Uniform proofs and natural deductions

Using some routine properties ([zu], [po]) of the Prawitz translation φ from (sequent calculus) derivations to (natural) deductions, and restricting ourselves to the language of first-order hereditary Harrop formulae, we show (i) that φ maps the simple uniform derivations of Miller et al onto the set of deductions in expanded normal form; and (ii) that φ identifies two such derivations iff they differ only by the order in which conjunctions and universal formulae on the left are broken up, thus factoring through a bijection between the set of uniform proofs with backchaining and the set of deductions in expanded normal form. Thus, the logic programmer’s restriction to the use of uniform proofs with backchaining is complete not merely w.r.t. derivability but also (in a bijective fashion) w.r.t. the construction of expanded normal deductions. (extended abstract, April 14, 1994. Caution: at present the results are not yet established to our satisfaction nor do we know the precise context, whether hH or wider, in which the results hold.)