages
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Citation Context ...tion 10. In the appendices we give two proofs of Higman’s theorem. The first essentially follows Higman’s original argument, and the second is original. This paper was inspired by papers of Hartmanis =-=[17]-=- and Hay [18]. In particular, the questions in Section 4 are similar to questions they asked. 2 Definitions 2.1 Language and Machine Conventions We fix a finite alphabet Σ. Definition 2.1 Let x, y ∈ Σ... |

8 | Computability and recursion, The Bulletin of Symbolic Logic 2 - Soare - 1996 |

5 | Efficient learning of real time onecounter automata
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(Show Context)
Citation Context ...ir properties studied by Valiant & Paterson [28]. Real-time counter machines were studied by Fischer, Meyer, & Rosenberg [12]. ROCAs were introduced in the context of machine learning by Fahmy & Roos =-=[9]-=-. 2.3 Rectangular Traces The concepts in this section will be used for the proofs of Lemma 4.4 and Theorem 4.5. We’ll use a standard notion of a Turing machine with a single one-way infinite tape. See... |

3 | editors. Handbook of Recursive Mathematics - Ershov, Goncharov, et al. - 1998 |

1 | Binary search and recursive graph problems. Theoretical Computer Science
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(Show Context)
Citation Context ...ertain problems. They will provide problems to reduce to in order to get lower bounds. Lemma 5.15 ([4]) # K n−1 ̸∈ EN(n − 1) The following lemma is a modification of a similar lemma from both [3] and =-=[13]-=-. It will enable us to prove lower bounds on the number of queries certain functions require to compute. Lemma 5.16 Let X ⊆ N, and h : N → N be any function. Let γ : N → N. If γ ∈ FQ(h(γ(x)), X) and (... |

1 |
On the recursion-theoretic complexity of relative succinctness of representations of languages
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(Show Context)
Citation Context ...he appendices we give two proofs of Higman’s theorem. The first essentially follows Higman’s original argument, and the second is original. This paper was inspired by papers of Hartmanis [17] and Hay =-=[18]-=-. In particular, the questions in Section 4 are similar to questions they asked. 2 Definitions 2.1 Language and Machine Conventions We fix a finite alphabet Σ. Definition 2.1 Let x, y ∈ Σ∗ . We say th... |

1 |
Recursive unsolvability of Post’s Problem of “Tag
- Minsky
- 1961
(Show Context)
Citation Context ...abet (except for a stack-empty marker), and at most one symbol pushed or popped at a time. Counter machines have been studied by many people. Two-counter machines were shown to be universal by Minsky =-=[21]-=- (improved by Fischer [11]). Deterministic one-counter automata (DOCAs) were first defined and their properties studied by Valiant & Paterson [28]. Real-time counter machines were studied by Fischer, ... |

1 | The complexity of learning SUBSEQ(A
- Fenner, Gasarch
- 2006
(Show Context)
Citation Context ... a representation for SUBSEQ(A) in the limit. This is similar to the Inductive Inference model of machine learning, and we explore in more depth the problem of learning SUBSEQ(A) in a companion paper =-=[10]-=-. The results of Sections 3, 4, and 7 suggest that finding SUBSEQ(A) in various representations is always of complexity either ∅ or ∅ ′′ . In Section 8 we construct, for every c.e. set X and for every... |