## The computational content of classical arithmetic (2009)

Citations: | 1 - 0 self |

### BibTeX

@MISC{Avigad09thecomputational,

author = {Jeremy Avigad},

title = {The computational content of classical arithmetic },

year = {2009}

}

### OpenURL

### Abstract

Almost from the inception of Hilbert’s program, foundational and structural efforts in proof theory have been directed towards the goal of clarifying the computational content of modern mathematical methods. This essay surveys various methods of extracting computational information from proofs in classical first-order arithmetic, and reflects on some of the relationships between them. Variants of the Gödel-Gentzen doublenegation translation, some not so well known, serve to provide canonical and efficient computational interpretations of that theory.

### Citations

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Citation Context ...rithm: 1. Gödel’s Dialectica interpretation [18, 6], in conjunction with a doublenegation interpretation that interprets PA in its intuitionistic counterpart, Heyting arithmetic (HA) 2. realizability =-=[22, 23, 27, 45]-=-, again in conjunction with a double-negation translation, and either the Friedman A-translation [14] (often also attributed to Dragalin and Leivant, independently) or a method due to Coquand and Hofm... |

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Citation Context ...eveloped in support of software verification [21], and a better understanding of the computational content of classical logic may support the development of better logical frameworks for that purpose =-=[40]-=-. Formal translations like the ones described here have also been effective in “proof mining,” the practice of using logical methods to extract mathematically useful information from nonconstructive p... |

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Citation Context ...alents. Since the translation relies on the negation-normal form representation of classical formulas, it shares many nice properties with a more complicated double-negation translation due to Girard =-=[17]-=-. It is this translation that I will use, in the next section, to provide an efficient computational interpretation of classical arithmetic. Let me close with one more translation, found in [3], which... |

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Citation Context ...for producing such a y from a given x, one that is more informative than blind search. There are four methods that are commonly used to extract such an algorithm: 1. Gödel’s Dialectica interpretation =-=[18, 6]-=-, in conjunction with a doublenegation interpretation that interprets PA in its intuitionistic counterpart, Heyting arithmetic (HA) 2. realizability [22, 23, 27, 45], again in conjunction with a doubl... |

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Citation Context ...mple, [40, 45]). It was Gödel [17] who first showed that the provably total computable functions of arithmetic can be characterized in terms of the primitive recursive functionals of finite type (see =-=[6, 19]-=-). The set of finite types can be defined to be the smallest set containing the symbol N, and closed under an operation which takes types σ and τ to a new type σ → τ. In the intended (“full”) interpre... |

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Citation Context ..., it is far from clear that classical arithmetic can be used as an effective programming language in its own right. But formal methods are actively being developed in support of software verification =-=[21]-=-, and a better understanding of the computational content of classical logic may support the development of better logical frameworks for that purpose [40]. Formal translations like the ones described... |

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Citation Context ...rithm: 1. Gödel’s Dialectica interpretation [18, 6], in conjunction with a doublenegation interpretation that interprets PA in its intuitionistic counterpart, Heyting arithmetic (HA) 2. realizability =-=[22, 23, 27, 45]-=-, again in conjunction with a double-negation translation, and either the Friedman A-translation [14] (often also attributed to Dragalin and Leivant, independently) or a method due to Coquand and Hofm... |

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Citation Context ...rithm: 1. Gödel’s Dialectica interpretation [18, 6], in conjunction with a doublenegation interpretation that interprets PA in its intuitionistic counterpart, Heyting arithmetic (HA) 2. realizability =-=[22, 23, 27, 45]-=-, again in conjunction with a double-negation translation, and either the Friedman A-translation [14] (often also attributed to Dragalin and Leivant, independently) or a method due to Coquand and Hofm... |

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Citation Context ...for producing such a y from a given x, one that is more informative than blind search. There are four methods that are commonly used to extract such an algorithm: 1. Gödel’s Dialectica interpretation =-=[18, 6]-=-, in conjunction with a doublenegation interpretation that interprets PA in its intuitionistic counterpart, Heyting arithmetic (HA) 2. realizability [22, 23, 27, 45], again in conjunction with a doubl... |

