Abstract
Logical formulas are naturally decomposed into their subformulas and circuits into their layers. How are these decompositions expressed in a purely language-theoretical setting? We address that question, and in doing so, introduce a product directly on languages that parallels formula composition. This framework makes an essential use of languages of higher-dimensional words, called hyperwords, of arbitrary dimensions. It is shown here that the product thus introduced is associative over classes of languages closed under the product itself; this translates back to extra freedom in the way formulas and circuits can be decomposed.
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Notes
- 1.
We only make scarce use of the variables with nonpositive indexes explicitly, with the notable exception of the first part of the proof of Theorem 21.
- 2.
This nomenclature stems from the algebraic operation bearing the same name. There is a precise relationship between block products of monoids and block products of languages of words (Definition 2) that will be made explicit in an extended version of this article.
- 3.
The reader not versed in that topic can think of block products of monoids as block products of the languages of dimension 1 recognized by them.
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Acknowledgments
We thank Charles Paperman for stimulating discussions.
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Cadilhac, M., Krebs, A., Lange, KJ. (2016). A Language-Theoretical Approach to Descriptive Complexity. In: Brlek, S., Reutenauer, C. (eds) Developments in Language Theory. DLT 2016. Lecture Notes in Computer Science(), vol 9840. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-53132-7_6
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