The most nonelementary theory
We give a direct proof by generic reduction that testing validity of formulas in a decidable rudimentary theory Ω of finite typed sets (Henkin, Fundamenta Mathematicæ 52 (1963) 323–344) requires space and time exceeding infinitely often (1) 2 · · · 2 exp ∞( exp(cn))=2 height 2 cm for some constant c...
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| Veröffentlicht in: | Information and computation Jg. 190; H. 2; S. 196 - 219 |
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| Abstract | We give a direct proof by generic reduction that testing validity of formulas in a decidable rudimentary theory Ω of finite typed sets (Henkin, Fundamenta Mathematicæ 52 (1963) 323–344) requires space and time exceeding infinitely often
(1)
2
·
·
·
2
exp
∞(
exp(cn))=2
height
2
cm
for
some
constant
c>0,
where
n denotes the length of input. This gives
the highest currently known lower bound for a decidable logical theory and affirmatively settles Problem 10.13 from (Compton and Henson, Ann. Pure Applied Logic 48 (1990) 1–79):
The highest previously known lower (and upper) bounds for “natural” decidable theories, like
WS1S,
S2S, are of the form exp
∞(
dn), with
just linearly growing stacks of twos.
Originally, the lower bound
(1) for Ω was settled in (12th Annual IEEE Symposium on Logic in Computer Science (LICS’97), 1997, 294–305) using the powerful uniform lower bounds method due to Compton and Henson, and probably would never be discovered otherwise. Although very concise, the original proof has certain gaps, because the method was pushed out of the limits it was originally designed and intended for, and some hidden assumptions were violated. This results in slightly weaker bounds—the stack of twos in
(1) grows subexponentially, but superpolynomially, namely, as
2
c
n
for formulas with fixed quantifier prefix, or as 2
cn/log(
n)
for formulas with varying prefix. The independent
direct proof presented in this paper closes the gaps and settles the originally claimed lower bound
(1) for the minimally typed, succinct version of Ω. |
|---|---|
| AbstractList | We give a direct proof by generic reduction that testing validity of formulas in a decidable rudimentary theory Ω of finite typed sets (Henkin, Fundamenta Mathematicæ 52 (1963) 323–344) requires space and time exceeding infinitely often
(1)
2
·
·
·
2
exp
∞(
exp(cn))=2
height
2
cm
for
some
constant
c>0,
where
n denotes the length of input. This gives
the highest currently known lower bound for a decidable logical theory and affirmatively settles Problem 10.13 from (Compton and Henson, Ann. Pure Applied Logic 48 (1990) 1–79):
The highest previously known lower (and upper) bounds for “natural” decidable theories, like
WS1S,
S2S, are of the form exp
∞(
dn), with
just linearly growing stacks of twos.
Originally, the lower bound
(1) for Ω was settled in (12th Annual IEEE Symposium on Logic in Computer Science (LICS’97), 1997, 294–305) using the powerful uniform lower bounds method due to Compton and Henson, and probably would never be discovered otherwise. Although very concise, the original proof has certain gaps, because the method was pushed out of the limits it was originally designed and intended for, and some hidden assumptions were violated. This results in slightly weaker bounds—the stack of twos in
(1) grows subexponentially, but superpolynomially, namely, as
2
c
n
for formulas with fixed quantifier prefix, or as 2
cn/log(
n)
for formulas with varying prefix. The independent
direct proof presented in this paper closes the gaps and settles the originally claimed lower bound
(1) for the minimally typed, succinct version of Ω. |
| Author | Vorobyov, Sergei |
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| Keywords | Inductive definition 03D15 Complexity of computation Lower complexity bound Reduction via length order MSC 68Q25 Analysis of algorithms and problem complexity Nonelementary theory Generic reduction Computer theory |
| Language | English |
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| References | Kuper, Vardi (BIB6) 1993; 116 A.R. Meyer, Weak monadic second-order theory of successor is not elementary-recursive, in: R. Parikh (Ed.), Logic Colloquium: Symposium on Logic Held at Boston, 1972–1973, Vol. 453 of Lect. Notes Math., Springer-Verlag, 1975, pp. 132–154 Hopcroft, Ullman (BIB4) 1979 Meyer (BIB9) 1974 L.J. Stockmeyer, A.R. Meyer, Word problems requiring exponential time: preliminary report, in: 5th Symposium on Theory of Computing, 1973, pp. 1–9 Stockmeyer (BIB15) 1987; 52 Statman (BIB11) 1979; 9 Hull, Su (BIB5) 1991; 43 Henkin (BIB3) 1963; 52 Lewis (BIB7) 1980; 21 Urquhart (BIB16) 1990 L.J. Stockmeyer, The complexity of decision problems in automata theory and logic, PhD thesis, MIT Lab for Computer Science, 1974 (Also /MIT/LCS Tech Rep 133) S. Vorobyov, A. Voronkov, Complexity of nonrecursive