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Citation Context ...and Leivant, independently) or a method due to Coquand and Hofmann ([12, 1]) to “repair” translated Π2 assertions 3. cut elimination ([15]; see, for example, [40]) 4. the epsilon substitution method (=-=[18, 8]-=-) These four approaches really come in two pairs: the Dialectica interpretation and realizability have much in common, and, indeed, Paulo Oliva [38] has recently shown that one can interpolate a range... |

59 | Classical Logic and Computation
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Citation Context ...racting witnesses from classical proofs yield different results, conveying the impression that there is something “nondeterministic” about classical logic. (There is a very nice discussion of this in =-=[47, 48]-=-. See also the discussion in Section 6 below.) Insofar as one has a natural translation from classical arithmetic to intuitionistic arithmetic, some of the canonicity of the associated computation is ... |

54 |
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Citation Context ...w, as a function of c, yields the following conclusions: Theorem 2.1. If classical arithmetic proves ∀x ¬¬∃y R(x, y), there is a term F of PR ω of type N to N such that HA ω proves ∀x R(x, F(x)). See =-=[45, 24]-=- for more about realizability, and [14] for the A-translation. Gödel’s Dialectica interpretation provides an alternative route to this result. In fact, one obtains a stronger conclusion, namely that t... |

49 |
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Citation Context ...he Friedman A-translation [14] (often also attributed to Dragalin and Leivant, independently) or a method due to Coquand and Hofmann ([12, 1]) to “repair” translated Π2 assertions 3. cut elimination (=-=[15]-=-; see, for example, [41]) 4. the epsilon substitution method ([19, 8]) These four approaches really come in two pairs: the Dialectica interpretation and realizability have much in common, and, indeed,... |

49 |
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Citation Context ...hile the second two do not. It is true that the Dialectica interpretation and realizability can be applied to classical calculi directly (see [42] for the Dialectica interpretation, and, for example, =-=[2, 38]-=- for realizability); but I know of no such interpretation that cannot be understood in terms of a passage through intuitionistic arithmetic [2, 5, 44]. In contrast, cut elimination and the epsilon sub... |

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Citation Context ...racting witnesses from classical proofs yield different results, conveying the impression that there is something “nondeterministic” about classical logic. (There is a very nice discussion of this in =-=[47, 48]-=-. See also the discussion in Section 6 below.) Insofar as one has a natural translation from classical arithmetic to intuitionistic arithmetic, some of the canonicity of the associated computation is ... |

29 |
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(Show Context)
Citation Context ...hat the simplicity of the theory is deceptive: via direct interpretation or more elaborate forms of proof-theoretic reduction, vast portions of mathematical reasoning can be understood in terms of PA =-=[4, 13, 43]-=-. Here, I will be concerned with the Π2, or “computational,” consequences of PA. Suppose PA proves ∀x ∃y R(x, y), where x and y range over the natural numbers and R(x, y) is a decidable (say, primitiv... |

26 | Local stability of ergodic averages
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(Show Context)
Citation Context ...al translations like the ones described here have also been effective in “proof mining,” the practice of using logical methods to extract mathematically useful information from nonconstructive proofs =-=[7, 24, 26]-=-. Grisha’s work has, primarily, addressed the general foundational question as to the computational content of classical methods. In that respect, the general metatheorems described here provide a sat... |

25 |
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(Show Context)
Citation Context ...ant DMS-0700174 and a grant from the John Templeton Foundation. 1([28, 29]); his method of continuous cut elimination, which provides a finitary interpretation of infinitary cut-elimination methods (=-=[32, 11]-=-); and his work on the epsilon substitution method (for example, [35, 37]). Grisha has also been a friend and mentor to me throughout my career. The characterization of the provably total computable f... |

