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Given two terms and their rewriting rules, an unreachability problem proves the non-existence of a reduction sequence from one term to another. This paper formalizes a method for solving unreachability problems by abstraction; i. e. , reducing an original concrete unreachability problem to a simpler abstract unreachability problem to prove the unreachability of the original concrete problem if the abstract unreachability is proved. The class of rewriting systems discussed in this paper is called β rewriting systems. The class of β rewriting systems includes very important systems such as semi-Thue systems and Petri Nets. Abstract rewriting systems are also a subclass of β rewriting systems. A β rewriting system is defined on axiomatically formulated base structures, called β structures, which are used to formalize the concepts of "contexts" and "replacement," which are common to many rewritten objects. Each domain underlying semi-Thue systems, Petri Nets, and other rewriting systems are formalized by a β structure. A concept of homomorphisms from a β structure (a concrete domain) to a β structure (an abstract domain) is introduced. A homomorphism theorem (Theorem1)is established for β rewriting systems, which states that concrete reachability implies abstract reachability. An unreachability theorem (Corollary1) is also proved for β rewriting systems. It is the contraposition of the homomorphism theorem, i. e. , it says that abstract unreachability implies concrete unreachability. The unreachability theorem is used to solve two unreachability problems: a coffee bean puzzle and a checker board puzzle.

- Publication
- IEICE TRANSACTIONS on Information Vol.E82-D No.2 pp.339-347

- Publication Date
- 1999/02/25

- Publicized

- Online ISSN

- DOI

- Type of Manuscript
- PAPER

- Category
- Automata,Languages and Theory of Computing

The copyright of the original papers published on this site belongs to IEICE. Unauthorized use of the original or translated papers is prohibited. See IEICE Provisions on Copyright for details.

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Kiyoshi AKAMA, Yoshinori SHIGETA, Eiichi MIYAMOTO, "Unreachability Proofs for β Rewriting Systems by Homomorphisms" in IEICE TRANSACTIONS on Information,
vol. E82-D, no. 2, pp. 339-347, February 1999, doi: .

Abstract: Given two terms and their rewriting rules, an unreachability problem proves the non-existence of a reduction sequence from one term to another. This paper formalizes a method for solving unreachability problems by abstraction; i. e. , reducing an original concrete unreachability problem to a simpler abstract unreachability problem to prove the unreachability of the original concrete problem if the abstract unreachability is proved. The class of rewriting systems discussed in this paper is called β rewriting systems. The class of β rewriting systems includes very important systems such as semi-Thue systems and Petri Nets. Abstract rewriting systems are also a subclass of β rewriting systems. A β rewriting system is defined on axiomatically formulated base structures, called β structures, which are used to formalize the concepts of "contexts" and "replacement," which are common to many rewritten objects. Each domain underlying semi-Thue systems, Petri Nets, and other rewriting systems are formalized by a β structure. A concept of homomorphisms from a β structure (a concrete domain) to a β structure (an abstract domain) is introduced. A homomorphism theorem (Theorem1)is established for β rewriting systems, which states that concrete reachability implies abstract reachability. An unreachability theorem (Corollary1) is also proved for β rewriting systems. It is the contraposition of the homomorphism theorem, i. e. , it says that abstract unreachability implies concrete unreachability. The unreachability theorem is used to solve two unreachability problems: a coffee bean puzzle and a checker board puzzle.

URL: https://global.ieice.org/en_transactions/information/10.1587/e82-d_2_339/_p

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@ARTICLE{e82-d_2_339,

author={Kiyoshi AKAMA, Yoshinori SHIGETA, Eiichi MIYAMOTO, },

journal={IEICE TRANSACTIONS on Information},

title={Unreachability Proofs for β Rewriting Systems by Homomorphisms},

year={1999},

volume={E82-D},

number={2},

pages={339-347},

abstract={Given two terms and their rewriting rules, an unreachability problem proves the non-existence of a reduction sequence from one term to another. This paper formalizes a method for solving unreachability problems by abstraction; i. e. , reducing an original concrete unreachability problem to a simpler abstract unreachability problem to prove the unreachability of the original concrete problem if the abstract unreachability is proved. The class of rewriting systems discussed in this paper is called β rewriting systems. The class of β rewriting systems includes very important systems such as semi-Thue systems and Petri Nets. Abstract rewriting systems are also a subclass of β rewriting systems. A β rewriting system is defined on axiomatically formulated base structures, called β structures, which are used to formalize the concepts of "contexts" and "replacement," which are common to many rewritten objects. Each domain underlying semi-Thue systems, Petri Nets, and other rewriting systems are formalized by a β structure. A concept of homomorphisms from a β structure (a concrete domain) to a β structure (an abstract domain) is introduced. A homomorphism theorem (Theorem1)is established for β rewriting systems, which states that concrete reachability implies abstract reachability. An unreachability theorem (Corollary1) is also proved for β rewriting systems. It is the contraposition of the homomorphism theorem, i. e. , it says that abstract unreachability implies concrete unreachability. The unreachability theorem is used to solve two unreachability problems: a coffee bean puzzle and a checker board puzzle.},

keywords={},

doi={},

ISSN={},

month={February},}

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TY - JOUR

TI - Unreachability Proofs for β Rewriting Systems by Homomorphisms

T2 - IEICE TRANSACTIONS on Information

SP - 339

EP - 347

AU - Kiyoshi AKAMA

AU - Yoshinori SHIGETA

AU - Eiichi MIYAMOTO

PY - 1999

DO -

JO - IEICE TRANSACTIONS on Information

SN -

VL - E82-D

IS - 2

JA - IEICE TRANSACTIONS on Information

Y1 - February 1999

AB - Given two terms and their rewriting rules, an unreachability problem proves the non-existence of a reduction sequence from one term to another. This paper formalizes a method for solving unreachability problems by abstraction; i. e. , reducing an original concrete unreachability problem to a simpler abstract unreachability problem to prove the unreachability of the original concrete problem if the abstract unreachability is proved. The class of rewriting systems discussed in this paper is called β rewriting systems. The class of β rewriting systems includes very important systems such as semi-Thue systems and Petri Nets. Abstract rewriting systems are also a subclass of β rewriting systems. A β rewriting system is defined on axiomatically formulated base structures, called β structures, which are used to formalize the concepts of "contexts" and "replacement," which are common to many rewritten objects. Each domain underlying semi-Thue systems, Petri Nets, and other rewriting systems are formalized by a β structure. A concept of homomorphisms from a β structure (a concrete domain) to a β structure (an abstract domain) is introduced. A homomorphism theorem (Theorem1)is established for β rewriting systems, which states that concrete reachability implies abstract reachability. An unreachability theorem (Corollary1) is also proved for β rewriting systems. It is the contraposition of the homomorphism theorem, i. e. , it says that abstract unreachability implies concrete unreachability. The unreachability theorem is used to solve two unreachability problems: a coffee bean puzzle and a checker board puzzle.

ER -