Learning from Łukasiewicz and Meredith: Investigations into Proof Structures

Abstract

The material presented in this paper contributes to establishing a basis deemed essential for substantial progress in Automated Deduction. It identifies and studies global features in selected problems and their proofs which offer the potential of guiding proof search in a more direct way. The studied problems are of the wide-spread form of “axiom(s) and rule(s) imply goal(s)”. The features include the well-known concept of lemmas. For their elaboration both human and automated proofs of selected theorems are taken into a close comparative consideration. The study at the same time accounts for a coherent and comprehensive formal reconstruction of historical work by Łukasiewicz, Meredith and others. First experiments resulting from the study indicate novel ways of lemma generation to supplement automated first-order provers of various families, strengthening in particular their ability to find short proofs.

CD Tools -- Condensed Detachment and Structure Generating Theorem Proving (System Description)

“…Operations on D-terms in CD Tools are based on the formal treatment and concepts from [43]. An important basic way to associate an atomic formula with a D-term is its most general theorem (MGT) (aka principal type scheme [12,11]) with respect to given axioms.…”

“…CD Tools includes implementations of many concepts and operations introduced or discussed in [43], including D-term comparison by ≥ c , various notions of regularity, the organic property (based on calling a SAT solver 7 or based on previously proven lemmas), the prime property, and n-simplification, that is, replacing complex subproofs of minor premises whose goal is irrelevant for the conclusion with a primitive subproof. D-terms allow precise proof size measures that are supported by CD Tools.…”

“…For SGCD, for each problem and size measure the minimal value of all the up to four proofs obtained with the four configurations of SGCD-1+2+3+7 is considered. The 𝑛 -superscript indicates values after n-simplification [43], which has a strong effect on the proofs by Prover9 but only a very minor effect on those by SGCD. In summary, the table indicates that SGCD returns smaller proofs than Prover9 in its default mode, moderately smaller with respect to compacted size and height, and drastically smaller with respect to tree size.…”

“…PSP was motivated by an analysis [43] of Meredith's variant [23] of Łukasiewicz's completeness proof for his shortest single axiom for implicational logic [18], where it was observed that most steps in Meredith's proof can be ascribed the relationship between levels that underlies PSP. The experiments summarized in Table 5 evaluate the potential of SGCD in configurations with PSP and with other structure generation methods for finding proofs, with emphasis on small compacted size.…”

“…Prover9 [21] appears to handle inputs that represent a CD problem in a special way by selecting hyperresolution as inference rule such that proofs may be viewed as CD proofs. Recently [43] it was observed that the connection method [3,5] suggests an ATP-oriented view on CD that is not centered on associating with (Det) an inference rule but on the proof structure, the D-term, as a whole. A D-term represents the structure formed by the connections.…”