Satisfiability in multi-valued circuits

Satisfiability of Boolean circuits is among the most known and important problems in theoretical computer science. This problem is NP-complete in general but becomes polynomial time when restricted either to monotone gates or linear gates. We go outside Boolean realm and consider circuits built of any fixed set of gates on an arbitrary large finite domain. From the complexity point of view this is strictly connected with the problems of solving equations (or systems of equations) over finite algebras. The research reported in this work was motivated by a desire to know for which finite algebras $\mathbf A$ there is a polynomial time algorithm that decides if an equation over $\mathbf A$ has a solution. We are also looking for polynomial time algorithms that decide if two circuits over a finite algebra compute the same function. Although we have not managed to solve these problems in the most general setting we have obtained such a characterization for a very broad class of algebras from congruence modular varieties. This class includes most known and well-studied algebras such as groups, rings, modules (and their generalizations like quasigroups, loops, near-rings, nonassociative rings, Lie algebras), lattices (and their extensions like Boolean algebras, Heyting algebras or other algebras connected with multi-valued logics including MV-algebras). This paper seems to be the first systematic study of the computational complexity of satisfiability of non-Boolean circuits and solving equations over finite algebras. The characterization results provided by the paper is given in terms of nice structural properties of algebras for which the problems are solvable in polynomial time.Comment: 50 page

The Complexity of Problems Connected with Two-element Algebras

This paper presents a complete classification of the complexity of the SAT and equivalence problems for two-element algebras. Cases of terms and polynomials are considered. We show that for any fixed two-element algebra the considered SAT problems are either in P or NP-complete and the equivalence problems are either in P or coNP-complete. We show that the complexity of the considered problems, parametrized by an algebra, are determined by the clone of term operations of the algebra and does not depend on generating functions for the clone

Bringing the margin to the focus: 10 challenges for riparian vegetation science and management

Riparian zones are the paragon of transitional ecosystems, providing critical habitat and ecosystem services that are especially threatened by global change. Following consultation with experts, 10 key challenges were identified to be addressed for riparian vegetation science and management improvement: (1) Create a distinct scientific community by establishing stronger bridges between disciplines; (2) Make riparian vegetation more visible and appreciated in society and policies; (3) Improve knowledge regarding biodiversity—ecosystem functioning links; (4) Manage spatial scale and context-based issues; (5) Improve knowledge on social dimensions of riparian vegetation; (6) Anticipate responses to emergent issues and future trajectories; (7) Enhance tools to quantify and prioritize ecosystem services; (8) Improve numerical modeling and simulation tools; (9) Calibrate methods and increase data availability for better indicators and monitoring practices and transferability; and (10) Undertake scientific validation of best management practices. These challenges are discussed and critiqued here, to guide future research into riparian vegetation