Parity function: Wikis

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In Boolean algebra, a parity function is a Boolean function whose value is 1 if the input vector has odd number of ones.

The parity function is notable for its role in theoretical investigation of circuit complexity of Boolean functions.

Properties

The parity is a symmetric Boolean function.

The n-variable parity function and its negation are the only ones for which all disjunctive normal forms have the maximal number of 2 n − 1 monomials of length n and all conjunctive normal forms have the maximal number of 2 n − 1 clauses of length n.[1]

Circuit complexity

In early 1980s Merrick Furst, James Saxe and Michael Sipser[2] and independently Miklós Ajtai[3] established super-polynomial lower bounds on the size of constant-depth Boolean circuits for the parity function,i.e., they have shown 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 to the parity function problem.[2] Until this time only linear lower bounds were known for various naturally arising functions.[2]

A number of increasing published lower bounds followed. Eventually, In 1989 Johan Håstad (who was awarded Gödel Prize for this work in 1994) established tight exponential lower bounds on the size of constant-depth Boolean circuits for the parity function. In particular, Håstad's Switching Lemma captured the technical difficulty of establishing of such lower bounds.[1]

Infinite version

An infinite parity function is a function $f\colon \{0, 1\}^{\omega} \to \{0, 1\}$ mapping every infinite binary string to 0 or 1, having the following property: if w and v are infinite binary strings differing only on finite number of coordinates then f(w) = f(v) if and only if w and v differ on even number of coordinates.

Assuming axiom of choice it can be easily proved that parity functions exist and there are $2^{\mathfrak{c}}$ many of them - as many as the number of all functions from {0,1}ω to {0,1}. It is enough to take one representative per equivalence class of relation $\approx$ defined as follows: $w \approx v$ iff w and v differ at finite number of coordinates. Having such representatives, we can map all of them to 0; the rest of f 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 f − 1[0] is a non-Borel set.

References

1. ^ a b Ingo Wegener, Randall J. Pruim, Complexity Theory, 2005, ISBN 3540210458, p. 260
2. ^ a b c Merrick Furst, James Saxe and Michael Sipser, "Parity, Circuits, and the Polynomial-Time Hierarchy", Annu. Intl. Symp. Found.Coimputer Sci., 1981, Theory of Computing Systems, vol. 17, no. 1, 1984, pp. 13–27, doi:10.1007/BF01744431
3. ^ Miklós Ajtai, "$\Sigma^1_1$-Formulae on Finite Structures", Annals of Pure and Applied Logic, 24 (1983) 1–48.