The computational complexity of prefix classes of logical theories

We derive upper and lower bounds on the computational complexity of prefix classes of several logical theories. The general method for obtaining lower bounds on the complexity of logical theories developed by Compton and Henson is adapted for their prefix classes. We are then able to show that for a fixed $r$, the prenex formulas in $\Pi\sb{2r+5}$ in the theory of finite trees of height at most $r$ have a $NTIME$(exp$\sb{r-2}(c\sqrt{n}/$log $n$)) lower bound. The same technique also gives a lower bound of $NTIME$(exp$\sb{m}(c\sb{m}n\sp{1/3}/$log$\sp2n$)) for the $\Sigma\sb{3m+3}$ formulas in the theory of any pairing function. We also get an upper bound for the prefix classes of the theory of $\langle$N, $<$, +, 2$\sp{x}\rangle$ by analyzing the quantifier elimination procedure for this theory described by Gottsch. We show that the formulas in $\Sigma\sb{m}\cup\Pi\sb{m}$ in this theory can be decided in $NSPACE$(exp$\sb{m}(cmn\sp3))$. Finally we get close upper and lower bounds for the prefix classes of the theory of finite linear orders with added unary predicates. By a direct coding of Turing machines we get that for $m\geq2$ the formulas in $\Pi\sb{m}$ have an $NSPACE$(exp$\sb{m}(d\sb{m}n$/log $n$)) lower bound. A careful analysis of a finite automata decision procedure for this theory gives that for $m\geq0$ the $\Sigma\sb{m+1}$ formulas can be decided in $NSPACE$(exp$\sb{m}(cn\sp2))$. We also show that the $\Sigma\sb1$ formulas in this theory are $NP$-complete and that the $\Pi\sb1$ and $\Sigma\sb2$ formulas are in $PSPACE$. An interpretation of this theory in the first-order theory of the binary tree with the prefix order and two successor functions shows that the formulas in $\Sigma\sb{m+1}$ have an $NSPACE$(exp$\sb{m}(c\sb{m}n/$log$\sp2n$)) lower bound.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Mathematics

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2011

Author dc:creator

• Streid, David Benjamin

Contributors dc:contributor

• Schupp, Paul E.

• Mathematics
• Computer Science

Statement dc:rights

• Copyright 1990 Streid, David Benjamin

Language dc:language

eng

Identifier

AAI9021763
(UMI)AAI9021763

OAI identifier oai:identifier

oai:www.ideals.illinois.edu:2142/20603