Model of computational complexity
Top 10 Circuit complexity related articles
In theoretical computer science, circuit complexity is a branch of computational complexity theory in which Boolean functions are classified according to the size or depth of the Boolean circuits that compute them. A related notion is the circuit complexity of a recursive language that is decided by a uniform family of circuits
Proving lower bounds on size of Boolean circuits computing explicit Boolean functions is a popular approach to separating complexity classes. For example, a prominent circuit class P/poly consists of Boolean functions computable by circuits of polynomial-size. Proving that
Circuit complexity Intro articles: 4
Size and Depth
A Boolean circuit with
There are two major notions of circuit complexity (these are outlined in Sipser (1997):324). The circuit-size complexity of a Boolean function
These notions generalize when one considers the circuit complexity of any language that contains strings with different bit lengths, especially infinite formal languages. Boolean circuits, however, only allow a fixed number of input bits. Thus no single Boolean circuit is capable of deciding such a language. To account for this possibility, one considers families of circuits
Hence, the circuit-size complexity of a formal language
Circuit complexity Size and Depth articles: 7
Boolean circuits are one of the prime examples of so-called non-uniform models of computation in the sense that inputs of different lengths are processed by different circuits, in contrast with uniform models such as Turing machines where the same computational device is used for all possible input lengths. An individual computational problem is thus associated with a particular family of Boolean circuits
A family of Boolean circuits
- M runs in polynomial time
- For all
, M outputs a description of on input
A family of Boolean circuits
- M runs in logarithmic space
- For all
, M outputs a description of on input
Circuit complexity Uniformity articles: 6
Circuit complexity goes back to Shannon (1949), who proved that almost all Boolean functions on n variables require circuits of size Θ(2n/n). Despite this fact, complexity theorists have only been able to prove superpolynomial circuit lower bounds on functions explicitly constructed for the purpose of being hard to calculate.
More commonly, superpolynomial lower bounds have been proved under certain restrictions on the family of circuits used. The first function for which superpolynomial circuit lower bounds were shown was the parity function, which computes the sum of its input bits modulo 2. The fact that parity is not contained in AC0 was first established independently by Ajtai (1983) and by Furst, Saxe and Sipser (1984). Later improvements by Håstad (1987) in fact establish that any family of constant-depth circuits computing the parity function requires exponential size. Extending a result of Razborov, Smolensky (1987) proved that this is true even if the circuit is augmented with gates computing the sum of its input bits modulo some odd prime p.
The k-clique problem is to decide whether a given graph on n vertices has a clique of size k. For any particular choice of the constants n and k, the graph can be encoded in binary using
Raz and McKenzie later showed that the monotone NC hierarchy is infinite (1999).
The Integer Division Problem lies in uniform TC0 (Hesse 2001).
Circuit complexity History articles: 7
Circuit lower bounds
Circuit lower bounds are generally difficult. Known results include
- Parity is not in nonuniform AC0, proved by Ajtai (1983) and by Furst, Saxe and Sipser.
- Uniform TC0 is strictly contained in PP, proved by Allender.
- The classes SP
2, PP and MA/1 (MA with one bit of advice) are not in SIZE(nk) for any constant k.
- While it is suspected that the nonuniform class ACC0 does not contain the majority function, it was only in 2010 that Williams proved that
It is open whether NEXPTIME has nonuniform TC0 circuits.
Proofs of circuit lower bounds are strongly connected to derandomization. A proof that
Razborov and Rudich (1997) showed that many known circuit lower bounds for explicit Boolean functions imply the existence of so called natural properties useful against the respective circuit class. On the other hand, natural properties useful against P/poly would break strong pseudorandom generators. This is often interpreted as a ``natural proofs" barrier for proving strong circuit lower bounds. Carmosino, Impagliazzo, Kabanets and Kolokolova (2016) proved that natural properties can be also used to construct efficient learning algorithms.
Circuit complexity Circuit lower bounds articles: 7
Many circuit complexity classes are defined in terms of class hierarchies. For each nonnegative integer i, there is a class NCi, consisting of polynomial-size circuits of depth
Relation to time complexity
- Sipser, M. (1997). 'Introduction to the theory of computation.' Boston: PWS Pub. Co.
