POLYNOMIAL SIZE DEEP-INFERENCE PROOFS INSTEAD OF EXPONENTIAL SIZE SHALLOW-INFERENCE PROOFS

Alessio Guglielmi (Bath) 10.8.2004 - updated on 22.9.2007 By a simple example, I show how deep inference can provide for an exponential speed-up in the size of proofs with respect to shallow inference. In particular, there are classes of tautologies whose cut-free proofs only grow polynomially with their size, instead of exponentially, as in the sequent calculus. There are two areas where the use of deep inference can lead to lower complexity: complexity of cut elimination and complexity in proof search. In both cases the greater liberality of deep inference rules is advantageous. In many important cases, the complexity passes from being exponential to being polynomial (with low exponent). Here we concentrate on the proof search case. The big picture is the following: • with shallow inference one has (seemingly!, relatively!) low nondeterminism and long proofs; • with (uncontrolled!, naif!) deep inference one has high nondeterminism and short proofs. One goal is to show that in the more liberal framework of deep inference one can develop search algorithms that drastically cut down on nondeterminism and still find short proofs (for example, something along the lines of a deep version of uniform provability [MNPS]). To this purpose, I’d like to remind the reader that very successful techniques like resolution are immediately available with deep inference, contrary to what happens in shallow inference [AG]. In any case, a preliminary technical point we should make is that the proofs available in deep inference are indeed shorter than those available in shallow inference. Of course, miracles are unlikely, so, since it's reasonable to believe that NP ≠ coNP, we cannot reasonably hope for polynomial-size proofs in all cases. But we can of course hope for