22 | The Warshall algorithm and Dickson’s lemma: Two examples of realistic program extraction
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(Show Context)
Citation Context ..., and so the process of extracting an algorithm from even a rather constructive mathematical argument can involve nondeterminism of sorts. And, despite some interesting explorations in this direction =-=[10]-=-, it is far from clear that classical arithmetic can be used as an effective programming language in its own right. But formal methods are actively being developed in support of software verification ... |

21 |
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(Show Context)
Citation Context ...in in conjunction with a double-negation translation, and either the Friedman A-translation [14] (often also attributed to Dragalin and Leivant, independently) or a method due to Coquand and Hofmann (=-=[12, 1]-=-) to “repair” translated Π2 assertions 3. cut elimination ([15]; see, for example, [41]) 4. the epsilon substitution method ([19, 8]) These four approaches really come in two pairs: the Dialectica int... |

18 | On the no-counterexample interpretation
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(Show Context)
Citation Context ...erpretation of ϕ awk , so the result follows from Theorem 3.6 as well. The no-counterexample interpretation can be viewed as a computational interpretation of arithmetic. But, in a remarkable article =-=[25]-=-, Kohlenbach has shown that it is not a very modular computational interpretation, in the sense that it does not have nice behavior with respect to modus ponens. To make this claim precise, note that ... |

17 | Number theory and elementary arithmetic
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(Show Context)
Citation Context ...hat the simplicity of the theory is deceptive: via direct interpretation or more elaborate forms of proof-theoretic reduction, vast portions of mathematical reasoning can be understood in terms of PA =-=[4, 13, 43]-=-. Here, I will be concerned with the Π2, or “computational,” consequences of PA. Suppose PA proves ∀x ∃y R(x, y), where x and y range over the natural numbers and R(x, y) is a decidable (say, primitiv... |

16 | A new method of establishing conservativity of classical systems over their intuitionistic version
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- 1999
(Show Context)
Citation Context ...in in conjunction with a double-negation translation, and either the Friedman A-translation [14] (often also attributed to Dragalin and Leivant, independently) or a method due to Coquand and Hofmann (=-=[12, 1]-=-) to “repair” translated Π2 assertions 3. cut elimination ([15]; see, for example, [41]) 4. the epsilon substitution method ([19, 8]) These four approaches really come in two pairs: the Dialectica int... |

11 |
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(Show Context)
Citation Context ...), one obtains a term a satisfying A(a). This provides a direct proof of Theorem 2.1. Details can be found in [2]. A more elaborate realizability relation, based on the A-translation, can be found in =-=[9]-=-. The corresponding variant of the Dialectica translation is similarly straightforward. As with the Shoenfield variant [42, 5], each formula ϕ is mapped to a formula ϕD′ of the form ∀x ∃y ϕD ′(x, y), ... |

9 |
Proof mining: a systematic way of analyzing proofs
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(Show Context)
Citation Context ...al translations like the ones described here have also been effective in “proof mining,” the practice of using logical methods to extract mathematically useful information from nonconstructive proofs =-=[7, 24, 26]-=-. Grisha’s work has, primarily, addressed the general foundational question as to the computational content of classical methods. In that respect, the general metatheorems described here provide a sat... |

9 |
Shoenfield is Gödel after Krivine
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(Show Context)
Citation Context ...for the Dialectica interpretation, and, for example, [2, 38] for realizability); but I know of no such interpretation that cannot be understood in terms of a passage through intuitionistic arithmetic =-=[2, 5, 44]-=-. In contrast, cut elimination and the epsilon substitution method apply to classical logic directly. That is not to deny that one can apply cut elimination methods to intuitionistic logic (see, for e... |

8 | Algebraic proofs of cut elimination
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(Show Context)
Citation Context ...irard [17]. It is this translation that I will use, in the next section, to provide an efficient computational interpretation of classical arithmetic. Let me close with one more translation, found in =-=[3]-=-, which is interesting in its own right. For reasons that will become clear later on, I will call it “the awkward translation”: if ϕ is any formula in negation-normal form, let ϕ awk denote ¬(∼ϕ). The... |

