## Applications of inductive definitions and choice principles to program synthesis

Citations: | 1 - 0 self |

### BibTeX

@MISC{Berger_applicationsof,

author = {Ulrich Berger and Monika Seisenberger},

title = {Applications of inductive definitions and choice principles to program synthesis},

year = {}

}

### OpenURL

### Abstract

Abstract. We describe two methods of extracting constructive content from classical proofs, focusing on theorems involving infinite sequences and nonconstructive choice principles. The first method removes any reference to infinite sequences and transforms the theorem into a system of inductive definitions, the other applies a combination of Gödel’s negativeand Friedman’s A-translation. Both approaches are explained by means of a case study on Higman’s Lemma and its well-known classical proof due to Nash-Williams. We also discuss some proof-theoretic optimizations that were crucial for the formalization and implementation of this work in the interactive proof system Minlog. 1

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Citation Context ...C using a special realizability interpretation based on infinite terms. Oliva and the first author [BO04] showed that Kreisel’s modified realizability [Kre59] together with Plotkin’s adequacy theorem =-=[Plo77]-=- can be used instead (thus avoiding infinite terms, the role of which is taken over by the Scott/Ershov model of partial continuous functionals). In our case study we worked with a realizer of DC A wh... |

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Citation Context ... Munich [BBS + 98]. We will focus on theorems about infinite sequences as they typically occur in analysis and infinitary combinatorics. Our running example and main case study will be Higman’s Lemma =-=[Hig52]-=- 1 and its classical proof due to NashWilliams [NW63] which we will briefly describe now. Higman’s Lemma is concerned with well-quasiorders (wqos), that is, binary relations (A, ≤A) on a set A of lett... |

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Citation Context ...ionals over the flat domains of partial booleans and partial natural numbers [Sco70,Ers77,Tro73,BO04]. It is important to note that models where all functionals are computable –like, for example, HEO =-=[Tro73]-=-, or, more generally, the effective topos [Hyl82]– cannot be used here, since in order for the argument given above to be valid the sequences g and h approximated by the xs and ys have to be (possibly... |

80 |
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(Show Context)
Citation Context ...and partial natural numbers [Sco70,Ers77,Tro73,BO04]. It is important to note that models where all functionals are computable –like, for example, HEO [Tro73], or, more generally, the effective topos =-=[Hyl82]-=-– cannot be used here, since in order for the argument given above to be valid the sequences g and h approximated by the xs and ys have to be (possibly non-computable) free choice sequences (a related... |

78 |
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(Show Context)
Citation Context ... above sketched computational interpretation of DC using a special realizability interpretation based on infinite terms. Oliva and the first author [BO04] showed that Kreisel’s modified realizability =-=[Kre59]-=- together with Plotkin’s adequacy theorem [Plo77] can be used instead (thus avoiding infinite terms, the role of which is taken over by the Scott/Ershov model of partial continuous functionals). In ou... |

74 |
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(Show Context)
Citation Context ... A with a decidable well-quasiorder. In section 3 we take another approach: we leave the statement of Higman’s Lemma as it is, but apply a combination of Gödel’s negative- and Friedman’ A-translation =-=[Fri78]-=- to constructivize the proof. This idea goes back to Constable and Murthy [CM91] who formalized Higman’s Lemma in a system of second order arithmetic representing infinite sequences by their graphs an... |

74 |
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(Show Context)
Citation Context ... contain function types (in the example of Higman’s Lemma ν is the type of pairs of integers). In [BO04] the functional Ψ is called modified bar recursion and it is shown that Spector’s bar recursion =-=[Spe62]-=-, which is the scheme SBR Ψ(xs) = � G(xs) if Y ( ˜xs) < |xs| H(xs, λx.Ψ(xs ∗ x)) if Y ( ˜xs) ≥ |xs| can be primitive recursively defined from MBR, but not vice versa. Berardi, Bezem and Coquand [BBC98... |

65 |
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(Show Context)
Citation Context ...finite sequences as they typically occur in analysis and infinitary combinatorics. Our running example and main case study will be Higman’s Lemma [Hig52] 1 and its classical proof due to NashWilliams =-=[NW63]-=- which we will briefly describe now. Higman’s Lemma is concerned with well-quasiorders (wqos), that is, binary relations (A, ≤A) on a set A of letters with the property that each infinite sequence (ai... |

63 |
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(Show Context)
Citation Context ...ed Higman’s Lemma in a system of second order arithmetic representing infinite sequences by their graphs and proving the necessary choice principles by impredicative comprehension and classical logic =-=[Mur90]-=-. Our formalization differs from theirs in that we work in a finite type system where infinite sequences are available as objects type N → ρ. This has the advantage that the formalization of Nash-Will... |

