Two-letter group codes that preserve aperiodicity of inverse finite automata

Jean Camille Birget; Stuart W. Margolis

doi:10.1007/s00233-007-9024-6

Two-letter group codes that preserve aperiodicity of inverse finite automata

Jean Camille Birget
, Stuart W. Margolis

Department of Mathematics

Rutgers - The State University of New Jersey, Camden

Research output: Contribution to journal › Article › peer-review

10 Scopus citations

Abstract

We construct group codes over two letters (i.e., bases of subgroups of a two-generated free group) with special properties. Such group codes can be used for reducing algorithmic problems over large alphabets to algorithmic problems over a two-letter alphabet. Our group codes preserve aperiodicity of inverse finite automata. As an application we show that the following problems are PSpace-complete for two-letter alphabets (this was previously known for large enough finite alphabets): The intersection-emptiness problem for inverse finite automata, the aperiodicity problem for inverse finite automata, and the closure-under-radical problem for finitely generated subgroups of a free group. The membership problem for 3-generated inverse monoids is PSpace-complete.

Original language	English
Pages (from-to)	159-168
Number of pages	10
Journal	Semigroup Forum
Volume	76
Issue number	1
DOIs	https://doi.org/10.1007/s00233-007-9024-6
State	Published - Jan 2008

Bibliographical note

Funding Information:
Both authors were supported in part by NSF grant DMS-9970471. The first author was also supported in part by NSF grant CCR-0310793. The second author acknowledges the support of the Excellency Center, “Group Theoretic Methods for the Study of Algebraic Varieties” of the Israeli Science Foundation.

Funding

Both authors were supported in part by NSF grant DMS-9970471. The first author was also supported in part by NSF grant CCR-0310793. The second author acknowledges the support of the Excellency Center, “Group Theoretic Methods for the Study of Algebraic Varieties” of the Israeli Science Foundation.

Funders	Funder number
National Science Foundation	DMS-9970471, CCR-0310793
Israel Science Foundation

Keywords

Free groups
Inverse automata
Inverse semigroups

Access to Document

10.1007/s00233-007-9024-6

Cite this

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title = "Two-letter group codes that preserve aperiodicity of inverse finite automata",

abstract = "We construct group codes over two letters (i.e., bases of subgroups of a two-generated free group) with special properties. Such group codes can be used for reducing algorithmic problems over large alphabets to algorithmic problems over a two-letter alphabet. Our group codes preserve aperiodicity of inverse finite automata. As an application we show that the following problems are PSpace-complete for two-letter alphabets (this was previously known for large enough finite alphabets): The intersection-emptiness problem for inverse finite automata, the aperiodicity problem for inverse finite automata, and the closure-under-radical problem for finitely generated subgroups of a free group. The membership problem for 3-generated inverse monoids is PSpace-complete.",

keywords = "Free groups, Inverse automata, Inverse semigroups",

author = "Birget, \{Jean Camille\} and Margolis, \{Stuart W.\}",

note = "Funding Information: Both authors were supported in part by NSF grant DMS-9970471. The first author was also supported in part by NSF grant CCR-0310793. The second author acknowledges the support of the Excellency Center, {\textquotedblleft}Group Theoretic Methods for the Study of Algebraic Varieties{\textquotedblright} of the Israeli Science Foundation.",

year = "2008",

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doi = "10.1007/s00233-007-9024-6",

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AU - Birget, Jean Camille

AU - Margolis, Stuart W.

N1 - Funding Information: Both authors were supported in part by NSF grant DMS-9970471. The first author was also supported in part by NSF grant CCR-0310793. The second author acknowledges the support of the Excellency Center, “Group Theoretic Methods for the Study of Algebraic Varieties” of the Israeli Science Foundation.

PY - 2008/1

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N2 - We construct group codes over two letters (i.e., bases of subgroups of a two-generated free group) with special properties. Such group codes can be used for reducing algorithmic problems over large alphabets to algorithmic problems over a two-letter alphabet. Our group codes preserve aperiodicity of inverse finite automata. As an application we show that the following problems are PSpace-complete for two-letter alphabets (this was previously known for large enough finite alphabets): The intersection-emptiness problem for inverse finite automata, the aperiodicity problem for inverse finite automata, and the closure-under-radical problem for finitely generated subgroups of a free group. The membership problem for 3-generated inverse monoids is PSpace-complete.

AB - We construct group codes over two letters (i.e., bases of subgroups of a two-generated free group) with special properties. Such group codes can be used for reducing algorithmic problems over large alphabets to algorithmic problems over a two-letter alphabet. Our group codes preserve aperiodicity of inverse finite automata. As an application we show that the following problems are PSpace-complete for two-letter alphabets (this was previously known for large enough finite alphabets): The intersection-emptiness problem for inverse finite automata, the aperiodicity problem for inverse finite automata, and the closure-under-radical problem for finitely generated subgroups of a free group. The membership problem for 3-generated inverse monoids is PSpace-complete.

KW - Free groups

KW - Inverse automata

KW - Inverse semigroups

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EP - 168

JO - Semigroup Forum

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ER -

Two-letter group codes that preserve aperiodicity of inverse finite automata

Abstract

Bibliographical note

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Keywords

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