8 |
Troelstra and Helmut Schwichtenberg. Basic Proof Theory (second edition
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- 2000
(Show Context)
Citation Context ...trast, cut elimination and the epsilon substitution method apply to classical logic directly. That is not to deny that one can apply cut elimination methods to intuitionistic logic (see, for example, =-=[46]-=-); but the arguments tend to be easier and more natural in the classical setting. Finally, there is the issue of canonicity. Algorithms extracted from proofs in intuitionistic arithmetic tend to produ... |

7 |
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Citation Context ...erprets PA in its intuitionistic counterpart, Heyting arithmetic (HA) 2. realizability [22, 23, 27, 45], again in conjunction with a double-negation translation, and either the Friedman A-translation =-=[14]-=- (often also attributed to Dragalin and Leivant, independently) or a method due to Coquand and Hofmann ([12, 1]) to “repair” translated Π2 assertions 3. cut elimination ([15]; see, for example, [41]) ... |

7 |
Mathematical Logic. Association for Symbolic Logic
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(Show Context)
Citation Context ...t two methods involve an intermediate translation to HA, while the second two do not. It is true that the Dialectica interpretation and realizability can be applied to classical calculi directly (see =-=[42]-=- for the Dialectica interpretation, and, for example, [2, 38] for realizability); but I know of no such interpretation that cannot be understood in terms of a passage through intuitionistic arithmetic... |

6 |
Proof theory: Some aspects of cut-elimination
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(Show Context)
Citation Context ...n [14] (often also attributed to Dragalin and Leivant, independently) or a method due to Coquand and Hofmann ([12, 1]) to “repair” translated Π2 assertions 3. cut elimination ([15]; see, for example, =-=[41]-=-) 4. the epsilon substitution method ([19, 8]) These four approaches really come in two pairs: the Dialectica interpretation and realizability have much in common, and, indeed, Paulo Oliva [39] has re... |

5 | A realizability interpretation for classical arithmetic
- Avigad
- 2000
(Show Context)
Citation Context ...hile the second two do not. It is true that the Dialectica interpretation and realizability can be applied to classical calculi directly (see [42] for the Dialectica interpretation, and, for example, =-=[2, 38]-=- for realizability); but I know of no such interpretation that cannot be understood in terms of a passage through intuitionistic arithmetic [2, 5, 44]. In contrast, cut elimination and the epsilon sub... |

5 |
The epsilon calculus. The Stanford Encyclopedia of Philosophy. CSLI
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- 2008
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Citation Context ...and Leivant, independently) or a method due to Coquand and Hofmann ([12, 1]) to “repair” translated Π2 assertions 3. cut elimination ([15]; see, for example, [40]) 4. the epsilon substitution method (=-=[18, 8]-=-) These four approaches really come in two pairs: the Dialectica interpretation and realizability have much in common, and, indeed, Paulo Oliva [38] has recently shown that one can interpolate a range... |

4 |
Infinity in mathematics: Is Cantor necessary
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(Show Context)
Citation Context ...hat the simplicity of the theory is deceptive: via direct interpretation or more elaborate forms of proof-theoretic reduction, vast portions of mathematical reasoning can be understood in terms of PA =-=[4, 13, 43]-=-. Here, I will be concerned with the Π2, or “computational,” consequences of PA. Suppose PA proves ∀x ∃y R(x, y), where x and y range over the natural numbers and R(x, y) is a decidable (say, primitiv... |

4 | Proof theory in the USSR 1925–1969 - Mints - 1991 |

4 |
Normalization of finite terms and derivations via infinite ones
- Mints
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Citation Context ...ant DMS-0700174 and a grant from the John Templeton Foundation. 1([28, 29]); his method of continuous cut elimination, which provides a finitary interpretation of infinitary cut-elimination methods (=-=[32, 11]-=-); and his work on the epsilon substitution method (for example, [35, 37]). Grisha has also been a friend and mentor to me throughout my career. The characterization of the provably total computable f... |

