The ultra-weak Ash conjecture and some particular cases

Abstract

Ash's functions N σ,k count the number of k-equivalence classes of σ-structures of size n. Some conditions on their asymptotic behavior imply the long standing spectrum conjecture. We present a new condition which is equivalent to this conjecture and we discriminate some easy and difficult particular cases.

Fifty years of the spectrum problem: survey and new results

“…For instance, he shows that if σ is a unary vocabulary, then for all k ≥ 1, Ash's function N σ,k is eventually constant. In [15], it is also proved that for all finite relational vocabulary σ, Ash's function N σ,2 is eventually constant. However, these results are of very limited interest because unary vocabularies or quantifier depth two (see [97]) only allow to define finite and cofinite spectra.…”

“…These ideas of Ash's have not been exploited afterwards, and his paper has remained isolated until recently. In 2006, Chateau and More [15] published a second paper related to Ash's counting functions. For all i ∈ N + , note N −1 σ,k (i) = {n ∈ N + |N σ,k (n) = i}, the inverse image of the positive integer i under the function N σ,k .…”

“…These ideas of Ash's have not been exploited afterwards, and his paper has remained isolated until recently. In 2006, Chateau and More [15] published a second paper related to Ash's counting functions. For all i ∈ N + , note…”

Some Estimated Likelihoods for Computational Complexity

“…Jones and Selman [JS74] showed that the spectrum problem has a yes answer if and only if NTIME[2 O(n) n) ], since in fact the class of all spectra (where the numbers are encoded in some finite alphabet) equals NTIME[2 O(n) ]. There are several interesting conjectures regarding spectra, any of which would imply a yes-answer [Ash94,CM06], and the conclusion would be stronger than proving NEXP = coNEXP. (For instance, one could have nondeterministic 2 O(n 9 )time algorithms for deciding the complements of nondeterministic O(2 n )-time problems, and this would still imply NEXP = coNEXP, but not necessarily the spectrum conjecture.)…”