3 research outputs found
Minimization for Generalized Boolean Formulas
The minimization problem for propositional formulas is an important
optimization problem in the second level of the polynomial hierarchy. In
general, the problem is Sigma-2-complete under Turing reductions, but
restricted versions are tractable. We study the complexity of minimization for
formulas in two established frameworks for restricted propositional logic: The
Post framework allowing arbitrarily nested formulas over a set of Boolean
connectors, and the constraint setting, allowing generalizations of CNF
formulas. In the Post case, we obtain a dichotomy result: Minimization is
solvable in polynomial time or coNP-hard. This result also applies to Boolean
circuits. For CNF formulas, we obtain new minimization algorithms for a large
class of formulas, and give strong evidence that we have covered all
polynomial-time cases
Equality on all #CSP Instances Yields Constraint Function Isomorphism via Interpolation and Intertwiners
A fundamental result in the study of graph homomorphisms is Lov\'asz's
theorem that two graphs are isomorphic if and only if they admit the same
number of homomorphisms from every graph. A line of work extending Lov\'asz's
result to more general types of graphs was recently capped by Cai and Govorov,
who showed that it holds for graphs with vertex and edge weights from an
arbitrary field of characteristic 0. In this work, we generalize from graph
homomorphism -- a special case of #CSP with a single binary function -- to
general #CSP by showing that two sets and of
arbitrary constraint functions are isomorphic if and only if the partition
function of any #CSP instance is unchanged when we replace the functions in
with those in . We give two very different proofs of
this result. First, we demonstrate the power of the simple Vandermonde
interpolation technique of Cai and Govorov by extending it to general #CSP.
Second, we give a proof using the intertwiners of the automorphism group of a
constraint function set, a concept from the representation theory of compact
groups. This proof is a generalization of a classical version of the recent
proof of the Lov\'asz-type result by Man\v{c}inska and Roberson relating
quantum isomorphism and homomorphisms from planar graphs.Comment: 21 pages, 2 figure