## Termination analysis for functional programs using term orderings (1995)

Venue: | IN PROCEEDINGS OF THE SECOND INTERNATIONAL STATIC ANALYSIS SYMPOSIUM, LNCS 983 |

Citations: | 31 - 12 self |

### BibTeX

@INPROCEEDINGS{Giesl95terminationanalysis,

author = {Jürgen Giesl},

title = {Termination analysis for functional programs using term orderings},

booktitle = {IN PROCEEDINGS OF THE SECOND INTERNATIONAL STATIC ANALYSIS SYMPOSIUM, LNCS 983},

year = {1995},

pages = {154--171},

publisher = {Springer}

}

### Years of Citing Articles

### OpenURL

### Abstract

To prove the termination of a functional program there has to be a well-founded ordering such that the arguments in each recursive call are smaller than the corresponding inputs. In this paper we present a procedure for automated termination proofs of functional programs. In contrast to previously presented methods a suited well-founded ordering does not have to be fixed in advance by the user, but can be synthesized automatically. For that purpose we use approaches developed in the area of term rewriting systems for the automated generation of suited well-founded term orderings. But unfortunately term orderings cannot be directly used for termination proofs of functional programs which call other algorithms in the arguments of their recursive calls. The reason is that while for the termination of term rewriting systems orderings between terms are needed, for functional programs we need orderings between objects of algebraic data types. Our method solves this problem and enables term orderings to be used for termination proofs of functional programs.

### Citations

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Citation Context ...ems and logic programs (for surveys on these topics see [Der87] and [SD93] resp.). Methods for termination proofs of functional programsshave for instance been developed by R. S. Boyer and J S. Moore =-=[BM79]-=-, C. Walther [Wal88], [Wal94] and F. and H. R. Nielson [NN95]. The procedure of Boyer and Moore has also been adapted for conditional rewrite systems [BL93]. But both the technique of Boyer and Moore ... |

469 | Termination of rewriting
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Citation Context ... disprove the termination of all algorithms. Most work on the automation of termination proofs has been done in the areas of term rewriting systems and logic programs (for surveys on these topics see =-=[Der87]-=- and [SD93] resp.). Methods for termination proofs of functional programsshave for instance been developed by R. S. Boyer and J S. Moore [BM79], C. Walther [Wal88], [Wal94] and F. and H. R. Nielson [N... |

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Citation Context ...for the automatedssynthesis of term orderings have been developed [Ste94]. For instance, there exist procedures for the automated generation of Knuth-Bendix orderings [Mar87], of polynomial orderings =-=[Col75]-=-, [BCL87], [Ste94], [Gie95a] and of path orderings [Ait85], [DF85], [DH93]. Our aim is to use these synthesis methods for automated termination proofs of functional programs. Unfortunately, while term... |

152 |
Collected Papers
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Citation Context ...ypotheses TH 0 of an algorithm are transformed into TH 1 , TH 2 etc. until we obtain a set of formulas THn containing no defined function symbols any more. This transformation is an abduction process =-=[Pei31]-=-, i.e. TH i+1 j= TH i 4 In this paper we only refer to those term orderings that are amenable to automation. There also exist classes of term orderings which can orient every terminating term rewritin... |

143 |
Termination of logic programs: the never-ending story
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Citation Context ...e termination of all algorithms. Most work on the automation of termination proofs has been done in the areas of term rewriting systems and logic programs (for surveys on these topics see [Der87] and =-=[SD93]-=- resp.). Methods for termination proofs of functional programsshave for instance been developed by R. S. Boyer and J S. Moore [BM79], C. Walther [Wal88], [Wal94] and F. and H. R. Nielson [NN95]. The p... |

130 |
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Citation Context ...2)) are sufficient for formulas like x 6= empty ! ffi rmsrm (: : :). To perform such proofs automatically we make use of an induction theorem proving system (e.g. those described in [BM79], [BHHW86], =-=[BHHS90]-=-). So to eliminate the defined strictness predicate ffisgg from a formula of the form ' ! tsssffisgg (: : :) we proceed as follows: 8 A proof for this observation can be found in the appendix. 1. Eith... |

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Citation Context ...ion of well-founded term orderings satisfying such constraints have been developed in the area of term rewriting systems. For instance, termination of flatten can be proved with a polynomial ordering =-=[Lan79]-=-, where every n-ary function symbol is associated with an n-ary polynomial over the natural numbers. Then a ground term t is greater than a ground term s with respect to the polynomial ordering iff th... |

84 | Natural termination
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- 1995
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Citation Context ...For instance, there exist procedures for the automated generation of Knuth-Bendix orderings [Mar87], of polynomial orderings [Col75], [BCL87], [Ste94], [Gie95a] and of path orderings [Ait85], [DF85], =-=[DH93]-=-. Our aim is to use these synthesis methods for automated termination proofs of functional programs. Unfortunately, while term orderings can easily be used for termination proofs of term rewriting sys... |

