CHAPTER Circuit Complexity The circuit complexity of a binary function is measured by the size or depth of the smallest or shallowest circuit for it. Circuit complexity derives its importance from the corollary to Theorem 3.9.2; namely, if a function has a large circuit size over a complete basis of fixed fan-in, then the time on a Turing machine required to compute it is large. The importance of ≥ (n) this observation is illustrated by the following fact. For n 1, let fL be the characteristic (n) function of an NP-complete language L,wherefL has value 1 on strings of length n in L (n) and value 0 otherwise. If fL has super-polynomial circuit size for all sufficiently large n,then P = NP. In this chapter we introduce methods for deriving lower bounds on circuit size and depth. Unfortunately, it is generally much more difficult to derive good lower bounds on circuit complexity than good upper bounds; an upper bound measures the size or depth of a particular circuit whereas a lower bound must rule out a smaller size or depth for all circuits. As a consequence, the lower bounds derived for functions realized by circuits over complete bases of bounded fan-in are often weak. In attempting to understand lower bounds for complete bases, researchers have studied monotone circuits over the monotone basis and bounded-depth circuits over the basis {AND, OR, NOT} in which the first two gates are allowed to have unbounded fan-in. Formula size, which is approximately the size of the smallest circuit with fan-out 1, has also been studied. Lower bounds to formula size also produce lower bounds to circuit depth, a measure of the parallel time needed for a function. Research on these restricted circuit models has led to some impressive results. Exponential lower bounds on circuit size have been derived for monotone functions over the monotone basis and functions such as parity when realized by bounded-depth circuits. Unfortunately, the methods used to obtain these results may not apply to complete bases of bounded fan-in. Fortunately, it has been shown that the slice functions have about the same circuit size over ? both the monotone and standard (non-monotone) bases. This may help resolve the P = NP question, since there are NP-complete slice problems. Despite the difficulty of deriving lower bounds, circuit complexity continues to offer one of the methods of highest potential for distinguishing between P and NP. 391 392 Chapter 9 Circuit Complexity Models of Computation 9.1 Circuit Models and Measures In this section we characterize types of logic circuits by their bases and the fan-in and fan- out of basis elements. We consider bases that are complete and incomplete and that have bounded and unbounded fan-in. We also consider circuits in which the fan-out is restricted and unrestricted. Each of these factors can affect the size and depth of a circuit. 9.1.1 Circuit Models The (general) logic circuit is the graph of a straight-line program in which the variables have value 0 or 1 and the operations are Boolean functions g : Bp →B, p ≥ 1. (Boolean functions have one binary value. Logic circuits are defined in Section 1.2 and discussed at length in Chapter 2.) The vertices in a logic circuit are labeled with Boolean operations and are called gates; the set of different gate types used in a circuit is called the basis (denoted Ω)forthe circuit. The fan-in of a basis is the maximal fan-in of any function in the basis. A circuit computes the binary function f : Bn →Bm, which is the mapping from the n circuit inputs to the m gate outputs designated as circuit outputs. The standard basis, denoted Ω0,istheset{AND, OR, NOT} in which AND and OR have fan-in 2. The full two-input basis, denoted B2, consists of all two-input Boolean functions. a b c The dyadic unate basis, denoted U2, consists of all Boolean functions of the form (x ∧ y ) for constants a, b, c in B.Herex1 = x and x0 = x. A basis Ω is complete if every binary function can be computed by a circuit over Ω.The bases Ω0, B2,andU2 are complete, as is the basis consisting of the NAND gate computing the function x NAND y = x ∧ y. (See Problem 2.5.) The bounded fan-out circuit model specifies a bound on the fan-out of a circuit. As we shall see, the fan-out-1 circuit plays a special role related to circuit depth. Each circuit of fan-out 1 corresponds to a formula in which the operators are the functions associated with vertices of the circuit. Figure 