95 lines
4.7 KiB
Markdown
95 lines
4.7 KiB
Markdown
---
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title: Abstract Rewriting Systems
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TARGET DECK: Obsidian::STEM
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FILE TAGS: ars
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tags:
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- ars
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---
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## Overview
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In an **abstract rewriting system** (ARS), an object is said to be in **normal form** if it cannot be rewritten any further, i.e. it is irreducible. An object is said to be in **canonical form** if it is presented in the "standard" representation (where "standard" is defined per field).
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In most fields, a canoncial form specifies a *unique* representation.
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%%ANKI
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Basic
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What is ARS an acronym for?
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Back: **A**bstract **R**ewriting **S**ystem.
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Reference: “Normal Form,” in _Wikipedia_, April 27, 2024, [https://en.wikipedia.org/w/index.php?title=Normal_form](https://en.wikipedia.org/w/index.php?title=Normal_form_(abstract_rewriting)&oldid=1221094193).
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<!--ID: 1719067812812-->
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END%%
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%%ANKI
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Basic
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Canonical/normal forms are associated with what general class of abstract systems?
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Back: Abstract rewriting systems.
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Reference: “Canonical Form,” in _Wikipedia_, January 7, 2024, [https://en.wikipedia.org/w/index.php?title=Canonical_form](https://en.wikipedia.org/w/index.php?title=Canonical_form&oldid=1194093963).
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<!--ID: 1719067812820-->
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END%%
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%%ANKI
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Basic
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What does it mean for an object to be in normal form?
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Back: It cannot be rewritten any further, i.e. it is irreducible.
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Reference: Normal Form,” in _Wikipedia_, April 27, 2024, [https://en.wikipedia.org/w/index.php?title=Normal_form](https://en.wikipedia.org/w/index.php?title=Normal_form_(abstract_rewriting)&oldid=1221094193).
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END%%
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%%ANKI
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Basic
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What does it mean for an object to be in canonical form?
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Back: It is presented in the "standard" way with respect to the field its presented in.
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Reference: “Canonical Form,” in _Wikipedia_, January 7, 2024, [https://en.wikipedia.org/w/index.php?title=Canonical_form](https://en.wikipedia.org/w/index.php?title=Canonical_form&oldid=1194093963).
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<!--ID: 1719067812824-->
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END%%
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%%ANKI
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Basic
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What two assumptions are usually associated with an object in canonical form?
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Back: It is as simply represented as possible and it is uniquely identifying.
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Reference: “Canonical Form,” in _Wikipedia_, January 7, 2024, [https://en.wikipedia.org/w/index.php?title=Canonical_form](https://en.wikipedia.org/w/index.php?title=Canonical_form&oldid=1194093963).
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<!--ID: 1719067812829-->
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END%%
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%%ANKI
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Basic
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How do "canonical" and "normal" forms differ in most fields?
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Back: The former usually specifies a unique representation for the object.
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Reference: “Canonical Form,” in _Wikipedia_, January 7, 2024, [https://en.wikipedia.org/w/index.php?title=Canonical_form](https://en.wikipedia.org/w/index.php?title=Canonical_form&oldid=1194093963).
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<!--ID: 1719067812833-->
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END%%
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## Confluence
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**Confluence** is the property by which two different terms can be further reduced to one common term. That is to say, confluence is a property of rewriting systems describing which terms in such a system can be rewritten in more than one way.
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%%ANKI
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Basic
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What is confluence?
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Back: A property by which two different terms can be further reduced to one common term.
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Reference: “Confluence (Abstract Rewriting),” in _Wikipedia_, May 22, 2024, [https://en.wikipedia.org/w/index.php?title=Confluence](https://en.wikipedia.org/w/index.php?title=Confluence_(abstract_rewriting)&oldid=1225041322).
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<!--ID: 1719578045810-->
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END%%
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%%ANKI
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Basic
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How might $(11 + 9) \times (2 + 4)$ be reduced to demonstrate confluence?
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Back: As $20 \times (2 + 4)$ and as $(11 + 9) \times 6$.
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Reference: “Confluence (Abstract Rewriting),” in _Wikipedia_, May 22, 2024, [https://en.wikipedia.org/w/index.php?title=Confluence](https://en.wikipedia.org/w/index.php?title=Confluence_(abstract_rewriting)&oldid=1225041322).
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<!--ID: 1719578045839-->
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END%%
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%%ANKI
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Basic
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How might $(\lambda x. (\lambda y. yx)z)v$ be reduced to demonstrate confluence?
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Back: As $(\lambda y.yv)z$ and as $(\lambda x. zx)v$.
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Reference: “Confluence (Abstract Rewriting),” in _Wikipedia_, May 22, 2024, [https://en.wikipedia.org/w/index.php?title=Confluence](https://en.wikipedia.org/w/index.php?title=Confluence_(abstract_rewriting)&oldid=1225041322).
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<!--ID: 1719578045843-->
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END%%
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## Bibliography
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* “Canonical Form,” in _Wikipedia_, January 7, 2024, [https://en.wikipedia.org/w/index.php?title=Canonical_form](https://en.wikipedia.org/w/index.php?title=Canonical_form&oldid=1194093963).
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* “Confluence (Abstract Rewriting),” in _Wikipedia_, May 22, 2024, [https://en.wikipedia.org/w/index.php?title=Confluence](https://en.wikipedia.org/w/index.php?title=Confluence_(abstract_rewriting)&oldid=1225041322).
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* Normal Form,” in _Wikipedia_, April 27, 2024, [https://en.wikipedia.org/w/index.php?title=Normal_form](https://en.wikipedia.org/w/index.php?title=Normal_form_(abstract_rewriting)&oldid=1221094193).
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