83 | Termination of rewriting systems by polynomial interpretations and its implementation
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Citation Context ...utomatedssynthesis of term orderings have been developed [Ste94]. For instance, there exist procedures for the automated generation of Knuth-Bendix orderings [Mar87], of polynomial orderings [Col75], =-=[BCL87]-=-, [Ste94], [Gie95a] and of path orderings [Ait85], [DF85], [DH93]. Our aim is to use these synthesis methods for automated termination proofs of functional programs. Unfortunately, while term ordering... |

75 |
Termination Proofs for Logic Programs
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(Show Context)
Citation Context ...elson [NN95]. The procedure of Boyer and Moore has also been adapted for conditional rewrite systems [BL93]. But both the technique of Boyer and Moore and the methods for logic programss(e.g. [UV88], =-=[Plu90]-=-, [SV91], [DSF93]) are only semi-automatic, i.e. for every termination proof at least the main characteristics of the suited wellfounded ordering have to be given in advance by the user. The methods o... |

68 |
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Citation Context ...H. R. Nielson [NN95]. The procedure of Boyer and Moore has also been adapted for conditional rewrite systems [BL93]. But both the technique of Boyer and Moore and the methods for logic programss(e.g. =-=[UV88]-=-, [Plu90], [SV91], [DSF93]) are only semi-automatic, i.e. for every termination proof at least the main characteristics of the suited wellfounded ordering have to be given in advance by the user. The ... |

51 |
Attempts for generalizing the recursive path ordering. Unpublished manuscript
- Kamin, Levy
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Citation Context ...only refer to those term orderings that are amenable to automation. There also exist classes of term orderings which can orient every terminating term rewriting system (e.g. semantical path orderings =-=[KL80]-=- or transformation orderings [BL90]). But the disadvantage of these powerful approaches is that up to now there are only very few suggestions for their automated generation [Ste95]. TH0 ! TH1 ! TH2 ! ... |

46 | Generating polynomial orderings for termination proofs
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- 1995
(Show Context)
Citation Context ... of term orderings have been developed [Ste94]. For instance, there exist procedures for the automated generation of Knuth-Bendix orderings [Mar87], of polynomial orderings [Col75], [BCL87], [Ste94], =-=[Gie95a]-=- and of path orderings [Ait85], [DF85], [DH93]. Our aim is to use these synthesis methods for automated termination proofs of functional programs. Unfortunately, while term orderings can easily be use... |

45 |
On proving the termination of algorithms by machine
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Citation Context ...surveys on these topics see [Der87] and [SD93] resp.). Methods for termination proofs of functional programsshave for instance been developed by R. S. Boyer and J S. Moore [BM79], C. Walther [Wal88], =-=[Wal94]-=- and F. and H. R. Nielson [NN95]. The procedure of Boyer and Moore has also been adapted for conditional rewrite systems [BL93]. But both the technique of Boyer and Moore and the methods for logic pro... |

41 |
Termination detection in logic programs using argument sizes
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Citation Context ...95]. The procedure of Boyer and Moore has also been adapted for conditional rewrite systems [BL93]. But both the technique of Boyer and Moore and the methods for logic programss(e.g. [UV88], [Plu90], =-=[SV91]-=-, [DSF93]) are only semi-automatic, i.e. for every termination proof at least the main characteristics of the suited wellfounded ordering have to be given in advance by the user. The methods of Walthe... |

33 |
A note on simplification orderings
- Dershowitz
(Show Context)
Citation Context ...t cannot be proved with a polynomial ordering which respects the semantics of min and rm 3 . Virtually all other term orderingssused in the area of term rewriting systems are simplification orderings =-=[Der79]-=-, [Ste94]. As these orderings possess the subterm property (i.e. f(: : : t : : :)st), simplification orderings do not respect the semantics of the algorithm min. The reason is that min(add(0; empty)) ... |

32 |
The Karlsruhe induction theorem proving system
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Citation Context ...e powerful than Walther's approach and has a higher degree of automation than the technique of Boyer and Moore. It has been implemented and integrated within the induction theorem proving system inka =-=[BHHW86]-=-. We introduce some refinements of our method in Section 5 and end up with a conclusion and an outlook on future work. 2 Functional Programs and Term Orderings In this paper we regard an eager functio... |

25 |
Termination by completion”, Applicable Algebra
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Citation Context ...that are amenable to automation. There also exist classes of term orderings which can orient every terminating term rewriting system (e.g. semantical path orderings [KL80] or transformation orderings =-=[BL90]-=-). But the disadvantage of these powerful approaches is that up to now there are only very few suggestions for their automated generation [Ste95]. TH0 ! TH1 ! TH2 ! : : : ! THn * * * * 0 / 1 / 2 / : :... |

22 |
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Citation Context ...tical path orderings [KL80] or transformation orderings [BL90]). But the disadvantage of these powerful approaches is that up to now there are only very few suggestions for their automated generation =-=[Ste95]-=-. TH0 ! TH1 ! TH2 ! : : : ! THn * * * * 0 / 1 / 2 / : : : / n Fig. 3. Elimination of defined function symbols from termination hypotheses. holds for all i. Therefore if a relationsi+1 satisfies the co... |

