In this paper we study families of resource aware logics that explore resource restriction on rules; in particular, we study the use of controlled cut-rule and introduce three families of parameterised logics that arise from different ways of controlling the use of cut. We start with a formulation of classical logic in which cut is non-eliminable and then impose restrictions on the use of cut. Three Cut-and-Pay families of logics are presented, and it is shown that each family provides an (...) approximation process for full propositional classical logic when the control over the use of cut is progressively weakened. A sound and complete semantics is given for each component of each of the three families of approximated logics. One of these families is shown to possess the uniform substitution property, a new result for approximated reasoning. A tableau based decision procedure is presented for each element of the approximation families and the complexity of each decision procedure is studied. We show that there are families in which every component logic can be decided polynomially. (shrink)

In this paper we explore a generalization of traditional abduction which can simultaneously perform two different tasks: given an unprovable sequent Γ ⊢ G, find a sentence H such that Γ, H ⊢ G is provable ; given a provable sequent Γ ⊢ G, find a sentence H such that Γ ⊢ H and the proof of Γ, H ⊢ G is simpler than the proof of Γ ⊢ G . We argue that the two tasks should not be distinguished, (...) and present a general procedure for finding suitable hypotheses or lemmas. When the original sequent is provable, the abduced formula can be seen as a cut formula with respect to Gentzen's sequent calculus, so the abduction method is cut-based. Our method is based on the tableau-like system KE and we argue for its advantages over existing abduction methods based on traditional Smullyan-style Tableaux. (shrink)

Any arbitrarily complicated non-oriented graph, that is a graph of arbitrarily large genus, can be encoded in a cut-free proof. This unpublished result of Statman was shown in the early seventies. We provide a proof of it, and of a number of other related facts.

We establish a connection between the geometric methods developed in the combinatorial theory of small cancellation and the propositional resolution calculus. We define a precise correspondence between resolution proofs in logic and diagrams in small cancellation theory, and as a consequence, we derive that a resolution proof is a 2-dimensional process. The isoperimetric function defined on diagrams corresponds to the length of resolution proofs.