## Translating I∆0+(exp) proofs into weaker systems (1999)

Citations: | 1 - 1 self |

### BibTeX

@MISC{Pollett99translatingi∆0+(exp),

author = {Chris Pollett},

title = { Translating I∆0+(exp) proofs into weaker systems},

year = {1999}

}

### OpenURL

### Abstract

The purpose of this paper is to explore the relationship between I∆0+(exp)

### Citations

227 |
Bounded Propositional Logic and Complexity Theory
- Kraj́ıcek
- 1995
(Show Context)
Citation Context ...S2 for ∪iS i 2. We mention here that it is straightforward using open-IND to prove the new XBASIC axiom, so XBASIC ⊆ IOpen. The theory S2 is a conservative extension of the Wilkie Paris theory I∆0+Ω1 =-=[15,9]-=-. The sequent calculus system for the former theory is often more convenient than for the latter, although the latter theory does have the virtue of being defined over the same language as I∆0+exp. Th... |

123 |
Bounded Arithmetic. Bibliopolis
- Buss
- 1986
(Show Context)
Citation Context ...er is sharply bounded. Next we define BASIC to be axiomatized by all subtitution instances of a finite set of quantifier free axioms for the non-logical symbols of L2. These axioms are listed in Buss =-=[2]-=- with the exception of the axioms for MSP and .− which are listed in Takeuti [14]. We will take BASIC1 to be the L1-theory 3saxiomatized by BASIC less the axioms not in L1. To present the rest of the ... |

115 |
Metamathematics of First-Order Arithmetics
- Hájek, Pudlák
- 1993
(Show Context)
Citation Context ... (∀x)A(x) where A is a bounded formula then I∆0 proves (∀x)((∃y)(y = 2 x k) ⊃ A(x)). Here 2 x k is a stack of 2’s k high with an x at the top. This result has both a simple compactness argument proof =-=[7]-=- as well as a proof using Herbrand’s theorem [4] which could in principle be used to give a bound on k in terms of the maximum nesting depths of exp in the I∆0+exp proof. Intuitively, this result says... |

46 |
On the scheme of induction for bounded arithmetic formulas
- Wilkie, Paris
- 1987
(Show Context)
Citation Context ...ts of arithmetic known to prove the Matiyasevic Robinson Davis Putnam Theorem (MRDP) Theorem that every Σ1-formula is equivalent to an ∃1-formula [6]. It is also known to be both finitely axiomatized =-=[15]-=- and equivalent to its IE1+exp fragment [8]. In contrast the bounded arithmetic theories I∆0 and I∆0+Ω1 (equivalent to Buss’ S2), which have induction on formulas involving only sub-exponential growth... |

31 |
RSUV isomorphisms
- Takeuti
- 1993
(Show Context)
Citation Context ...re |x|n is the length function (⌈log 2(x + 1)⌉) applied n times to x. The precise definition of our translations is closely related to the RSUV-translation of second-order bounded arithmetic theories =-=[14]-=-. We show that if IEi(exp) proves A with a proof P in which no function symbols appear in bounding terms of (∀ ≤: right) or (∃ ≤: left) inferences then there is an n ≤ exp-rank(P ) + 1 such that IE m ... |

26 |
On spectra
- BENNETT
- 1962
(Show Context)
Citation Context ..., and MSP ; however, since these functions are all ∆0-definable 2 in the usual I∆0 [3], our theory will be a conservative extension of that theory. The graph of exp(x, y) := xy is ∆0-definable in I∆0 =-=[1,6]-=-. Using this ∆0-definition, we can define the define exp axiom (∀x)(∀y)(∃z)(z = exp(x, y)). The theory I∆0+exp is axiomatized by I∆0 together with the exp axiom. The theory I∆0+Ω1 is axiomatized by I∆... |

22 | Structure and definability in general bounded arithmetic theories
- Pollett
- 1999
(Show Context)
Citation Context ...s often more convenient than for the latter, although the latter theory does have the virtue of being defined over the same language as I∆0+exp. The above axiomatization of S i 2 was given in Pollett =-=[11]-=- and shown to be equivalent to the original one given in Buss [2]. It is known from the latter reference that Si 2 ⊆ T i 2 ⊆ S i+1 2 and it follows from the above definitions that S i−1 2 ⊆ IE 0 i ⊆ T... |

