The Completeness of Propositional Resolution: A Simple and Constructive<br> Proof

It is well known that the resolution method (for propositional logic) is complete. However, completeness proofs found in the literature use an argument by contradiction showing that if a set of clauses is unsatisfiable, then it must have a resolution refutation. As a consequence, none of these proofs actually gives an algorithm for producing a resolution refutation from an unsatisfiable set of clauses. In this note, we give a simple and constructive proof of the completeness of propositional resolution which consists of an algorithm together with a proof of its correctness.Comment: 7 pages, submitted to LMC

Remarks on the Cayley Representation of Orthogonal Matrices and on Perturbing the Diagonal of a Matrix to Make it Invertible

This note contains two remarks. The first remark concerns the extension of the well-known Cayley representation of rotation matrices by skew symmetric matrices to rotation matrices admitting -1 as an eigenvalue and then to all orthogonal matrices. We review a method due to Hermann Weyl and another method involving multiplication by a diagonal matrix whose entries are +1 or -1. The second remark has to do with ways of flipping the signs of the entries of a diagonal matrix, C, with nonzero diagonal entries, obtaining a new matrix, E, so that E + A is invertible, where A is any given matrix (invertible or not).Comment: 7 page

Fast and Simple Methods For Computing Control Points

The purpose of this paper is to present simple and fast methods for computing control points for polynomial curves and polynomial surfaces given explicitly in terms of polynomials (written as sums of monomials). We give recurrence formulae w.r.t. arbitrary affine frames. As a corollary, it is amusing that we can also give closed-form expressions in the case of the frame (r, s) for curves, and the frame ((1, 0, 0), (0, 1, 0), (0, 0, 1) for surfaces. Our methods have the same low polynomial (time and space) complexity as the other best known algorithms, and are very easy to implement.Comment: 15 page

Democracy and compliance in public goods games

I investigate if, how, and why the effect of a contribution rule in a public goods game depends on how it is implemented: endogenously chosen or externally imposed. The rule prescribes full contributions to the public good backed by a nondeterrent sanction for those who do not comply. My experimental design allows me to disentangle to what extent the effect of the contribution rule under democracy is driven by self-selection of treatments, information transmitted via the outcome of the referendum, and democracy per se. In case treatments are endogenously chosen via a democratic decision-making process, the contribution rule significantly increases contributions to the public good. However, democratic participation does not affect participants’ contribution behavior directly, after controlling for self-selection of treatments and the information transmitted by voting