Skip to main content

semi-Thue system

semi-Thue system An important concept in formal language theory that underlies the notion of a grammar. It was defined and investigated by Axel Thue from about 1904. A semi-Thue system over the alphabet Σ is a finite set of ordered pairs of Σ-words: {〈l1,r1〉,…,〈ln,rn〉}

Each pair 〈li,ri〉 is a rule, referred to as a production, with left-hand side li and right-hand side ri; it is usually written li ri

Let u and v be Σ-words, and lr be a production, then the word ulv is said to directly derive the word urv; this is written ulvurv

So w directly derives w′ if w′ is the result of applying a production to some substring of w. If w1w2 ⇒ … ⇒ wn–1wn

then w1 is said to derive wn; this is written w1 ⇒∗ wn

So w derives w′ if w′ is obtained from w by a sequence of direct derivations.

As one example, let Σ be {a,b} and let the productions be {abba, baab}

then aabba derives baaab by the sequence aabbaabababaababaaab

It is clear that w derives any permutation of w.

As a second example, with productions {abba, ba → Λ}

w derives Λ (the empty word) if and only if w has the same number of as as bs.

The question of whether w derives w′ is algorithmically undecidable.

Cite this article
Pick a style below, and copy the text for your bibliography.

  • MLA
  • Chicago
  • APA

"semi-Thue system." A Dictionary of Computing. . 19 Mar. 2019 <>.

"semi-Thue system." A Dictionary of Computing. . (March 19, 2019).

"semi-Thue system." A Dictionary of Computing. . Retrieved March 19, 2019 from

Learn more about citation styles

Citation styles gives you the ability to cite reference entries and articles according to common styles from the Modern Language Association (MLA), The Chicago Manual of Style, and the American Psychological Association (APA).

Within the “Cite this article” tool, pick a style to see how all available information looks when formatted according to that style. Then, copy and paste the text into your bibliography or works cited list.

Because each style has its own formatting nuances that evolve over time and not all information is available for every reference entry or article, cannot guarantee each citation it generates. Therefore, it’s best to use citations as a starting point before checking the style against your school or publication’s requirements and the most-recent information available at these sites:

Modern Language Association

The Chicago Manual of Style

American Psychological Association

  • Most online reference entries and articles do not have page numbers. Therefore, that information is unavailable for most content. However, the date of retrieval is often important. Refer to each style’s convention regarding the best way to format page numbers and retrieval dates.
  • In addition to the MLA, Chicago, and APA styles, your school, university, publication, or institution may have its own requirements for citations. Therefore, be sure to refer to those guidelines when editing your bibliography or works cited list.