- Ajtai, Miklós; Komlós, János; Szemerédi, Endre (1983). An 0(n log n) sorting network. STOC '83 Proceedings of the Fifteenth Annual ACM Symposium on Theory of Computing. pp. 1–9. ISBN 978-0-89791-099-6.
- Furst, Merrick; Saxe, James B.; Sipser, Michael (1984). "Parity, circuits, and the polynomial-time hierarchy". Mathematical Systems Theory. 17 (1): 13–27. doi:10.1007/BF01744431. MR 0738749.
- See proof
- Santhanam, Rahul (2007). "Circuit lower bounds for Merlin-Arthur classes". STOC 2007: Proceedings of the thirty-ninth annual ACM symposium on Theory of computing. pp. 275–283. CiteSeerX 10.1.1.92.4422. doi:10.1145/1250790.1250832.
- Williams, Ryan (2011). "Non-Uniform ACC Circuit Lower Bounds" (PDF). CCC 2011: Proceedings of the 26th Annual IEEE Conference on Computational Complexity. pp. 115–125. doi:10.1109/CCC.2011.36.
- Kabanets, V.; Impagliazzo, R. (2004). "Derandomizing polynomial identity tests means proving circuit lower bounds". Computational Complexity. 13 (1): 1–46. doi:10.1007/s00037-004-0182-6.
- Razborov, Alexander; Rudich, Stephen (1997). "Natural proofs". Journal of Computer and System Sciences. 55. pp. 24–35.
- Carmosino, Marco; Impagliazzo, Russell; Kabanets, Valentine; Kolokolova, Antonina (2016). "Learning algorithms from natural proofs". Computational Complexity Conference.
- Ajtai, Miklós (1983). "
-formulae on finite structures". Annals of Pure and Applied Logic. 24: 1–24. doi:10.1016/0168-0072(83)90038-6.
- Alon, Noga; Boppana, Ravi B. (1987). "The monotone circuit complexity of Boolean functions". Combinatorica. 7 (1): 1–22. CiteSeerX 10.1.1.300.9623. doi:10.1007/bf02579196.
- Furst, Merrick L.; Saxe, James B.; Sipser, Michael (1984). "Parity, circuits, and the polynomial-time hierarchy". Mathematical Systems Theory. 17 (1): 13–27. doi:10.1007/bf01744431.
- Håstad, Johan (1987), Computational limitations of small depth circuits (PDF), Ph.D. thesis, Massachusetts Institute of Technology.
- Hesse, William (2001). "Division is in uniform TC0". Proc. 28th International Colloquium on Automata, Languages and Programming. Springer. pp. 104–114.
- Raz, Ran; McKenzie, Pierre (1999). "Separation of the monotone NC hierarchy". Combinatorica. 19 (3): 403–435. doi:10.1007/s004930050062.
- Razborov, Alexander A. (1985). "Lower bounds on the monotone complexity of some Boolean functions". Mathematics of the USSR, Doklady. 31: 354–357.
- Rossman, Benjamin (2008). "On the constant-depth complexity of k-clique". STOC 2008: Proceedings of the 40th annual ACM symposium on Theory of computing. ACM. pp. 721–730. doi:10.1145/1374376.1374480.
- Shannon, Claude E. (1949). "The synthesis of two-terminal switching circuits". Bell System Technical Journal. 28 (1): 59–98. doi:10.1002/j.1538-7305.1949.tb03624.x.
- Smolensky, Roman (1987). "Algebraic methods in the theory of lower bounds for Boolean circuit complexity". Proc. 19th Annual ACM Symposium on Theory of Computing. ACM. pp. 77–82. doi:10.1145/28395.28404.
- Vollmer, Heribert (1999). Introduction to Circuit Complexity: a Uniform Approach. Springer Verlag. ISBN 978-3-540-64310-4.
- Wegener, Ingo (1987). The Complexity of Boolean Functions. John Wiley and Sons Ltd, and B. G. Teubner, Stuttgart. ISBN 978-3-519-02107-0. At the time an influential textbook on the subject, commonly known as the "Blue Book". Also available for download (PDF) at the Electronic Colloquium on Computational Complexity.
- Lecture notes for a course of Uri Zwick on circuit complexity
- Circuit Complexity before the Dawn of the New Millennium, a 1997 survey of the field by Eric Allender slides.