## Unfolding feasible arithmetic and weak truth (2012)

Citations: | 1 - 1 self |

### BibTeX

@MISC{Eberhard12unfoldingfeasible,

author = {Sebastian Eberhard and Thomas Strahm},

title = {Unfolding feasible arithmetic and weak truth},

year = {2012}

}

### OpenURL

### Abstract

In this paper we continue Feferman’s unfolding program initiated in [11] which uses the concept of the unfolding U(S) of a schematic system S in order to describe those operations, predicates and principles concerning them, which are implicit in the acceptance of S. The program has been carried through for a schematic system of non-finitist arithmetic NFA in Feferman and Strahm [13] and for a system FA (with and without Bar rule) in Feferman and Strahm [14]. The present contribution elucidates the concept of unfolding for a basic schematic system FEA of feasible arithmetic. Apart from the operational unfolding U0(FEA) of FEA, we study two full unfolding notions, namely the predicate unfolding U(FEA) and a more general truth unfolding UT(FEA) of FEA, the latter making use of a truth predicate added to the language of the operational unfolding. The main results obtained are that the provably convergent functions on binary words for all three unfolding systems are precisely those being computable in polynomial time. The upper bound computations make essential use of a specific theory of truth TPT over combinatory logic, which has recently been introduced in Eberhard and Strahm [7] and Eberhard [6] and whose involved proof-theoretic analysis is due to Eberhard [6]. The results of this paper were first announced in [8].

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(Show Context)
Citation Context ...ded. Lemma 1 The polynomial time computable functions are provably total in the operational unfolding U0(FEA). Proof. We use Cobham’s characterization of the polynomial time computable functions (cf. =-=[5, 4]-=-): starting off from the initial functions of L and arbitrary projections, the polynomial time computable functions can be generated by closing under composition and bounded recursion. First of all, t... |

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Citation Context ...ements of an underlying combinatory algebra which is extended by a binary relation ∈ for elementship, so predicates are represented via classifications in the sense of Feferman’s explicit mathematics =-=[9, 10]-=-. We additionally use a relation Π to single out the operations representing predicates one is committed to by accepting FEA. The language L2 of U(FEA) is an extension of L1 by new individual constant... |

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Citation Context ...ements of an underlying combinatory algebra which is extended by a binary relation ∈ for elementship, so predicates are represented via classifications in the sense of Feferman’s explicit mathematics =-=[9, 10]-=-. We additionally use a relation Π to single out the operations representing predicates one is committed to by accepting FEA. The language L2 of U(FEA) is an extension of L1 by new individual constant... |

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(Show Context)
Citation Context ...ded. Lemma 1 The polynomial time computable functions are provably total in the operational unfolding U0(FEA). Proof. We use Cobham’s characterization of the polynomial time computable functions (cf. =-=[5, 4]-=-): starting off from the initial functions of L and arbitrary projections, the polynomial time computable functions can be generated by closing under composition and bounded recursion. First of all, t... |

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Citation Context ...asse 10, CH-3012 Bern, Switzerland. Email: strahm@iam.unibe.ch. Homepage: http://www.iam.unibe.ch/~strahm 11 Introduction The notion of unfolding a schematic formal system was introduced in Feferman =-=[11]-=- in order to answer the following question: Given a schematic system S, which operations and predicates, and which principles concerning them, ought to be accepted if one has accepted S? A paradigmati... |

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Citation Context ...tion rule. Let us sketch the theory TPT; for a more detailed description, the reader is referred to [7, 6]. For a more extensive survey on similar kinds of theories in a stronger setting, see Cantini =-=[1]-=- and Kahle [16]. TPT is based on a total version of the basic applicative theory B for words which was developed in Strahm [17]. In particular, we have a word predicate W which is interpreted as the t... |

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Citation Context ...ve survey on similar kinds of theories in a stronger setting, see Cantini [1] and Kahle [16]. TPT is based on a total version of the basic applicative theory B for words which was developed in Strahm =-=[17]-=-. In particular, we have a word predicate W which is interpreted as the type of binary strings, constants for some simple functions on the words and a computationally complete combinatory algebra. TPT... |

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Citation Context ...m T(πix) ↔ Pi(x) for every i ∈ N. Since no other axioms for the P 11 We note that TPT can be seen as a polynomial time analogue of a theory of truth of primitive recursive strength studied in Cantini =-=[2, 3]-=-. 12 As usual for applicative systems, we call a function F : W n → W provably total in TPT, if there exists a closed term tF such that (i) tF defines F pointwise, i.e. on each standard word, and, mor... |

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Citation Context ...m T(πix) ↔ Pi(x) for every i ∈ N. Since no other axioms for the P 11 We note that TPT can be seen as a polynomial time analogue of a theory of truth of primitive recursive strength studied in Cantini =-=[2, 3]-=-. 12 As usual for applicative systems, we call a function F : W n → W provably total in TPT, if there exists a closed term tF such that (i) tF defines F pointwise, i.e. on each standard word, and, mor... |

7 |
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Citation Context ... the following rule of term substitution: (S0) Γ[α] → A[α] Γ[τ] → A[τ] The non-logical axioms of FEA state the usual defining equations for the function symbols of the language L, see, e.g., Ferreria =-=[15]-=- for similar axioms. Finally, we have the schematic induction rule formulated for a free predicate P as follows: (Ind) Γ → P (ɛ) Γ, P (α) → P (Si(α)) (i = 0, 1) Γ → P (α) In the various unfolding syst... |

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(Show Context)
Citation Context ...predicates and principles concerning them, which are implicit in the acceptance of S. The program has been carried through for a schematic system of non-finitist arithmetic NFA in Feferman and Strahm =-=[13]-=- and for a system FA (with and without Bar rule) in Feferman and Strahm [14]. The present contribution elucidates the concept of unfolding for a basic schematic system FEA of feasible arithmetic. Apar... |

4 |
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(Show Context)
Citation Context ... us sketch the theory TPT; for a more detailed description, the reader is referred to [7, 6]. For a more extensive survey on similar kinds of theories in a stronger setting, see Cantini [1] and Kahle =-=[16]-=-. TPT is based on a total version of the basic applicative theory B for words which was developed in Strahm [17]. In particular, we have a word predicate W which is interpreted as the type of binary s... |

3 | Unfolding finitist arithmetic
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(Show Context)
Citation Context ...nce of S. The program has been carried through for a schematic system of non-finitist arithmetic NFA in Feferman and Strahm [13] and for a system FA (with and without Bar rule) in Feferman and Strahm =-=[14]-=-. The present contribution elucidates the concept of unfolding for a basic schematic system FEA of feasible arithmetic. Apart from the operational unfolding U0(FEA) of FEA, we study two full unfolding... |

2 |
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(Show Context)
Citation Context ...n polynomial time. The upper bound computations make essential use of a specific theory of truth TPT over combinatory logic, which has recently been introduced in Eberhard and Strahm [7] and Eberhard =-=[6]-=- and whose involved proof-theoretic analysis is due to Eberhard [6]. The results of this paper were first announced in [8]. ∗Institut für Informatik und angewandte Mathematik, Universität Bern, Neubrü... |

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2 |
The Oxford Handbook of the Philosophy of Mathematics and
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(Show Context)
Citation Context ...s proof-theoretically equivalent to predicative analysis. For more information on the path to the unfolding program, especially with regard to predicativity and the implicitness program, see Feferman =-=[12]-=-. More recently, the unfolding notions for a basic schematic system of finitist arithmetic, FA, and for an extension of that by a form BR of the so-called bar rule have been worked out in Feferman and... |