In Boolean algebra, a parity function is a Boolean function whose value is 1 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.
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 − 1 monomials of length n and all conjunctive normal forms have the maximal number of 2 n − 1 clauses of length n.
In the early 1980s, Merrick Furst, James Saxe and Michael Sipser and independently Miklós Ajtai 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.
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 .
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.
- Ingo Wegener, Randall J. Pruim, Complexity Theory, 2005, ISBN 3-540-21045-8, p. 260
- 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
- Miklós Ajtai, "-Formulae on Finite Structures", Annals of Pure and Applied Logic, 24 (1983) 1–48.