14 |
Automatisierung von Terminierungsbeweisen fur rekursiv definierte Algorithmen
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- 1995
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Citation Context ...no well-founded term ordering satisfying the constraints at the leafs can be found. To improve the efficiency of our method, we have also developed heuristics for choosing the "right" deriva=-=tion tree [Gie95b]-=-. These heuristics have proved successful in practice. An alternative method for termination proofs with user provided orderings which avoids such choice points is presented in [Gie95c]. Our method ca... |

14 |
Argument-Bounded Algorithms as a Basis for Automated Termination Proofs. 9th CADE, LNCS310
- Walther
- 1988
(Show Context)
Citation Context ...ams (for surveys on these topics see [Der87] and [SD93] resp.). Methods for termination proofs of functional programsshave for instance been developed by R. S. Boyer and J S. Moore [BM79], C. Walther =-=[Wal88]-=-, [Wal94] and F. and H. R. Nielson [NN95]. The procedure of Boyer and Moore has also been adapted for conditional rewrite systems [BL93]. But both the technique of Boyer and Moore and the methods for ... |

13 |
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Citation Context ...Ste94]. For instance, there exist procedures for the automated generation of Knuth-Bendix orderings [Mar87], of polynomial orderings [Col75], [BCL87], [Ste94], [Gie95a] and of path orderings [Ait85], =-=[DF85]-=-, [DH93]. Our aim is to use these synthesis methods for automated termination proofs of functional programs. Unfortunately, while term orderings can easily be used for termination proofs of term rewri... |

10 | Automated termination proofs with measure functions
- Giesl
- 1995
(Show Context)
Citation Context ...erivation tree [Gie95b]. These heuristics have proved successful in practice. An alternative method for termination proofs with user provided orderings which avoids such choice points is presented in =-=[Gie95c]-=-. Our method can easily be extended to algorithms with several formal parameters. For that purpose we introduce a new undefined function symbolsand instead of two tuples (t 1 ; : : : ; t n ) and (s 1 ... |

10 |
Termination of Rewriting: Extensions, Comparison and Automatic Generation of Simplification Orderings
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Citation Context ...alysis Symposium, Glasgow, Scotland, Springer--Verlag, LNCS 983, 1995. To prove termination of term rewriting systems several methods for the automatedssynthesis of term orderings have been developed =-=[Ste94]-=-. For instance, there exist procedures for the automated generation of Knuth-Bendix orderings [Mar87], of polynomial orderings [Col75], [BCL87], [Ste94], [Gie95a] and of path orderings [Ait85], [DF85]... |

8 |
How to choose the weights in the Knuth-Bendix ordering
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Citation Context ...ewriting systems several methods for the automatedssynthesis of term orderings have been developed [Ste94]. For instance, there exist procedures for the automated generation of Knuth-Bendix orderings =-=[Mar87]-=-, of polynomial orderings [Col75], [BCL87], [Ste94], [Gie95a] and of path orderings [Ait85], [DF85], [DH93]. Our aim is to use these synthesis methods for automated termination proofs of functional pr... |

6 |
Proving Termination of (Conditional) Rewrite Systems
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Citation Context ...een developed by R. S. Boyer and J S. Moore [BM79], C. Walther [Wal88], [Wal94] and F. and H. R. Nielson [NN95]. The procedure of Boyer and Moore has also been adapted for conditional rewrite systems =-=[BL93]-=-. But both the technique of Boyer and Moore and the methods for logic programss(e.g. [UV88], [Plu90], [SV91], [DSF93]) are only semi-automatic, i.e. for every termination proof at least the main chara... |

4 |
De Schreye and M. Fabris, Automatic inference of norms: A missing link in automatic termination analysis
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Citation Context ... procedure of Boyer and Moore has also been adapted for conditional rewrite systems [BL93]. But both the technique of Boyer and Moore and the methods for logic programss(e.g. [UV88], [Plu90], [SV91], =-=[DSF93]-=-) are only semi-automatic, i.e. for every termination proof at least the main characteristics of the suited wellfounded ordering have to be given in advance by the user. The methods of Walther and of ... |

4 | Termination Analysis based on Operational Semantics
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(Show Context)
Citation Context ...7] and [SD93] resp.). Methods for termination proofs of functional programsshave for instance been developed by R. S. Boyer and J S. Moore [BM79], C. Walther [Wal88], [Wal94] and F. and H. R. Nielson =-=[NN95]-=-. The procedure of Boyer and Moore has also been adapted for conditional rewrite systems [BL93]. But both the technique of Boyer and Moore and the methods for logic programss(e.g. [UV88], [Plu90], [SV... |

2 |
An Algorithm for Finding a Minimal Recursive Path Ordering. RAIRO
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(Show Context)
Citation Context ...veloped [Ste94]. For instance, there exist procedures for the automated generation of Knuth-Bendix orderings [Mar87], of polynomial orderings [Col75], [BCL87], [Ste94], [Gie95a] and of path orderings =-=[Ait85]-=-, [DF85], [DH93]. Our aim is to use these synthesis methods for automated termination proofs of functional programs. Unfortunately, while term orderings can easily be used for termination proofs of te... |