Graham Wrightson

a counterexample to W. Bibel's and E. Eder's strong completeness result for connection graph resolution

The connection graph proof procedure (or clause graph resolution as it is more commonly called today) is a theorem-proving technique due to Robert Kowalski, [1975]. It is a negative test calculus (a refutation procedure) based on resolution. Due to an intricate deletion mechanism that generalizes the well-known purity principle, it substantially refines the usual notions of resolution-based systems and leads to a largely reduced search space. The dynamic nature of the clause graph upon which this refutation procedure is based, poses novel problems previously unencountered in logical deduction systems. Ever since its invention in 1975, the soundness, confluence and (strong) completeness of the procedure have been in doubt in spite of many partial results to the positive. Bibel and Eder [1997] claim to have solved the long-outstanding open strong completeness problem of the procedure. However, as we point out below, there is a well-known Permission to make digital / hard copy of part or all of this work for personal or classroom use is granted without fee provided that the copies are not made or distributed for profit or commercial advantage, the copyright notice, the title of the publication, and its date appear, and notice is given that copying is by permission of the Association for Computing Machinery (ACM), Inc. To copy otherwise, to republish, to post on servers, or to redistribute to lists, requires prior specific permission and / or a fee.

Why do stored value systems fail

Stored value systems are the most recent form of electronic payment technology. They are meant to coexist with credit and debit technology, by primarily targeting the low value area of the transaction market. Being targeted at low value transactions, they are designed to have very low transaction cost.Stored value systems rely on creating a form of electronic value, on smart cards or as computer files. Such value can be bought (withdrawn) at any one time, and spent in arbitrary fractions at later times. When the technology emerged for its first implementations in the first half of the 1990s, it was much celebrated as a replacement for cash with many benefits over existing payment technologies.Many of these systems have subsequently been set up as trials, and the commercial rollout of some systems has started. However, actual usage of stored value systems is still low, much lower than was expected by the operators of the systems. Several years after the first trials were implemented, it is still unclear whether and when they will play a relevant role in the payments system market. And none of the trials that have been run can be considered a commercial success. This makes it necessary not only to assess the technology, but also the commercial future and user uptake of these systems.

On finding short resolution refutations and small unsatisfiable subsets

We consider the parameterized problems of whether a given set of clauses can be refuted within k resolution steps, and whether a given set of clauses contains an unsatisfiable subset of size at most k. We show that both problems are complete for the class W[1], the first level of the W-hierarchy of fixed-parameter intractable problems. Our results remain true if restricted to 3-SAT formulas and/or to various restricted versions of resolution including tree-like resolution, input resolution, and read-once resolution.

An Open Research Problem: Strong Completeness of R. Kowalski's Connection Graph Proof Procedure

The connection graph proof procedure (or clause graph resolution as it is more commonly called today) is a theorem proving technique due to Robert Kowalski. It is a negative test calculus (a refutation procedure) based on resolution.Due to an intricate deletion mechanism that generalises the well-known purity principle, it substantially refines the usual notions of resolution-based systems and leads to a largely reduced search space. The dynamic nature of the clause graph upon which this refutation procedure is based, poses novel meta-logical problems previously unencountered in logical deduction systems. Ever since its invention in 1975 the soundness, confluence and (strong) completeness of the procedure have been in doubt in spite of many partial results.Th is paper provides an introduction to the problem as well as an overview of the main results that have been obtained in the last twenty-five years.