## A Propositional Proof System for Log Space

Citations: | 1 - 1 self |

### BibTeX

@MISC{Perron_apropositional,

author = {Steven Perron},

title = {A Propositional Proof System for Log Space},

year = {}

}

### OpenURL

### Abstract

Abstract. The proof system G ∗ 0 of the quantified propositional calculus corresponds to NC 1, and G ∗ 1 corresponds to P, but no formula-based proof system that corresponds log space reasoning has ever been developed. This paper does this by developing GL ∗. We begin by defining a class ΣCNF (2) of quantified formulas that can be evaluated in log space. Then GL ∗ is defined as G ∗ 1 with cuts restricted to ΣCNF (2) formulas and no cut formula that is not quantifier free contains a non-parameter free variable. To show that GL ∗ is strong enough to capture log space reasoning, we translate theorems of Σ B 0-rec into a family of tautologies that have polynomial size GL ∗ proofs. Σ B 0-rec is a theory of bounded arithmetic that is known to correspond to log space. To do the translation, we find an appropriate axiomatization of Σ B 0-rec, and put Σ B 0-rec proofs into a new normal form. To show that GL ∗ is not too strong, we prove the soundness of GL ∗ in such a way that it can be formalized in Σ B 0-rec. This is done by giving a log space algorithm that witnesses GL ∗ proofs. 1

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(Show Context)
Citation Context ...re able to prove the following theorem. Theorem 1. H ∗ p-simulates G ∗ 1 for Σ q 1 formulas. A complete proof can be found in [9]. 3 Propositional Translations This section is motivated by results in =-=[11]-=-. In that paper, Cook showed that theorems of the equational theory P V can be translated into a family of tautologies that have polynomial size extended Frege proofs. We will show that theorems of Σ ... |

78 | editor. Handbook of Proof Theory
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Citation Context ... complexity class. This paper is based on my Masters thesis [9], where more details can be found. 2 The Proof System The proof system P K is the Gentzen-style sequent calculus for propositional logic =-=[5, 6]-=-. The initial sequents are ⊥ →, → ⊤, and A → A, for any propositional formula A. The rules of inference include structural rules, which are weakening, contraction, and exchange; propositional rules, w... |

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(Show Context)
Citation Context ... give a log space algorithm that directs the edges appropriately, and prove the correctness of the algorithm in V L. This also gives us the function W we are looking for. We use a trick first used in =-=[13]-=- to find a cycle in a graph. For each edge e = (u, v) in G, we call its two end points e u and e v , with the obvious meaning. Given an end point p, we the edge can be obtain by e(p) and the vertex by... |

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Citation Context ...ting the proof system G ∗ to get a proof system GL ∗ , which corresponds to log space. The proof system G ∗ is a tree-like proof system for the quantified propositional calculus based on Gentzen’s LK =-=[2]-=-. This affirms the belief of Cook that there exists a formula-based proof system that corresponds to log space [1]. Before this the only proof system for log space was based on liar games [3], and it ... |

30 | Theories for complexity classes and their propositional translations
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(Show Context)
Citation Context ...tly there has been lots of research looking into the connection between computational complexity, bounded arithmetic, and propositional proof complexity. A recent survey on this topic can be found at =-=[1]-=-. In this paper, we give a method of restricting the proof system G ∗ to get a proof system GL ∗ , which corresponds to log space. The proof system G ∗ is a tree-like proof system for the quantified p... |

10 |
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Citation Context ...to formulas in the sequents; and the cut rule, which infers Γ → ∆ from A, Γ → ∆ and Γ → ∆, A. We can then extend P K to the proof system G by adding rules for quantifiers over propositional variables =-=[7]-=-. Anytime you deal with quantifiers, you will have problems with the substituting terms or, in this case, formulas for variables. To help avoid these problems, we use the following convention. Notatio... |

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(Show Context)
Citation Context ... on the cut formulas. Section 3 is devoted to translating theorems of Σ B 0 -rec into tautologies with polynomial size GL ∗ proofs. Σ B 0 -rec is a theory of bounded arithmetic introduced by Zambella =-=[12]-=- that is known to correspond to log space [1]. This proves GL ∗ is strong enough to capture log space reasoning. In section 4, we prove in Σ B 0 -rec that GL ∗ is sound. This is sometimes called the r... |

8 |
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(Show Context)
Citation Context ... complexity class. This paper is based on my Masters thesis [9], where more details can be found. 2 The Proof System The proof system P K is the Gentzen-style sequent calculus for propositional logic =-=[5, 6]-=-. The initial sequents are ⊥ →, → ⊤, and A → A, for any propositional formula A. The rules of inference include structural rules, which are weakening, contraction, and exchange; propositional rules, w... |

7 |
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(Show Context)
Citation Context ...L is a conservative extension of Σ B 0 -rec [9]. We start the proof by giving a log space algorithm that witnesses ΣCNF (2) formulas when the formula is true. This algorithm is the algorithm given in =-=[10]-=- with a few additions to find the satisfying assignment. This algorithm can be formalized in V L since it is a log space algorithm, and V L proves that it is correct. This means we can use this functi... |

3 |
GL ⋆ : A Propositional Proof System For Logspace
- Perron
- 2005
(Show Context)
Citation Context ... -rec that GL ∗ is sound. This is sometimes called the reflection principle. This tells us that GL ∗ does not capture reasoning for a higher complexity class. This paper is based on my Masters thesis =-=[9]-=-, where more details can be found. 2 The Proof System The proof system P K is the Gentzen-style sequent calculus for propositional logic [5, 6]. The initial sequents are ⊥ →, → ⊤, and A → A, for any p... |

2 |
A Survey of Complexity Classes and their Associated Propositional Proof Systems and Theories, and a Proof System for Log Space Slides for Edinburgh talk, presented at
- Cook
- 2001
(Show Context)
Citation Context ...zen’s LK [2]. This affirms the belief of Cook that there exists a formula-based proof system that corresponds to log space [1]. Before this the only proof system for log space was based on liar games =-=[3]-=-, and it has never been well developed. The definition of GL ∗ is similar to the definition of G ∗ i : it is obtain from G∗ by restricting cuts. We restrict cuts to the class of formulas ΣCNF (2). Thi... |

1 |
A propositional proof system for R i 2
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(Show Context)
Citation Context ...formulas where no variable appears more than twice. Another attempt to capture reasoning between NC 1 and P resorted to putting a bound on the depth of the proof and the number of cuts along a branch =-=[4]-=-, but it is not obvious how to capture other complexity classes using this type of restriction. In contrast to that, it seems plausible that a proof system for NL could be defined in the same spirit a... |