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词条 Bar recursion
释义

  1. Technical definition

  2. References

Bar recursion is a generalized form of recursion developed by C. Spector in his 1962 paper.[1] It is related to bar induction in the same fashion that primitive recursion is related to ordinary induction, or transfinite recursion is related to transfinite induction.

Technical definition

Let V, R, and O be types, and i be any natural number, representing a sequence of parameters taken from V. Then the function sequence f of functions fn from Vi+nR to O is defined by bar recursion from the functions Ln : RO and B with Bn : ((Vi+nR) x (VnR)) → O if:

  • fn((λα:Vi+n)r) = Ln(r) for any r long enough that Ln+k on any extension of r equals Ln. Assuming L is a continuous sequence, there must be such r, because a continuous function can use only finitely much data.
  • fn(p) = Bn(p, (λx:V)fn+1(cat(p, x))) for any p in Vi+nR.

Here "cat" is the concatenation function, sending p, x to the sequence which starts with p, and has x as its last term.

(This definition is based on the one by Escardó and Oliva.[2])

Provided that for every sufficiently long function (λα)r of type ViR, there is some n with Ln(r) = Bn((λα)r, (λx:V)Ln+1(r)), the bar induction rule ensures that f is well-defined.

The idea is that one extends the sequence arbitrarily, using the recursion term B to determine the effect, until a sufficiently long node of the tree of sequences over V is reached; then the base term L determines the final value of f. The well-definedness condition corresponds to the requirement that every infinite path must eventually pass through a sufficiently long node: the same requirement that is needed to invoke a bar induction.

The principles of bar induction and bar recursion are the intuitionistic equivalents of the axiom of dependent choices.[3]

References

1. ^{{cite book|author=C. Spector|chapter=Provably recursive functionals of analysis: a consistency proof of analysis by an extension of principles in current intuitionistic mathematics|editor=F. D. E. Dekker|title=Recursive Function Theory: Proc. Symposia in Pure Mathematics|volume=5|pages=1–27|year=1962|publisher=American Mathematical Society}}
2. ^{{cite journal|author=Martín Escardó|author2=Paulo Oliva|title=Selection functions, Bar recursion, and Backwards Induction|journal=Math. Struct. in Comp.Science|url=http://www.cs.bham.ac.uk/~mhe/papers/selection-escardo-oliva.pdf}}
3. ^{{cite book|author=Jeremy Avigad|author-link=Jeremy Avigad|author2=Solomon Feferman|author2-link=Solomon Feferman|chapter=VI: Gödel's functional ("Dialectica") interpretation|title=Handbook of Proof Theory|editor=S. R. Buss|editor-link=S. R. Buss|year=1999|url=http://math.stanford.edu/~feferman/papers/dialectica.pdf}}
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1 : Recursion

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