20 |
Fragments of Peano’s arithmetic and the MRDP theorem
- Gaifman, Dimitracopoulos
- 1982
(Show Context)
Citation Context ...the more interesting. It is one of the weakest fragments of arithmetic known to prove the Matiyasevic Robinson Davis Putnam Theorem (MRDP) Theorem that every Σ1-formula is equivalent to an ∃1-formula =-=[6]-=-. It is also known to be both finitely axiomatized [15] and equivalent to its IE1+exp fragment [8]. In contrast the bounded arithmetic theories I∆0 and I∆0+Ω1 (equivalent to Buss’ S2), which have indu... |

17 | Algorithms for Boolean formula evaluation and for tree-contraction
- Buss
- 1993
(Show Context)
Citation Context ... In this section we will assume we have carried out a formalization of the syntax of bounded arithmetic proofs within I∆0+exp. The reader is invited to consult Buss [2], Hajek and Pudlak [7], or Buss =-=[3]-=- for details on how this may be carried out. We write T hmF CFT ( ⌈ φ ⌉ ) for the formula which says “φ codes a formula which is a free-cut free theorem of theory T ”. We write F CF Con(T ) for the fo... |

10 | Arithmetic Theories with Prenex Normal Form Induction - Pollett - 1997 |

7 |
Bounded existential induction
- Wilmers
- 1985
(Show Context)
Citation Context .... This indicates IOpen(exp) may be substantially stronger than IOpen which has recursive models. As far as the author knows it is still open if IOpen(exp) has recursive models. It is known by Wilmers =-=[16]-=- that IE1 does not. We show IOpen(exp) is not equal to I∆0+exp also holds if one adds new function symbols and axioms that respect our translation and if the theory that is translated to is in interpr... |

4 |
Bounded arithmetic, complexity and cryptography
- Buss
- 1998
(Show Context)
Citation Context ...proves (∀x)((∃y)(y = 2 x k) ⊃ A(x)). Here 2 x k is a stack of 2’s k high with an x at the top. This result has both a simple compactness argument proof [7] as well as a proof using Herbrand’s theorem =-=[4]-=- which could in principle be used to give a bound on k in terms of the maximum nesting depths of exp in the I∆0+exp proof. Intuitively, this result says: given x, if I∆0 knows a big enough y exists th... |

4 |
Exponential time and bounded arithmetic (extended abstract
- Clote, Takeuti
- 1986
(Show Context)
Citation Context ...r an IOpen(exp) proof P . Given this function I∆0+exp can define a function which translates an IOpen(exp) proof P of A into an IOpen proof of A n for this n. To see this recall the folklore 9sresult =-=[5]-=- that I∆0+exp can ∆0-define any elementary function. In this case, the case the proof of Theorem 10 gives a translation which can be computed in polynomial time. Notice for any n our translation has (... |

1 |
Diophantine and parameter free induction
- Kaye
- 1987
(Show Context)
Citation Context ...vic Robinson Davis Putnam Theorem (MRDP) Theorem that every Σ1-formula is equivalent to an ∃1-formula [6]. It is also known to be both finitely axiomatized [15] and equivalent to its IE1+exp fragment =-=[8]-=-. In contrast the bounded arithmetic theories I∆0 and I∆0+Ω1 (equivalent to Buss’ S2), which have induction on formulas involving only sub-exponential growth rate functions are not known to prove the ... |

1 |
Multifunction algebras and the provability of PH⇓. To appear Annals of Pure and Applied Logic
- Pollett
(Show Context)
Citation Context ...ng function symbols. The reason why we are interested in such weak theories is that recently it has been shown that when m − i ≥ 4 these theories cannot prove the collapse of the polynomial hierarchy =-=[10]-=-. We hope our translations might be useful for separation results. To see that this may be possible, in this paper we show IOpen(exp) � I∆0(exp). This is done by showing if IOpen(exp) proves an open f... |

1 |
Non-standard models for fragments of number theory
- Shepherson
- 1965
(Show Context)
Citation Context ...≤ exp-rank(P )+1 such that IOpen proves A n . We 2suse this to show I∆0(exp) ⊢ F CF Con(IOpen(exp)). Since I∆0(exp) does not prove its own free-cut-free consistency this gives the result. Shepherdson =-=[13]-=- has noted that IOpen(exp) can prove the irrationality of √ 2, which is not provable in IOpen. This indicates IOpen(exp) may be substantially stronger than IOpen which has recursive models. As far as ... |

1 | Multifunction algebras and the provability of PH+. Submitted. Available at: http://aleph0.clarku.edu/~cpollett/index.html - Pollett |