54 | Program extraction from classical proofs
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(Show Context)
Citation Context ... by (C → A) → A where A is an existential formula. This has the effect that higher types and many case distinctions come up in the extracted program which may lead to complex and inefficient code. In =-=[BBS02]-=- a refined A-translation is introduced that minimizes double negations and hence reduce these negative effects. These refinements are implemented in Minlog and we have tested them in our case study. A... |

53 |
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(Show Context)
Citation Context ...at ensure the soundness of the system. The distinction between logical constructs with and without computational content is similar to the distinction Set/Prop in intuitionistic type theory (see e.g. =-=[PW93]-=-), but seems to be more flexible. 3 Classical dependent choice Now we show how to extract computational content directly from Nash-Williams’ proof. Recall that in this proof one derives a contradictio... |

37 |
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(Show Context)
Citation Context ...classical proof via A-translation described in section 3 could be extended in a straightforward way to a corresponding classical proof of Kruskal’s Theorem, even in its strong form with gap condition =-=[Sim85]-=-. The latter would be interesting because then we could extract a program from a theorem for which no constructive proof is known so far. On the other hand, the inductive method of section 2 seems to ... |

34 | On the computational content of the axiom of choice
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(Show Context)
Citation Context ...on here, but instead informally explain how to directly interpret DC A computationally in terms of realizability. The idea of the following interpretation of DC A is due to Berardi, Bezem and Coquand =-=[BBC98]-=-. In order to realize DC A we assume we are given realizers G1, G2, G3 of the hypotheses Hyp 1, Hyp 2, Hyp 3 respectively. We have to compute a realizer of A. We use G3. So, we need to compute some fu... |

29 | Model C of partial continuous functionals - Ershov - 1977 |

26 | Modified bar recursion and classical dependent choice
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(Show Context)
Citation Context ...) where xs varies over finite sequences of some type ρ and the equation is of some type ν that doesn’t contain function types (in the example of Higman’s Lemma ν is the type of pairs of integers). In =-=[BO04]-=- the functional Ψ is called modified bar recursion and it is shown that Spector’s bar recursion [Spe62], which is the scheme SBR Ψ(xs) = � G(xs) if Y ( ˜xs) < |xs| H(xs, λx.Ψ(xs ∗ x)) if Y ( ˜xs) ≥ |x... |

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25 | Proof theory at work: Program development in the Minlog system - Benl, Berger, et al. - 1998 |

24 | Ordinal numbers and the Hilbert basis theorem - Simpson - 1988 |

23 | The Warshall Algorithm and Dickson’s Lemma: Two Examples of Realistic Program Extraction - Berger, Schwichtenberg, et al. |

19 | A constructive proof of Higman’s lemma - Murthy, Russell - 1990 |

17 |
Finding computational content in classical proofs
- Constable, Murthy
- 1991
(Show Context)
Citation Context ...ave the statement of Higman’s Lemma as it is, but apply a combination of Gödel’s negative- and Friedman’ A-translation [Fri78] to constructivize the proof. This idea goes back to Constable and Murthy =-=[CM91]-=- who formalized Higman’s Lemma in a system of second order arithmetic representing infinite sequences by their graphs and proving the necessary choice principles by impredicative comprehension and cla... |

13 |
Eine in der reinen Zahlentheorie unbeweisbarer Satz über endliche Folgen von natürlichen Zahlen
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(Show Context)
Citation Context ...]). The interest in our methods lies in the fact that they both yield proofs and programs that make constructive use of the crucial ideas in Nash-Williams’ nonconstructive proof, whereas the proof in =-=[SS85]-=- does not seem to be related to Nash-Williams’ proof. The same applies to another inductive proof given by Fridlender [Fri97] which is based on an intuitionistic proof of Veldman (published in [Vel04]... |

9 | A proof of Higman's lemma by structural induction
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(Show Context)
Citation Context ...objects is avoided. In section 2 we will do this, following 1 Higman’s lemma is used, for example, in term rewriting theory for termination proofs [CTB94,Tou02].san approach of Coquand and Fridlender =-=[CF94]-=-. The main idea is to express the property of being a well-quasiorder by an inductive definition. In [CF94] the case A = {0, 1} was treated whereas we allow an arbitrary alphabet A with a decidable we... |

