Circuit complexity

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 Boolean circuits that compute them. One speaks of the circuit complexity of a Boolean circuit. A related notion is the circuit complexity of a recursive language that is decided by a family of circuits $C_{1},C_{2},\ldots$ (see below).

A Boolean circuit with $n$ input bits is a directed acyclic graph in which every node (usually called gates in this context) is either an input node of in-degree 0 labeled by one of the $n$ input bits, an AND gate, an OR gate, or a NOT gate. One of these gates is designated as the output gate. Such a circuit naturally computes a function of its $n$ inputs. The size of a circuit is the number of gates it contains and its depth is the maximal length of a path from an input gate to the output gate.

There are two major notions of circuit complexity (these are outlined in Sipser (1997)^[1]:324). The circuit-size complexity of a Boolean function $f$ is the minimal size of any circuit computing $f$. The circuit-depth complexity of a Boolean function $f$ is the minimal depth of any circuit computing $f$.

These notions generalize when one considers the circuit complexity of a recursive language: A formal language may contain strings with many different bit lengths. 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 $C_{1},C_{2},\ldots$ where each $C_{n}$ accepts inputs of size $n$. Each circuit family will naturally generate a recursive language by outputting $1$ when a string is a member of the family, and $0$ otherwise. We say that a family of circuits is size minimal if there is no other family that decides on inputs of any size, $n$, with a circuit of smaller size than $C_n$ (respectively for depth minimal families).

Hence, the circuit-size complexity of a recursive language $A$ is defined as the function $t:\mathbb{N}\to\mathbb{N}$, that relates a bit length of an input, $n$, to the circuit-size complexity of a minimal circuit $C_{n}$ that decides whether inputs of that length are in $A$. The circuit-depth complexity is defined similarly.

Complexity classes defined in terms of Boolean circuits include AC⁰, AC, TC⁰ and NC.

Uniformity

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 $C_1, C_2, \dots$ where each $C_n$ is the circuit handling inputs of n bits. A uniformity condition is often imposed on these families, requiring the existence of some resource-bounded Turing machine which, on input n, produces a description of the individual circuit $C_n$. When this Turing machine has a running time polynomial in n, the circuit family is said to be P-uniform. The stricter requirement of DLOGTIME-uniformity is of particular interest in the study of shallow-depth circuit-classes such as AC⁰ or TC⁰.

Polynomial-time uniform

A family of Boolean circuits $\{C_n:n \in \mathbb{N}\}$ is polynomial-time uniform if there exists a deterministic Turing machine M, such that

• M runs in polynomial time
• For all $n \in \mathbb{N}$, M outputs a description of $C_n$ on input $1^n$

Logspace uniform

A family of Boolean circuits $\{C_n:n \in \mathbb{N}\}$ is logspace uniform if there exists a deterministic Turing machine M, such that

• M runs in logarithmic space
• For all $n \in \mathbb{N}$, M outputs a description of $C_n$ on input $1^n$

History

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 not been able to prove superpolynomial circuit lower bounds for specific Boolean functions.

On the other hand, 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)^[2] and by Furst, Saxe and Sipser (1984).^[3] Later improvements by Håstad (1987) in fact establish that any family of constant-depth circuits computing the parity function requires exponential size. 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 ${n \choose 2}$ bits which indicate for each possible edge whether it is present. Then the k-clique problem is formalized as a function $f_k:\{0,1\}^{{n \choose 2}}\to\{0,1\}$ such that $f_k$ outputs 1 if and only if the graph encoded by the string contains a clique of size k. This family of functions is monotone and can be computed by a family of circuits, but it has been shown that it cannot be computed by a polynomial-size family of monotone circuits (that is, circuits with AND and OR gates but without negation). The original result of Razborov (1985) was later improved to an exponential-size lower bound by Alon and Boppana (1987). Rossman (2008) shows that constant-depth circuits with AND, OR, and NOT gates require size $\Omega(n^{k/4})$ to solve the k-clique problem even in the average case. Moreover, there is a circuit of size $n^{k/4+O(1)}$ which computes $f_k$.

Raz and McKenzie later showed that the monotone NC hierarchy is infinite (1999).

The Integer Division Problem lies in uniform TC⁰ (Hesse 2001).

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 not contained in PP, proved by Allender.
• The classes SP
  2, PP^[4] and MA/1^[5] (MA with one bit of advice) are not in SIZE(n^k) 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 $\mathsf{NEXP} \not \subseteq \mathsf{ACC}^0$.^[6]

It is open whether NEXPTIME has nonuniform TC⁰ circuits.

Proofs of circuit lower bounds are strongly connected to derandomization. A proof that P = BPP would imply that either $\mathsf{NEXP} \not \subseteq \mathsf{P/poly}$ or that permanent cannot be computed by nonuniform arithmetic circuits (polynomials) of polynomial size and polynomial degree.^[7]

Complexity classes

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 $O(\log^i(n))$, using bounded fan-in AND, OR, and NOT gates. We can talk about the union NC of all of these classes. By considering unbounded fan-in gates, we construct the classes ACⁱ and AC. We construct many other circuit complexity classes with the same size and depth restrictions by allowing different sets of gates.

Relation to time complexity^[1]

Say that a certain language, $A$, belongs to the time-complexity class $\text{TIME}(t(n))$ for some function $t:\mathbb{N}\to\mathbb{N}$. Then $A$ has circuit complexity $\mathcal{O}(t^{2}(n))$

References

1. Sipser, M. (1997). 'Introduction to the theory of computation.' Boston: PWS Pub. Co.
2. 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: 1–9. ISBN 0-89791-099-0.
3. Furst, Merrick; Saxe, James B.; Sipser, Michael (1984). "Parity, circuits, and the polynomial-time hierarchy". Math. Syst. Theory 17: 13–27. doi:10.1007/bf01744431. ISSN 0025-5661. Zbl 0534.94008.
4. See proof
5. 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. doi:10.1145/1250790.1250832.
6. 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.
7. 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.

• Ajtai, Miklós (1983). "$\Sigma^1_1$-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. 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 3-540-64310-9.
• Wegener, Ingo (1987). The Complexity of Boolean Functions. John Wiley and Sons Ltd, and B. G. Teubner, Stuttgart. ISBN 3-519-02107-2. 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.