Cut-elimination is one of the most famous problems in proof theory, and it was defined and solved for first-order sequent calculus by Gentzen in his celebrated Hauptsatz. Ceres is a different cut-elimination algorithm for firstand higher-order classical logic. It is based on the notion of a characteristic set of clauses which is extracted from a proof in sequent calculus and is always unsatisfi...

In molecular resolution refutation, the detection of empty clauses is important. We propose a rolling circle amplification-based detection method for resolution refutation. The rolling circle amplification (RCA) technique is known to be able to distinguish and amplify circular DNA. In this paper, we describe the representation of clauses and the RCA-based detection method. Bio-lab experiments s...

Connections between Muggleton's stochastic logic programs (SLPs) and (undirected) Bayesian nets are explored. Logical variables map to random variables, predicates map to joint distributions and Markov net structure maps to clausal structure. Naive methods for answering proba-bilistic queries using SLPs are given, as well as some suggestions for non-naive approaches. Multi-clause SLPs deene mix...

Complete deductive systems are constructed for the non-valid (refutable) formulae and sequents of some propositional modal logics. Thus, complete syntactic characterizations in the sense of Lukasiewicz are established for these logics and, in particular, purely syntactic decision procedures for them are obtained. The paper also contains some historical remarks and a general discussion on refuta...

Local search is widely applied to satisfiable SAT problems, and on some problem classes outperforms backtrack search. An intriguing challenge posed by Selman, Kautz and McAllester in 1997 is to use it instead to prove unsatisfiability. We design a greedy randomised resolution algorithm called RANGER that will eventually refute any unsatisfiable instance while using only bounded memory. RANGER c...

A Refutation of Skeptical Theism David Kyle Johnson The evidential problem of evil suggests that our awareness of the existence of seemingly unjustified evils reduces the epistemic probability of God’s existence. Arguments to this effect have been most famously and successfully championed by William Rowe. 1 His defense of the argument has developed over the years, but the basics of the argument...

Explaining the causes of infeasibility of Boolean formulas has many practical applications in various fields. A small unsatisfiable subformula provides a succinct explanation of infeasibility and is valuable for applications. In recent years finding unsatisfiable subformulas has been addressed frequently by research works, mostly based on the SAT solvers with DPLL backtrack-search algorithm. Ho...

We consider the problem of finding the smallest proof of unsatisfiability of a 2CNF formula. In particular, we look at Resolution refutations and at minimum unsatisfiable subsets of the clauses of the CNF. We give a characterization of minimum tree-like Resolution refutations that explains why, to find them, it is not sufficient to find shortest paths in the implication graph of the CNF. The ch...