## Walther Recursion (1996)

Venue: | Proceedings CADE 13, Springer LNCS |

Citations: | 18 - 0 self |

### BibTeX

@INPROCEEDINGS{Mcallester96waltherrecursion,

author = {David Mcallester and Kostas Arkoudas},

title = {Walther Recursion},

booktitle = {Proceedings CADE 13, Springer LNCS},

year = {1996},

pages = {643--657},

publisher = {Springer-Verlag}

}

### OpenURL

### Abstract

. Primitive recursion is a well known syntactic restriction on recursive definitions which guarantees termination. Unfortunately many natural definitions, such as the most common definition of Euclid's GCD algorithm, are not primitive recursive. Walther has recently given a proof system for verifying termination of a broader class of definitions. Although Walther's system is highly automatible, the class of acceptable definitions remains only semi-decidable. Here we simplify Walther's calculus and give a syntactic criterion on definitions which guarantees termination. This syntactic criteria generalizes primitive recursion and handles most of the examples given by Walther. We call the corresponding class of acceptable definitions "Walther recursive". 1 Introduction One of the central problems in verification logics, such as the Boyer-Moore theorem prover [2], [10], is the need to prove termination for recursive definitions. Many logics, such as that of Boyer and Moore, assu...

### Citations

779 | Rewrite Systems
- Dershowitz, Jouannaud
- 1990
(Show Context)
Citation Context ...(x) ? x will still require proving that f terminates. Some definitions can be proved to terminate by translation into a term rewriting system and the application of standard term rewriting techniques =-=[4, 3]-=-. Unfortunately, all automated systems for termination analysis for rewrite systems use well founded term orders such that usv implies f(: : : u : : :)sf(: : : v : : :) and usf(: : : u : : :). There a... |

506 |
Automata on infinite objects
- Thomas
- 1990
(Show Context)
Citation Context ... the form Y ! X c or X c ! c(Y 1 ; : : : ; Yn ) where Y and each Y i are aggregate nonterminals. 5 Regular types can also be characterized as the term languages accepted by finite state tree automata =-=[7, 8]-=-. We find the grammar notation clearer. Properties of tree automata can always be stated directly on grammars. More specifically, a tree automaton can be defined to be a set of productions of the form... |

272 | Orderings for term-rewriting systems
- Dershowitz
- 1982
(Show Context)
Citation Context ...(x) ? x will still require proving that f terminates. Some definitions can be proved to terminate by translation into a term rewriting system and the application of standard term rewriting techniques =-=[4, 3]-=-. Unfortunately, all automated systems for termination analysis for rewrite systems use well founded term orders such that usv implies f(: : : u : : :)sf(: : : v : : :) and usf(: : : u : : :). There a... |

192 | Soft typing with conditional types
- Aiken, Wimmers, et al.
- 1994
(Show Context)
Citation Context ...ted with input and output types. This simplifies the system and allows our formulation to focus on Walther's reducerconserver analysis independent of the type inference problem for regular types. See =-=[1, 6] for algor-=-ithms for inferring regular types. Section 2 describes the concept of a regular type and introduces the class of "monomorphic" regular types. The monomorphism assumption allows types to be v... |

122 |
Decidability of second order theories and automata on infinite trees
- Rabin
- 1969
(Show Context)
Citation Context ... the form Y ! X c or X c ! c(Y 1 ; : : : ; Yn ) where Y and each Y i are aggregate nonterminals. 5 Regular types can also be characterized as the term languages accepted by finite state tree automata =-=[7, 8]-=-. We find the grammar notation clearer. Properties of tree automata can always be stated directly on grammars. More specifically, a tree automaton can be defined to be a set of productions of the form... |

83 | IMPS: An interactive mathematical proof system
- Farmer, Guttman, et al.
- 1993
(Show Context)
Citation Context ...ursive definitions. Many logics, such as that of Boyer and Moore, assume that all function symbols define total functions. Even in systems where partial functions are allowed, such as the IMPS system =-=[11]-=-, proofs of termination are still important. For example, proving a lemma of the form 8x f(x) ? x will still require proving that f terminates. Some definitions can be proved to terminate by translati... |

45 |
Set Based Analysis of ML Programs
- Heintze
- 1993
(Show Context)
Citation Context ...ted with input and output types. This simplifies the system and allows our formulation to focus on Walther's reducerconserver analysis independent of the type inference problem for regular types. See =-=[1, 6] for algor-=-ithms for inferring regular types. Section 2 describes the concept of a regular type and introduces the class of "monomorphic" regular types. The monomorphism assumption allows types to be v... |

45 |
On proving the termination of algorithms by machine
- Walther
- 1994
(Show Context)
Citation Context ...ere DIFF is the set difference functions (which is itself primitive recursive). Again, the problem is not the choice of ordering, it is the method of verifying that the arguments are reduced. Walther =-=[9]-=- has developed a calculus which is quite effective at deriving assertions of the form u ! x where u is a term containing the variable x. For example, if x and y are nonzero then MINUS(x; y) ! x. If x ... |

7 |
Eyal Yardeni. Logic programs as types for logic programs
- Fruhwirth, Shapiro, et al.
- 1991
(Show Context)
Citation Context ...nalogous operations on the types being represented. Determining emptiness of an intersection of a set of nonterminals in an arbitrary (nonmonomorphic) regular term grammar is known to be EXPTIME hard =-=[5]-=-. Monomorphic regular types yield a considerable simplification. Throughout this paper we assume a fixed user declared grammar defining the types of constructors. The user provides a grammar in which ... |

1 |
page for computational logic incorporated. http://www.cli.com/index.html
- Home
(Show Context)
Citation Context ... Walther. We call the corresponding class of acceptable definitions "Walther recursive". 1 Introduction One of the central problems in verification logics, such as the Boyer-Moore theorem pr=-=over [2], [10]-=-, is the need to prove termination for recursive definitions. Many logics, such as that of Boyer and Moore, assume that all function symbols define total functions. Even in systems where partial funct... |