3 | Proofs and Types, Translated and with appendices by Paul Taylor and Yves - Girard - 1989 |

3 |
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Citation Context ...9]); his method of continuous cut elimination, which provides a finitary interpretation of infinitary cut-elimination methods ([32, 11]); and his work on the epsilon substitution method (for example, =-=[35, 37]-=-). Grisha has also been a friend and mentor to me throughout my career. The characterization of the provably total computable functions of IΣ1 just mentioned was, in fact, also discovered by Charles P... |

3 | Unifying functional interpretations
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(Show Context)
Citation Context ...ample, [41]) 4. the epsilon substitution method ([19, 8]) These four approaches really come in two pairs: the Dialectica interpretation and realizability have much in common, and, indeed, Paulo Oliva =-=[39]-=- has recently shown that one can interpolate a range of methods between the two; and, 2similarly, cut elimination and the epsilon substitution method have a lot in common, as work by Grisha (e.g. [36... |

2 |
A variant of the double-negation translation. Carnegie Mellon
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(Show Context)
Citation Context ...for the Dialectica interpretation, and, for example, [2, 38] for realizability); but I know of no such interpretation that cannot be understood in terms of a passage through intuitionistic arithmetic =-=[2, 5, 44]-=-. In contrast, cut elimination and the epsilon substitution method apply to classical logic directly. That is not to deny that one can apply cut elimination methods to intuitionistic logic (see, for e... |

2 |
On E-theorems
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(Show Context)
Citation Context ...in the classical setting. Finally, there is the issue of canonicity. Algorithms extracted from proofs in intuitionistic arithmetic tend to produce canonical witnesses to Π2 assertions; work by Grisha =-=[33, 34]-=- shows, for example, that algorithms extracted by various methods yield the same results. In contrast, different ways of extracting witnesses from classical proofs yield different results, conveying t... |

1 |
first volume, 1934, second volume
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(Show Context)
Citation Context ...and Leivant, independently) or a method due to Coquand and Hofmann ([12, 1]) to “repair” translated Π2 assertions 3. cut elimination ([15]; see, for example, [41]) 4. the epsilon substitution method (=-=[19, 8]-=-) These four approaches really come in two pairs: the Dialectica interpretation and realizability have much in common, and, indeed, Paulo Oliva [39] has recently shown that one can interpolate a range... |

1 |
Grigori Mints (Minc). Quantifier-free and one-quantifier systems
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(Show Context)
Citation Context ...bly total computable functions of IΣ1 as exactly the primitive recursive functions ∗ This work has been partially supported by NSF grant DMS-0700174 and a grant from the John Templeton Foundation. 1(=-=[28, 29]-=-); his method of continuous cut elimination, which provides a finitary interpretation of infinitary cut-elimination methods ([32, 11]); and his work on the epsilon substitution method (for example, [3... |

1 |
Grigori Mints (Minc). What can be done in primitive recursive arithmetic
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(Show Context)
Citation Context ...bly total computable functions of IΣ1 as exactly the primitive recursive functions ∗ This work has been partially supported by NSF grant DMS-0700174 and a grant from the John Templeton Foundation. 1(=-=[28, 29]-=-); his method of continuous cut elimination, which provides a finitary interpretation of infinitary cut-elimination methods ([32, 11]); and his work on the epsilon substitution method (for example, [3... |

1 | Cut elimination for a simple formulation of epsilon calculus
- Mints
- 2008
(Show Context)
Citation Context ...38] has recently shown that one can interpolate a range of methods between the two; and, 2similarly, cut elimination and the epsilon substitution method have a lot in common, as work by Grisha (e.g. =-=[35]-=-) shows. That is not to say that there aren’t significant differences between the methods in each pairing, but the differences between the two pairs are much more pronounced. For one thing, they produ... |