On computations with integer division: Extended abstract

Bettina Just; Fb Mathematik; Friedhelm Meyer auf der Heide; Fb Informatik; Avi Wigderson

doi:10.1007/BFb0035829

On computations with integer division: Extended abstract

Bettina Just
, Fb Mathematik
, Friedhelm Meyer auf der Heide
, Fb Informatik
, Avi Wigderson

The Rachel and Selim Benin School of Engineering and Computer Science

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

2 Scopus citations

Abstract

We consider computation trees (CT's) with operations S ⊂ {+, −, *, DIV, DIV_C}, where DIV denotes integer division and DIV_C integer division by constants. We characterize the families of languages L ⊂ ℕ that can be recognized over {+, −, DIV_C}, {+, −, DIV}, and {+, −, *, DIV}, resp. and show that they are identical. Furthermore we prove lower bounds for CT's with operations {+, −, DIV_C} for languages L ⊂ ℕ which only contain short arithmetic progressions. We cannot apply the classical component counting arguments as for operation sets S ⊂ {+, −, *,./.} because of the DIV_C - operation. Such bounds are even no longer true. Instead we apply results from the Geometry of Numbers about arithmetic progressions on integer points in high-dimensional convex sets for our lower bounds.

Original language	English
Title of host publication	STACS 1988 - 5th Annual Symposium on Theoretical Aspects of Computer Science, Proceedings
Editors	Robert Cori, Martin Wirsing
Publisher	Springer Verlag
Pages	29-37
Number of pages	9
ISBN (Print)	9783540188346
DOIs	https://doi.org/10.1007/BFb0035829
State	Published - 1988
Event	5th Annual Symposium on Theoretical Aspects of Computer Science, STACS 1988 - Bordeaux, France Duration: 11 Feb 1988 → 13 Feb 1988

Publication series

Name	Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume	294 LNCS
ISSN (Print)	0302-9743
ISSN (Electronic)	1611-3349

Conference

Conference	5th Annual Symposium on Theoretical Aspects of Computer Science, STACS 1988
Country/Territory	France
City	Bordeaux
Period	11/02/88 → 13/02/88

Bibliographical note

Publisher Copyright:
© Springer-Verlag Berlin Heidelberg 1988.

Access to Document

10.1007/BFb0035829

Cite this

Just, B., Mathematik, F., Meyer auf der Heide, F., Informatik, F., & Wigderson, A. (1988). On computations with integer division: Extended abstract. In R. Cori, & M. Wirsing (Eds.), STACS 1988 - 5th Annual Symposium on Theoretical Aspects of Computer Science, Proceedings (pp. 29-37). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 294 LNCS). Springer Verlag. https://doi.org/10.1007/BFb0035829

Just, Bettina ; Mathematik, Fb ; Meyer auf der Heide, Friedhelm et al. / On computations with integer division : Extended abstract. STACS 1988 - 5th Annual Symposium on Theoretical Aspects of Computer Science, Proceedings. editor / Robert Cori ; Martin Wirsing. Springer Verlag, 1988. pp. 29-37 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)).

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title = "On computations with integer division: Extended abstract",

abstract = "We consider computation trees (CT's) with operations S ⊂ \{+, −, *, DIV, DIVC\}, where DIV denotes integer division and DIVC integer division by constants. We characterize the families of languages L ⊂ ℕ that can be recognized over \{+, −, DIVC\}, \{+, −, DIV\}, and \{+, −, *, DIV\}, resp. and show that they are identical. Furthermore we prove lower bounds for CT's with operations \{+, −, DIVC\} for languages L ⊂ ℕ which only contain short arithmetic progressions. We cannot apply the classical component counting arguments as for operation sets S ⊂ \{+, −, *,./.\} because of the DIVC - operation. Such bounds are even no longer true. Instead we apply results from the Geometry of Numbers about arithmetic progressions on integer points in high-dimensional convex sets for our lower bounds.",

author = "Bettina Just and Fb Mathematik and \{Meyer auf der Heide\}, Friedhelm and Fb Informatik and Avi Wigderson",

note = "Publisher Copyright: {\textcopyright} Springer-Verlag Berlin Heidelberg 1988.; 5th Annual Symposium on Theoretical Aspects of Computer Science, STACS 1988 ; Conference date: 11-02-1988 Through 13-02-1988",

year = "1988",

doi = "10.1007/BFb0035829",

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series = "Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)",

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Just, B, Mathematik, F, Meyer auf der Heide, F, Informatik, F & Wigderson, A 1988, On computations with integer division: Extended abstract. in R Cori & M Wirsing (eds), STACS 1988 - 5th Annual Symposium on Theoretical Aspects of Computer Science, Proceedings. Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 294 LNCS, Springer Verlag, pp. 29-37, 5th Annual Symposium on Theoretical Aspects of Computer Science, STACS 1988, Bordeaux, France, 11/02/88. https://doi.org/10.1007/BFb0035829

On computations with integer division: Extended abstract. / Just, Bettina; Mathematik, Fb; Meyer auf der Heide, Friedhelm et al.
STACS 1988 - 5th Annual Symposium on Theoretical Aspects of Computer Science, Proceedings. ed. / Robert Cori; Martin Wirsing. Springer Verlag, 1988. p. 29-37 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 294 LNCS).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

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AB - We consider computation trees (CT's) with operations S ⊂ {+, −, *, DIV, DIVC}, where DIV denotes integer division and DIVC integer division by constants. We characterize the families of languages L ⊂ ℕ that can be recognized over {+, −, DIVC}, {+, −, DIV}, and {+, −, *, DIV}, resp. and show that they are identical. Furthermore we prove lower bounds for CT's with operations {+, −, DIVC} for languages L ⊂ ℕ which only contain short arithmetic progressions. We cannot apply the classical component counting arguments as for operation sets S ⊂ {+, −, *,./.} because of the DIVC - operation. Such bounds are even no longer true. Instead we apply results from the Geometry of Numbers about arithmetic progressions on integer points in high-dimensional convex sets for our lower bounds.

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Just B, Mathematik F, Meyer auf der Heide F, Informatik F, Wigderson A. On computations with integer division: Extended abstract. In Cori R, Wirsing M, editors, STACS 1988 - 5th Annual Symposium on Theoretical Aspects of Computer Science, Proceedings. Springer Verlag. 1988. p. 29-37. (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)). doi: 10.1007/BFb0035829

On computations with integer division: Extended abstract

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