An exponential separation between regular and general resolution

Abstract

This paper gives two distinct proofs of an exponential separation between regular resolution and unrestricted resolution. The previous best known separation between these systems was quasi-polynomial.

An unexpected separation result in Linearly Bounded Arithmetic

“…Hence M i (n, x, 1) is equivalent to M i+1 (n, x, 0, n), and the latter follows from our meta induction hypothesis (2). This serves as the induction base.…”

Width and size of regular resolution proofs

“…Recent results [1,13] showing near-exponential separations between the size of regular and general refutations of certain sets of clauses also show a separation of general and regular resolution width. That is to say, the examples used in showing the size separation have large regular resolution width, but bounded general resolution width.…”

“…If Σ is a set of clauses on a set V of variables, then a non-empty family A of V -assignments is an extendible k-family for Σ if it satisfies the following conditions: (1) No assignment in A falsifies a clause in Σ; (2) Each assignment α in A satisfies the condition |α| ≤ k; (3) If α ∈ A, and β ⊆ α, then β ∈ A; (4) If α ∈ A, |α| < k, and x ∈ V , then there is a β ∈ A, so that α ⊆ β, and β(x) is defined.…”

“…Proof. The paper [1] implicitly contains such a separation. More specifically the family of clauses GT ′ n,ρ defined in §3 of [1] fulfil the conditions of the theorem.…”

Small Stone in Pool

“…On one hand, if Γ is a set of clauses, and DPLL with clause learning can show that Γ is unsatisfiable in n steps, then Γ has a resolution refutation with size polynomially bounded by n (see [BKS04]). On the other hand, the results of [AJPU07,Urq11,BB12,BBJ14] imply that the length of DPLL with clause learning proof searches can be nearly exponentially smaller than the length of the shortest regular resolution proofs. Systems designed to correspond to DPLL with clause learning, such as pool resolution ( [VG05]) and regRTI ( [BHJ08]), are therefore simulated (i ′ , i) ∈ E from i ′ to i then i ′ > i, and so that the source nodes of G are exactly vertices n+1, n+2, .…”