In computational complexity theory, Håstad's switching lemma is a key tool for proving lower bounds on the size of constant-depth Boolean circuits. It was first introduced by Johan Håstad to prove that AC0 Boolean circuits of depth k require size to compute the parity function on bits.[1] He was later awarded the Gödel Prize for this work in 1994.
The switching lemma describes the behavior of a depth-2 circuit under random restriction, i.e. when randomly fixing most of the coordinates to 0 or 1. Specifically, from the lemma, it follows that a formula in conjunctive normal form (that is, an AND of ORs) becomes a formula in disjunctive normal form (an OR and ANDs) under random restriction, and vice versa. This "switching" gives the lemma its name.
^Håstad, Johan (1986). "Almost optimal lower bounds for small depth circuits". STOC '86: Proceedings of the Eighteenth Annual ACM Symposium on Theory of Computing. ACM Press: 6–20. doi:10.1145/12130.12132. ISBN 978-0-89791-193-1.
In computational complexity theory, Håstad's switchinglemma is a key tool for proving lower bounds on the size of constant-depth Boolean circuits. It...
switching Message switching Packet switching Fast packet switching Network switching subsystem 5ESS Switching System Number One Electronic Switching System...
constant-depth Boolean circuits for the parity function. Håstad's SwitchingLemma is the key technical tool used for these lower bounds and Johan Håstad...
the latter class can compute parity. More precise bounds follow from switchinglemma. Using them, it has been shown that there is an oracle separation between...
Tautology Problem, which is complete for co-NP. Circuit satisfiability Switchinglemma Samuel R. Buss (Jan 1987). "The Boolean formula value problem is in...
Retrieved 2021-05-03. "switching function". TheFreeDictionary.com. Retrieved 2021-05-03. Davies, D. W. (December 1957). "Switching Functions of Three Variables"...
characterizations of complexity classes. Chapter thirteen uses the switchinglemma for lower bounds, and chapter fourteen concerns applications to databases...
{\displaystyle {\dot {f}}\to 0} as t → ∞ {\displaystyle t\to \infty } . Barbalat's Lemma says: If f ( t ) {\displaystyle f(t)} has a finite limit as t → ∞ {\displaystyle...
of Γ[a,b]. The generalisation of Morse's lemma to CAT(-1) spaces is often referred to as the Morse–Mostow lemma and can be proved by a straightforward generalisation...
then to 2 {\displaystyle 2} by Håstad (1998). Additionally, Håstad's Switchinglemma (1987) applied the same technique to the much richer model of constant...
Marrickville, where the Labor candidate was Education Minister Carmel Tebbutt (switching from the Legislative Council), the Labor primary vote increased in the...
operations (respectively, column operations): Row switching A row within the matrix can be switched with another row. R i ↔ R j {\displaystyle R_{i}\leftrightarrow...
studying switching circuits, Claude Shannon observed that one could also apply the rules of Boole's algebra in this setting, and he introduced switching algebra...
can simply be interpreted as switching the time domain and the frequency domain, with the inverse Fourier transform switching them back, more geometrically...
love with his mother Tefnut, takes her as his chief queen-consort. "Tfn.t (Lemma ID 171880)". Thesaurus Linguae Aegyptiae. Love, Edward O. D. (2021). "Innovative...
productions". 9th Annual Symposium on Switching and Automata Theory (Swat 1968). 9th Annual Symposium on Switching and Automata Theory (swat 1968). pp. 131–142...
them to construct switching networks for computer hardware, replacing both buses and the faster, but more expensive, crossbar switches. Since the 2000s...
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