## Ordinal arithmetic with list structures (1992)

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Venue: | In Logical Foundations of Computer Science |

Citations: | 5 - 0 self |

### BibTeX

@INPROCEEDINGS{Dershowitz92ordinalarithmetic,

author = {Nachum Dershowitz and Edward M. Reingold},

title = {Ordinal arithmetic with list structures},

booktitle = {In Logical Foundations of Computer Science},

year = {1992},

pages = {117--126},

publisher = {SpringerVerlag}

}

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### Abstract

We provide a set of \natural " requirements for well-orderings of (binary) list structures. We showthat the resultant order-type is the successor of the rst critical epsilon number. The checker has to verify that the process comes to an end. Here again he should be assistedbytheprogrammer giving a further de nite assertion to be veri ed. This may take the form of a quantity which is asserted todecrease continually and vanish when the machine stops. To the pure mathematician it is natural to give an ordinal number. In this problem the ordinal might be (n, r)! 2 +(r, s)! + k. A less highbrow form of the same thing would be to give the integer 2 80 (n, r)+2 40 (r, s)+k. |Alan M. Turing (1949) 1

### Citations

566 | Assigning meaning to programs - Floyd - 1967 |

273 | Ordering by divisibility in abstract algebras - Higman - 1952 |

198 |
Z.: Proving termination with multiset orderings
- Dershowitz, Manna
- 1979
(Show Context)
Citation Context ... the natural numbers [ Dijkstra, 1976; Gries, 1981 ] , but lexicographic orderings (! n ) also play an important part [ Manna, 1974 ] . Occasionally, "larger" orderings have been used (for e=-=xample, [ Dershowitz and Manna, 1979-=-; Dershowitz, 1987 ] ); see [ Dershowitz, 1987; Dershowitz and Okada, 1988; Cichon, 1990 ] . The riddle above is a termination question on binary trees, one of the most pervasive data structures used ... |

193 |
Mathematical Theory of Computation
- Manna
- 1974
(Show Context)
Citation Context ...e quotation above). The well-ordering most commonly used is !, the natural ordering of the natural numbers [Dijkstra, 1976; Gries, 1981], but lexicographic orderings (! n ) also play an importantpart[=-=Manna, 1974-=-]. Occasionally, \larger" orderings have been used (for example, [Dershowitz and Manna, 1979; Dershowitz, 1987]); see [Dershowitz, 1987; Dershowitz and Okada, 1988; Cichon, 1990]. The riddle above is ... |

119 |
the tree theorem, and Vazsonyi's conjecture
- Well-quasi-ordering
- 1960
(Show Context)
Citation Context ...eferman, 1968; Schmidt, 1976]. Less natural orderings on (nonbinary) ordered trees correspond to much larger ordinals in that hierarchy. In particular, some orderings based on Kruskal's Tree Theorem [=-=Kruskal, 1960-=-] correspond to the rst impredicative ordinal, , 0, and even to larger ones [Friedman, 19??; Simpson, 1985; Smorynski, 1986; Dershowitz, 1987; Gallier, 1991]. The signi cance of , 0 for computer scien... |

39 | Some independence results for Peano arithmetic - Paris - 1978 |

31 |
Rapidly growing Ramsey functions
- Ketonen, Solovay
- 1981
(Show Context)
Citation Context ...)) 7! cons(x; pred n (y)) if y 6j nil For example, this binary-tree data structure could be used in implementing the computation of the various extensions of Ackermann's function (see, for example, [ =-=Ketonen and Solovay, 1981-=- ] ). An ordinal-indexed function A ff (n) can be defined for ordinals ff and natural numbers n by A ff (n) = 8 ? ! ? : 2n if ff = 0; ns1, A (n) fi (1) if ff is a successor ordinal fi + 1, A pred n (f... |

20 |
A Discipline of Programming. Prentice-Hall, Engewood Cliffs
- Dijkstra
- 1976
(Show Context)
Citation Context ...function decreases strictly with each repetition of a loop, as did Turing before him (see the quotation above). The well-ordering most commonly used is !, the natural ordering of the natural numbers [=-=Dijkstra, 1976-=-; Gries, 1981], but lexicographic orderings (! n ) also play an importantpart[Manna, 1974]. Occasionally, \larger" orderings have been used (for example, [Dershowitz and Manna, 1979; Dershowitz, 1987]... |

