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A degree-decreasing Lemma for (MOD q, MOD p) circuits

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Automata, Languages and Programming (ICALP 1998)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 1443))

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Abstract

Consider a (MODq,MODp) circuit, where the inputs of the bottom MODp gates are degree-d polynomials of the input variables (p, q are different primes). Using our main tool — the Degree Decreasing Lemma — we show that this circuit can be converted to a (MODq, MODp) circuit with linear polynomials on the input-level with the price of increasing the size of the circuit. This result has numerous consequences: for the Constant Degree Hypothesis of Barrington, Straubing and Thérien [3], and generalizing the lower bound results of Yan and Parberry [21], Krause and Waack [13] and Krause and Pudlák [12]. Perhaps the most important application is an exponential lower bound for the size of (MODq, MODp) circuits computing the n fan-in AND, where the input of each MODp gate at the bottom is an arbitrary integer valued function of cn variables (c < 1) plus an arbitrary linear function of n input variables.

We believe that the Degree Decreasing Lemma becomes a standard tool in modular circuit theory.

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Authors and Affiliations

  1. Department of Computer Science, Eötvös University, Budapest MÚzeum krt.6-8, H-1088, Budapest, Hungary

    Vince Grolmusz

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  1. Vince Grolmusz
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Editor information

Kim G. Larsen Sven Skyum Glynn Winskel

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© 1998 Springer-Verlag Berlin Heidelberg

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Grolmusz, V. (1998). A degree-decreasing Lemma for (MOD q, MOD p) circuits. In: Larsen, K.G., Skyum, S., Winskel, G. (eds) Automata, Languages and Programming. ICALP 1998. Lecture Notes in Computer Science, vol 1443. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0055055

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  • DOI: https://doi.org/10.1007/BFb0055055

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-64781-2

  • Online ISBN: 978-3-540-68681-1

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