Abstract
Descriptive complexity is inherently parallel in nature. This is a particularly delightful dividend of applying this form of logic to computer science. The time to compute a query on a certain parallel computer corresponds exactly to the depth of a first-order induction needed to describe the query. There is also a close relationship between the amount of hardware used — memory and processors — and the number of variables in the inductive definition.
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Notes
This is obvious if n is a power of 2. If not, we can just let each processor break its processor number into k [log n]-tuples of bits. If any of these is greater than or equal to n, then the processor should do nothing during the entire computation.
This is analogous to a result we will see later: The polynomial-time hierarchy (PH) is equal to the set of boolean queries expressible in second-order logic (SO), (Corollary 7.22).
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© 1999 Springer Science+Business Media New York
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Immerman, N. (1999). Parallelism. In: Descriptive Complexity. Graduate Texts in Computer Science. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-0539-5_6
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DOI: https://doi.org/10.1007/978-1-4612-0539-5_6
Publisher Name: Springer, New York, NY
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