9 |
Systems of predicative analysis II: Representation of ordinals
- Feferman
- 1968
(Show Context)
Citation Context ...of +1. A similar analysis of in nite, not necessarily rational, binary trees may also be possible. 4 Bigger Ordinals The epsilon number is 2(0) in the Veblen-Feferman-Schutte hierarchy [Veblen, 1908; =-=Feferman, 1968-=-; Schmidt, 1976]. Less natural orderings on (nonbinary) ordered trees correspond to much larger ordinals in that hierarchy. In particular, some orderings based on Kruskal's Tree Theorem [Kruskal, 1960... |

8 |
Nichtbeweisbarkeit von gewissen kombinatorischen Eigenschaften endlicher BĂ¤ume
- Simpson
- 1985
(Show Context)
Citation Context ...er ordinals in that hierarchy. In particular, some orderings based on Kruskal's Tree Theorem [Kruskal, 1960] correspond to the rst impredicative ordinal, , 0, and even to larger ones [Friedman, 19??; =-=Simpson, 1985-=-; Smorynski, 1986; Dershowitz, 1987; Gallier, 1991]. The signi cance of , 0 for computer science is discussed in [Gallier, 1991]. 115 Conclusions It has been argued [Gries, 1979] that the natural num... |

7 |
Proof-theoretic techniques and the theory of rewriting
- Dershowitz, Okada
- 1988
(Show Context)
Citation Context ...rderings (! n ) also play an important part [ Manna, 1974 ] . Occasionally, "larger" orderings have been used (for example, [ Dershowitz and Manna, 1979; Dershowitz, 1987 ] ); see [ Dershowi=-=tz, 1987; Dershowitz and Okada, 1988-=-; Cichon, 1990 ] . The riddle above is a termination question on binary trees, one of the most pervasive data structures used in computer science. Like numbers, binary trees can be well-ordered in man... |

4 | Ordering structures and the Knuth-Bendix completion algorithm - Okada, Steele - 1988 |

4 |
Is Sometimes Ever Better than Always
- Gries
- 1979
(Show Context)
Citation Context ...es [Friedman, 19??; Simpson, 1985; Smorynski, 1986; Dershowitz, 1987; Gallier, 1991]. The signi cance of , 0 for computer science is discussed in [Gallier, 1991]. 115 Conclusions It has been argued [=-=Gries, 1979-=-] that the natural numbers su ce for termination proofs, since the (maximum) number of iterations of any terminating deterministic (or bounded nondeterministic) program loop is xed, depending only on ... |

4 | Nash-Williams: On wellquasiordering nite trees - A - 1963 |

3 |
Built-up systems of fundamental sequences and hierarchies of number-theoretic functions
- Schmidt
- 1976
(Show Context)
Citation Context ... analysis of in nite, not necessarily rational, binary trees may also be possible. 4 Bigger Ordinals The epsilon number is 2(0) in the Veblen-Feferman-Schutte hierarchy [Veblen, 1908; Feferman, 1968; =-=Schmidt, 1976-=-]. Less natural orderings on (nonbinary) ordered trees correspond to much larger ordinals in that hierarchy. In particular, some orderings based on Kruskal's Tree Theorem [Kruskal, 1960] correspond to... |

2 | The metamathematics of the Higman and Kruskal theorems - Friedman - 1991 |

1 | a Dover edition - Jourdain - 1952 |

1 |
Canonical simpli cation of nite objects well quasi-ordered by tree embedding
- Brown
- 1979
(Show Context)
Citation Context .... . . . x 1 . . . xn,1 . . xn can be replaced by the double-self-loop corresponding to the full tree or by a self-loop z 0 cons(maxfx ig;z 0 ). An attempt to prove a result like Theorem 3 appears in [=-=Brown, 1979-=-]. Proposition 2. There is an order-preserving mapping from normalized lists, under the above ordering, onto the ordinals up to and including . Proof. The mapping from lists to ordinals is: [nil ] = 0... |

1 | Contributions to the Founding of the Theory of Trans nite Numbers. The Open Court - Cantor - 1915 |