Parity function – Wikipedia
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In Boolean algebra, a parity function is a Boolean function whose value is one if and only if the input vector has an odd number of ones. The parity function of two inputs is also known as the XOR function.
The parity function is notable for its role in theoretical investigation of circuit complexity of Boolean functions.
The output of the parity function is the parity bit.
Definition[edit]
The
-variable parity function is the Boolean function
with the property that
if and only if the number of ones in the vector
is odd.
In other words,
is defined as follows:
where
denotes exclusive or.
Properties[edit]
Parity only depends on the number of ones and is therefore a symmetric Boolean function.
The n-variable parity function and its negation are the only Boolean functions for which all disjunctive normal forms have the maximal number of 2 n − 1monomials of length n and all conjunctive normal forms have the maximal number of 2 n − 1 clauses of length n.[1]
Computational complexity[edit]
Some of the earliest work in computational complexity was 1961 bound of Bella Subbotovskaya showing the size of a Boolean formula computing parity must be at least
. This work uses the method of random restrictions. This exponent of
has been increased through careful analysis to
by Paterson and Zwick (1993) and then to
by Håstad (1998). [2]
In the early 1980s, Merrick Furst, James Saxe and Michael Sipser[3] and independently Miklós Ajtai[4] established super-polynomial lower bounds on the size of constant-depth Boolean circuits for the parity function, i.e., they showed that polynomial-size constant-depth circuits cannot compute the parity function. Similar results were also established for the majority, multiplication and transitive closure functions, by reduction from the parity function.[3]
Håstad (1987) established tight exponential lower bounds on the size of constant-depth Boolean circuits for the parity function. Håstad’s Switching Lemma is the key technical tool used for these lower bounds and Johan Håstad was awarded the Gödel Prize for this work in 1994.
The precise result is that depth-k circuits with AND, OR, and NOT gates require size
to compute the parity function.
This is asymptotically almost optimal as there are depth-k circuits computing parity which have size
.
Infinite version[edit]
An infinite parity function is a function
mapping every infinite binary string to 0 or 1, having the following property: if
and
are infinite binary strings differing only on finite number of coordinates then
if and only if
and
differ on even number of coordinates.
Assuming axiom of choice it can be easily proved that parity functions exist and there are
many of them – as many as the number of all functions from
to
. It is enough to take one representative per equivalence class of relation
defined as follows:
if
and
differ at finite number of coordinates. Having such representatives, we can map all of them to 0; the rest of
values are deducted unambiguously.
Infinite parity functions are often used in theoretical Computer Science and Set Theory because of their simple definition and – on the other hand – their descriptive complexity. For example, it can be shown that an inverse image
is a non-Borel set.
See also[edit]
Related topics:
References[edit]
- ^ Ingo Wegener, Randall J. Pruim, Complexity Theory, 2005, ISBN 3-540-21045-8, p. 260
- ^ Jukna, Stasys (Jan 6, 2012). Boolean Function Complexity: Advances and Frontiers. Springer Science & Business Media. pp. 167–173. ISBN 978-3642245084.
- ^ a b Merrick Furst, James Saxe and Michael Sipser, “Parity, Circuits, and the Polynomial-Time Hierarchy”, Annu. Intl. Symp. Found.Computer Sci., 1981, Theory of Computing Systems, vol. 17, no. 1, 1984, pp. 13–27, doi:10.1007/BF01744431
- ^ Miklós Ajtai, “-Formulae on Finite Structures”, Annals of Pure and Applied Logic, 24 (1983) 1–48.
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