## An Improved Lower Bound for the Elementary Theories of Trees (1996)

Citations: | 27 - 3 self |

### BibTeX

@INPROCEEDINGS{Vorobyov96animproved,

author = {Sergei Vorobyov},

title = {An Improved Lower Bound for the Elementary Theories of Trees},

booktitle = {},

year = {1996},

pages = {275--287},

publisher = {Springer-Verlag}

}

### Years of Citing Articles

### OpenURL

### Abstract

. The first-order theories of finite and rational, constructor and feature trees possess complete axiomatizations and are decidable by quantifier elimination [15, 13, 14, 5, 10, 3, 20, 4, 2]. By using the uniform inseparability lower bounds techniques due to Compton and Henson [6], based on representing large binary relations by means of short formulas manipulating with high trees, we prove that all the above theories, as well as all their subtheories, are non-elementary in the sense of Kalmar, i.e., cannot be decided within time bounded by a k- story exponential function 1 exp k (n) for any fixed k. Moreover, for some constant d ? 0 these decision problems require nondeterministic time exceeding exp 1 (bdnc) infinitely often. 1 Introduction Trees are fundamental in Computer Science. Different tree structures are used as underlying domains in automated theorem proving, term rewriting, functional and logic programming, constraint solving, symbolic computation, knowledge re...

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326 |
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74 |
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Citation Context ...ion. The larger is the size, the higher is the respective lower bound. Technically, the method of Compton and Henson has a great advantage over the pioneer methods of Meyer, Stockmeyer, Fisher, Rabin =-=[16, 23, 9, 8]-=-, since it allows one to avoid tedious encodings of Turing machines. To estimate the size of a binary relation representable by short formulas of a theory is much simpler than to prove that all Turing... |

72 | A Feature-Based Constraint System for Logic Programming with Entailment, Fifth Generation Computer Systems
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Citation Context ... rational trees, and proved decidability by quantifier elimination. The last result is also proved for finite signatures by quantifier elimination in [5]. Feature Trees introduced and investigated in =-=[1, 19, 3, 20, 11, 2]-=- allow for non-fixed and variable arities of tree nodes and constitute a convenient formalism for logic and constraint programming, knowledge representation. Intuitively, saying f(y) yff 1 ; : : : ; f... |

57 |
Separating nondeterministic time complexity classes
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- 1978
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Citation Context ... all elements x of B satisfying B j= ffi n(x; m) and the binary predicate P (x; y) defined on the elements x, y of the carrier satisfying B j=sB n (x; y; m). 4 By the following well-known result from =-=[18]-=-: Let t1 (n) and t2 (n) be functions such that limn!1 t 1 (n+1) t 2 (n) = 0: Then there is a problem in NTIME(t2(n))nNTIME (t1 (n)). 2) Let A 2 C belong to NTIME(exp 1 (c 2 n))nNTIME(exp 1 (c 1 n)) fo... |

55 |
A uniform method for proving lower bounds on the computational complexity of logical theories
- Compton, Henson
- 1990
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Citation Context ...es possess complete axiomatizations and are decidable by quantifier elimination [15, 13, 14, 5, 10, 3, 20, 4, 2]. By using the uniform inseparability lower bounds techniques due to Compton and Henson =-=[6]-=-, based on representing large binary relations by means of short formulas manipulating with high trees, we prove that all the above theories, as well as all their subtheories, are non-elementary in th... |

48 |
Classical recursion theory, volume 125
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- 1989
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Citation Context ...eas of Trakhtenbrot and Vaught dating back to 50-- 60, used for proving the recursive inseparability of the sets of valid and finitely refutable sets of formulas of the first-order predicate calculus =-=[7, 21, 17]. Compton -=-and Henson refined the idea of recursive inseparability by replacing it with "inseparable by NTIME (t(n))-recognizable sets". The lower bounds method of Compton and Henson relies on the abil... |

47 |
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- 1961
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36 |
Weak monadic second-order theory of successor is not elementary recursive
- Meyer
- 1972
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Citation Context ...exp 1 (dn)). In other words, both require nondeterministic non-elementary time exp 1 (dn) infinitely often. ut The first and the best known proof of a non-elementary lower bound was given by A. Meyer =-=[16]-=- for the weak monadic second-order logic of one successor function (Buchi arithmetic). Outline of the Paper. After giving a brief account in the next section of Compton-Henson's uniform lower bound me... |

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25 |
Answer Sets and Negation as Failure
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Citation Context ...er elimination. Contribution of the Paper. None of the papers cited above suggest any lower or upper complexity bounds for the theories of trees and the proposed quantifier elimination procedures. In =-=[12]-=- Kunen conjectured and gave a sketch of the proof that the decision problem for TA (for infinite signatures) is PSPACE - complete. In this paper by applying Compton-Henson's uniform lower bound techni... |

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19 |
Enumeration Theory
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- 1977
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Citation Context ... sides of these definitions all the occurrences of Stn and Matchn should be understood as abbreviations for the right-hand sides of (5) and (6). Min 0 (p; s) = 9xy i p = s = f(x; y)s[x = Lsx = R] j ; =-=(7)-=- Minn+1 (p; s) = 9a h Pntn (a)sStn (p; a; s) i 8bt h Pntn (b)sStn (p; b; t) ) :Succ(t; s) i ; (8) Max 0 (p; s) = 9xy i p = s = f(x; y)s[x = Lsx = R] j ; (9) Maxn+1 (p; s) = 9a h Pntn (a)sStn (p; a; s)... |

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2 | Theory of finite trees revisited: Application of model-theoretic algebra
- Vorobyov
- 1994
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Citation Context ...tures, Malcev in 1961--62 [15] proved the decidability of (Ax1)--(Ax3) and (Ax1)--(Ax3), (DCA) by quantifier elimination. Other proofs are given in [13, 14, 5, 10], also by quantifier elimination. In =-=[24]-=- (Ax1)-- (Ax3), (DCA) is proved decidable by A. Robinson's model completeness test. Rational Trees. The theory of rational trees is obtained from TA by replacing the acyclicity scheme (Ax3) with the f... |