9.1 shows an example of a circuit of fan-out 1 over the standard basis and its associated formula. (See also Problem 9.9.) Although each input variable appears once in this example, Boolean functions generally require multiple instances of variables (have fan-out greater than 1). Formula size is studied at length in Section 9.4. To define the monotone circuits, we need an ordering of binary n-tuples. Two such tuples, x =(x1, x2, ..., xn) and y =(y1, y2, ..., yn), are in the relation x ≤ y if for all 1 ≤ i ≤ n, xi ≤ yi,where0≤ 0, 1 ≤ 1, and 0 ≤ 1, but 1 ≤ 0. (Thus, 001011 ≤ 101111, but 011011 ≤ 101111.) A monotone circuit is a circuit over the monotone basis Ωmon = {AND, OR} in which the fan-in is 2. There is a direct correspondence between monotone circuits and monotone functions. A monotone function is a function f : Bn →Bm that is either monotone increasing,thatis,forallx, y ∈Bn,ifx ≤ y,thenf(x) ≤ f(y),orismonotone decreasing,thatis,forallx, y ∈Bn,ifx ≤ y,thenf(x) ≥ f(y). Unless stated explicitly, a monotone function will be understood to be a monotone increasing function. A monotone Boolean function has the following expansion on the first variable, as the reader can show. (See Problem 9.10.) A similar expansion is possible on any variable. f(x1, x2, ..., xn)=f(0, x2, ..., xn) ∨ (x1 ∧ f(1, x2, ..., xn)) By applying this expansion to every variable in succession, we see that each monotone function can be realized by a circuit over the monotone basis. Furthermore, the monotone basis Ωmon c John E Savage 9.1 Circuit Models and Measures 393 y = ((((x7 ∨ x6) ∧ (x5 ∨ x4)) ∨ x3) ∧ (x2 ∧ x1)) x3 x2 x1 x7 x6 x5 x4 Figure 9.1 A circuit of fan-out 1 over a basis with fan-in 2 and a corresponding formula. The value y at the root is the AND of the value (((x7 ∨ x6) ∧ (x5 ∨ x4)) ∨ x3) of the left subtree with the value (x2 ∧ x1) of the right subtree. is complete for the monotone functions, that is, every monotone function can be computed by a circuit over the basis Ωmon. (See Problem 2.) In Section 9.6 we show that some monotone functions on n variables require monotone circuitswhosesizeisexponentialinn. In particular, some monotone functions requiring exponential-size monotone circuits can be realized by polynomial-size circuits over the standard basis Ω0. Thus, the absence of negation can result in a large increase in circuit size. The bounded-depth circuit is a circuit over the standard basis Ω0 where the fan-in of AND and OR gates is allowed to be unbounded, but the circuit depth is bounded. The conjunctive and disjunctive normal forms and the product-of-sums and sum-of-products normal forms realize arbitrary Boolean functions by circuits of depth 2 over Ω0.(SeeSection2.3.) In these normal forms negations are used only on the input variables. Note that any circuit over the standard basis can be converted to a circuit in which the NOT gates are applied only to the input variables. (See Problem 9.11.) 9.1.2 Complexity Measures We now define the measures of complexity studied in this chapter. The depth of a circuit is the number of gates of fan-in 2 or more on the longest path in the circuit. (Note that NOT gates do not affect the depth measure.) DEFINITION 9.1.1 The circuit size of a binary function f : Bn →Bm with respect to the basis Ω, denoted CΩ(f),isthesmallestnumberofgatesinanycircuitforf over the basis Ω.Thecircuit size with fan-out s, denoted Cs,Ω(f),isthecircuitsizeoff when the circuit fan-out is limited to at most s. 394 Chapter 9 Circuit Complexity Models of Computation n m The circuit depth of a binary function f : B →B with respect to the basis Ω, DΩ(f),is the depth of the smallest depth circuit for f over the basis Ω.Thecircuit depth with fan-out s, denoted Ds,Ω(f),isthecircuitdepthoff when the circuit fan-out is limited to at most s. n The formula size of a Boolean function f : B →Bwith respect to a basis Ω, LΩ(f),isthe minimal number of input vertices in any circuit of fan-out 1 for f over the basis Ω. It is important to note the distinction between formula and circuit size: in the former the number of input vertices is counted, whereas in the latter it is the number of gates. A relationship between the two is shown in Lemma 9.2.2. 9.2 Relationships Among Complexity Measures In this section we explore the effect on circuit complexity measures of a change in either the basis or the fan-out of a circuit. We also establish relationships between circuit depth and formula size.