logic programs with complex values, in: J. Paredaens, L. Colby (Eds.), Seventeenth ACM SIGACT-SIGMOD-SIGART Symposium on Principles of Database Systems (PODS’98), 1998, pp. 244–253 Compton, Henson (BIB1) 1990; 48 in: Lect. Notes Math., vol. 718, Springer-Verlag, Berlin, 1979 Mairson (BIB8) 1992; 103 J. Ferrante, C. W. Rackoff, The computational complexity of logical theories Stockmeyer (BIB14) 1977; 3 S. Vorobyov, The “hardest” natural decidable theory, in: G. Winskel (Ed.), 12th Annual IEEE Symp. on Logic in Computer Science (LICS’97), 1997, pp. 294–305 Hopcroft (10.1016/j.ic.2004.02.002_BIB4) 1979 Lewis (10.1016/j.ic.2004.02.002_BIB7) 1980; 21 Meyer (10.1016/j.ic.2004.02.002_BIB9) 1974 Stockmeyer (10.1016/j.ic.2004.02.002_BIB14) 1977; 3 10.1016/j.ic.2004.02.002_BIB13 Henkin (10.1016/j.ic.2004.02.002_BIB3) 1963; 52 10.1016/j.ic.2004.02.002_BIB10 10.1016/j.ic.2004.02.002_BIB2 10.1016/j.ic.2004.02.002_BIB12 Mairson (10.1016/j.ic.2004.02.002_BIB8) 1992; 103 Statman (10.1016/j.ic.2004.02.002_BIB11) 1979; 9 Compton (10.1016/j.ic.2004.02.002_BIB1) 1990; 48 10.1016/j.ic.2004.02.002_BIB17 10.1016/j.ic.2004.02.002_BIB18 Stockmeyer (10.1016/j.ic.2004.02.002_BIB15) 1987; 52 Hull (10.1016/j.ic.2004.02.002_BIB5) 1991; 43 Urquhart (10.1016/j.ic.2004.02.002_BIB16) 1990 Kuper (10.1016/j.ic.2004.02.002_BIB6) 1993; 116 |
| References_xml | – start-page: 61 year: 1990 end-page: 76 ident: BIB16 article-title: The complexity of decision procedures in relevance logic publication-title: Truth or Consequences: Essays in Honor of Nuel Belnap – reference: in: Lect. Notes Math., vol. 718, Springer-Verlag, Berlin, 1979 – volume: 9 start-page: 73 year: 1979 end-page: 81 ident: BIB11 article-title: The typed publication-title: Theor. Comput. Sci. – volume: 52 start-page: 323 year: 1963 end-page: 344 ident: BIB3 article-title: A theory of propositional types publication-title: Fundamenta Mathematicæ – volume: 103 start-page: 387 year: 1992 end-page: 394 ident: BIB8 article-title: A simple proof of a theorem of Statman publication-title: Theor. Comput. Sci. – start-page: 477 year: 1974 end-page: 482 ident: BIB9 article-title: The inherent computational complexity of theories of ordered sets publication-title: International Congress of Mathematicians – reference: A.R. Meyer, Weak monadic second-order theory of successor is not elementary-recursive, in: R. Parikh (Ed.), Logic Colloquium: Symposium on Logic Held at Boston, 1972–1973, Vol. 453 of Lect. Notes Math., Springer-Verlag, 1975, pp. 132–154 – reference: L.J. Stockmeyer, A.R. Meyer, Word problems requiring exponential time: preliminary report, in: 5th Symposium on Theory of Computing, 1973, pp. 1–9 – volume: 52 start-page: 1 year: 1987 end-page: 43 ident: BIB15 article-title: Classifying the computational complexity of problems publication-title: J. Symb. Logic – reference: S. Vorobyov, A. Voronkov, Complexity of nonrecursive logic programs with complex values, in: J. Paredaens, L. Colby (Eds.), Seventeenth ACM SIGACT-SIGMOD-SIGART Symposium on Principles of Database Systems (PODS’98), 1998, pp. 244–253 – volume: 116 start-page: 33 year: 1993 end-page: 57 ident: BIB6 article-title: On the complexity of queries in the logical data model publication-title: Theor. Comput. Sci. – reference: L.J. Stockmeyer, The complexity of decision problems in automata theory and logic, PhD thesis, MIT Lab for Computer Science, 1974 (Also /MIT/LCS Tech Rep 133) – reference: J. Ferrante, C. W. Rackoff, The computational complexity of logical theories, – volume: 43 start-page: 219 year: 1991 end-page: 267 ident: BIB5 article-title: On the expressive power of database queries with intermediate types publication-title: J. Comput. Syst. Sci. – volume: 21 start-page: 317 year: 1980 end-page: 353 ident: BIB7 article-title: Complexity results for classes of quantificational formulas publication-title: J. Comput. Syst. Sci. – reference: S. Vorobyov, The “hardest” natural decidable theory, in: G. 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Sci. doi: 10.1016/0304-3975(93)90219-J – ident: 10.1016/j.ic.2004.02.002_BIB10 doi: 10.1007/BFb0064872 – ident: 10.1016/j.ic.2004.02.002_BIB2 doi: 10.1007/BFb0062837 – volume: 21 start-page: 317 year: 1980 ident: 10.1016/j.ic.2004.02.002_BIB7 article-title: Complexity results for classes of quantificational formulas publication-title: J. Comput. Syst. Sci. doi: 10.1016/0022-0000(80)90027-6 – ident: 10.1016/j.ic.2004.02.002_BIB17 doi: 10.1109/LICS.1997.614956 |
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| SubjectTerms | Applied sciences Computer science; control theory; systems Exact sciences and technology Generic reduction Inductive definition Lower complexity bound Miscellaneous Nonelementary theory Reduction via length order Theoretical computing |
| Title | The most nonelementary theory |
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