Descriptive Minicomplexity
F. Lamana Mulaffer (CS 2015) Christos A. Kapoutsis (Adviser)
Context: Computational Complexity vs. Descriptive Complexity
Problems can be solved either by algorithms or by logical formulae. Example: Imagine five policemen were assigned to guard all roads. Are there five roundabouts where they can stand at and achieve this?
Abstractly, given a graph G = (V,E ) and a number k, are there k vertices that cover all edges?
for every set S of k vertices: for every edge e: if e is touched by S: mark e if all edges are marked: return TRUE return FALSE
Researchers have established equivalences between some types of algorithms and some types of logical formulae. Example: NP = ƎSO (Fagin’s Theorem).
Problem: Nondeterministic Read-Only Algorithms
In our research, we focused on a special kind of algorithms: “Small” Non-deterministic Read-Only Algorithms. We asked what kind of formulae are equivalent to these algorithms. We identifed such a class and proved the equivalence. Example: Imagine that many roads in West Bay are blocked due to construction. Is there a way to enter from ENTRANCE and exit from EXIT?
Abstractly, given a graph G = (V,E ) and vertices s and t, is there a path from s to t in G ?
start at v=s repeat: if v=t: return TRUE if v has no neighbours: return FALSE nondeterministically: select a neighbour u set v to u
What kinds of formulae are equivalent to these algorithms?
Solution: 2Way Nondeterministic Finite Automata vs. 1D Graph-Accessibility DNF We modelled NRO Algorithms using 2NFAs.
Every 2NFA with k states can be converted to a 1D GADNF of length polynomial in k.
A 1D GADNF is a formula of the following form:
2NFA
A 2NFA encodes the problem in a string and determines the solution by scanning along the string multiple times.
Every 1D GADNF of length k can be converted to a 2NFA where the number of states is polynomial in k.
The formula is true iff there’s a path from vertex +1 to vertex -1 in the graph described by φ.
Future research includes finding formulae that are equivalent to Deterministic Algorithms.
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4/23/14 12:41 PM