9 | On the logical strength of Nash-Williams’ theorem on transfinite sequences
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- 1996
(Show Context)
Citation Context ...ses the idea of Higman’s original proof. The proof-theoretic strength of a form of the minimal bad sequence argument that is sufficiently general for Nash-Williams’ proof has been analyzed by Marcone =-=[Mar96]-=-.s2 Inductive definitions In this section we describe how Nash-Williams’ classical proof of Higman’s Lemma given in the introduction may be transformed into a constructive inductive argument. The cruc... |

8 | Normalization by evaluation
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- 1998
(Show Context)
Citation Context ...hese negative effects. These refinements are implemented in Minlog and we have tested them in our case study. Another improvement is specific to Minlog’s implementation of normalization by evaluation =-=[BES98]-=-. We introduced the functional Ψ in Minlog as a program constant together with a rewrite rule corresponding to MBR. Normalization by evaluation means that in order to normalize a term, it is evaluated... |

8 | An intuitionistic proof of Kruskal’s theorem
- Veldman
- 2000
(Show Context)
Citation Context ... [SS85] does not seem to be related to Nash-Williams’ proof. The same applies to another inductive proof given by Fridlender [Fri97] which is based on an intuitionistic proof of Veldman (published in =-=[Vel04]-=-). Veldman’s proof does not require decidability of ≤A and uses the idea of Higman’s original proof. The proof-theoretic strength of a form of the minimal bad sequence argument that is sufficiently ge... |

7 | A computational interpretation of open induction - Berger |

7 |
A constructive proof of Higman’s lemma in Isabelle
- Berghofer
- 2004
(Show Context)
Citation Context ...undertaken such an implementation. The inductive approach presented in this paper may be carried out in any theorem prover supporting program extraction from inductive definitions (see, for instance, =-=[Berg04]-=- for an implementation of Higman’s Lemma in Isabelle and [Fri97] for a formalization of a different proof in Alf). The second approach is more specific to the Minlog system, since it requires an imple... |

6 | Well quasi-ordered sets - Richman, Stolzenberg - 1990 |

6 |
On the Constructive Content of Proofs
- Seisenberger
- 2003
(Show Context)
Citation Context ...duction step the decidability of equality on A (≤A in the general case) is used. An inductive proof of Higman’s Lemma for an arbitrary well-quasiordered alphabet - not only a finite one - is given in =-=[Sei03]-=- (see also [Sei01] for an earlier version). In this case, the induction on the number of letters that do not occur as last letters becomes an induction on the predicate BarA. Furthermore, the sequence... |

5 | Tahhan Bittar, “Ordinal recursive bounds for Higman’s Theorem - Cichoń, E - 1998 |

5 | Higman’s lemma in type theory
- Fridlender
- 1998
(Show Context)
Citation Context ...crucial ideas in Nash-Williams’ nonconstructive proof, whereas the proof in [SS85] does not seem to be related to Nash-Williams’ proof. The same applies to another inductive proof given by Fridlender =-=[Fri97]-=- which is based on an intuitionistic proof of Veldman (published in [Vel04]). Veldman’s proof does not require decidability of ≤A and uses the idea of Higman’s original proof. The proof-theoretic stre... |

4 |
A program from an A-translated impredicative proof of Higman’s Lemma
- Herbelin
(Show Context)
Citation Context ...h an analysis, although it is considerably shorter than the program extracted by Murthy [Mur90]. It also remains unclear how our programs are related to those extracted by Murthy [Mur90] and Herbelin =-=[Her94]-=-. Since Higman’s Lemma is provable in a system which is conservative over first-order arithmetic, we know that wellfounded recursion is not needed at all, but Gödel primitive recursion (of type N → N ... |

3 | An inductive version of Nash-Williams’ minimal-badsequence argument for Higman’s Lemma
- Seisenberger
- 2002
(Show Context)
Citation Context ...ecidability of equality on A (≤A in the general case) is used. An inductive proof of Higman’s Lemma for an arbitrary well-quasiordered alphabet - not only a finite one - is given in [Sei03] (see also =-=[Sei01]-=- for an earlier version). In this case, the induction on the number of letters that do not occur as last letters becomes an induction on the predicate BarA. Furthermore, the sequence structure, given ... |

3 | A characterisation of multiply recursive functions with Higman’s lemma - Touzet |

2 | Constructive toplogy and combinators - Coquand - 1991 |

2 | Well–orderings and their maximal order types - Schmidt - 1979 |

1 | A Note on the Open Induction Principle - Coquand - 1997 |

1 | Proof theory and complexity - Girard - 1987 |

1 | Well partial orderings and their order types - Jongh, Parikh - 1977 |