MATH 262A LECTURE 1: AN INTRODUCTION TO BOOLEAN CIRCUITS AND FORMULAE

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1 MATH 262A LECTURE 1: AN INTRODUCTION TO BOOLEAN CIRCUITS AND FORMULAE SCRIBE: CHRISTIAN WOODS 1. Admiistrivia MATH 262A - Circuit Compleit WF 3:30-5:00, APM B412 Istructor: Dr. Sam Buss, sbuss@math.ucsd.edu Course Website: Temporar Office Hour: Moda, September 30, 2:30-3:30 No course meetig o Frida, November 8 2. Basic Structures ad Eamples Boolea fuctios computed b Boolea circuits or Boolea formulae take o biar values of True/False, 0/1, etc. Gates logical coectives, like Date: September 27,

2 2 SCRIBE: CHRISTIAN WOODS As a eample, let us write the Boolea fuctio usig ol the coectives,, ad : = ( ) ( ). We ca write this as a circuit i the followig wa: Figure 1. Circuit (1). Notice that both the circuit ad the writte formula use the same umber of logic gates. This eed ot alwas be the case, because a circuit (ulike a formula) ma reuse a subformula. A eample where this ca happe is the fuctio P arit 3 (,, z) := z. We ma use the followig circuit to compute the fuctio: z z Figure 2. Circuit (2). This depictio is a circuit, but ot a formula. We observe the followig Fact. P arit ( 1, 2,..., ) := has circuits of size O().

3 MATH 262A LECTURE 1: AN INTRODUCTION TO BOOLEAN CIRCUITS AND FORMULAE 3 Proof. B a simple iductio argumet, we eed ol add o a cop of circuit (1) to the output of the circuit for P arit 1 ( 1, 2,..., 1 ) ad to achieve the desired circuit. Our aïve formula for P arit ( 1,..., ) has size O(2 ). We ca actuall improve this to get a formula of size O( 2 ) b oticig that P arit ( 1,..., ) = P arit 2 ( 1,..., 2 ) P arit 2 ( 2 +1,..., ). 3. Itroductor Defiitios ad Theorems Defiitio. A (famil of) Boolea fuctio(s) is a mappig f : {0, 1} {0, 1}. A -ar Boolea fuctio is a mappig f : {0, 1} {0, 1}. Notice that we ca decompose a Boolea fuctio ito a class {f 0, f 1, f 2,...} of -ar Boolea fuctios, where f k is k-ar. Defiito. A basis Ω is a set of Boolea fuctios. Defiitio. A Boolea circuit over a basis Ω is a directed acclic graph (DAG) such that for each verte v i the graph, either (i) v has idegree 0 ad is labeled with a iput i for some i N or (ii) v has idegree > 0 ad is labeled with g Ω, where g is a -ar Boolea fuctio ad with a orderig o the icomig edges to v. Usuall a Boolea circuit has a sigle ouput, represeted with a gate with outdegree 0. Note: If a circuit has iputs amog 1,..., it computes a -ar fuctio f : {0, 1} {0, 1} i the obvious wa. Defiitio. A gate is a o-iput verte i a Boolea circuit. Defiitio. A Boolea circuit is a Boolea formula if each gate has outdegree less tha or equal to 1. We ma thik of a Boolea formula as a directed tree, ecept perhaps at the leaves (sice a iput i ca occur at multiple leaves). Defiitio. The size of a Boolea circuit is the umber of gates i the circuit. To discuss Boolea circuits, it is ecessar to choose a basis of Boolea fuctios. Some commo choices that are studied are Ω = {,, }, the DeMorga basis

4 4 SCRIBE: CHRISTIAN WOODS Ω = B 2, the set of all fuctios f : {0, 1} 2 {0, 1} Ω = {, ubouded fa-i, ubouded fa-i }, where a or or ad gate ma take i a fiite umber of iputs. A useful observatio is Theorem. There are 2 2 -ar Boolea fuctios (i.e., B = 2 2 ). Proof. The size of the domai of a -ar Boolea fuctio is 2. For each iput we must specif oe of 2 outputs, so there are 2 2 possible fuctios i total. Whe we do ot wish to cout gates i the size of a give circuit over a basis cotaiig, we will call the resultig cout the -size of the circuit. Theorem. Ever -ar Boolea fuctio has a circuit (ad formula) of -size at most 2 over the basis Ω = {,, }. Proof. Let f be a -ar Boolea fuctio. The truth table for f has 2 rows. For each row of the truth table let us defie a vector a i {0, 1} where a i = 1 if ad ol if i = 1 i that row. Defie the smbol a i i = { i if a i = 1 i if a i = 0. The f() = as.t.f( a)=1 ( a 1 1 a a ). Let us cout how ma gates ad gates it takes to create the circuit (which is actuall a formula) correspodig to this smbolic represetatio of f. We see that there are at most 2 s required, because this is the umber of rows of the truth table. For each of these 2 rows, we eed 1 s to coect the idividual iputs. So at most 2 ( 1) gates are required. I total, the -size of such a formula is ( 1) = 2. Note: This is ot a optimal size. I fact, we ca easil reduce the upper boud b a factor of 2 with a simple observatio: if there are at most half of the rows of f with ouput 1, the we eed ol use 2 1 rows (istead of 2 ) i the calculatios above. Otherwise, there are at most half false rows. I this case, we create the formula for the egatio of f i the stle of the proof. The we add a fial gate, which does ot cotribute to the -size.

5 MATH 262A LECTURE 1: AN INTRODUCTION TO BOOLEAN CIRCUITS AND FORMULAE 5 Defiitio. The C Ω -size of a Boolea fuctio f is the size of the smallest circuit over basis Ω that computes f. The L Ω -size of a Boolea fuctio f is the size of the smallest formula over basis Ω that computes f. Theorem (Shao, 1949). Most -ar Boolea fuctios have C B2 -size at least 2 for sufficietl large. Proof. Let F (s, ) be the umber of -ar Boolea fuctios that ca be computed b a B 2 circuit of size s. To completel describe a circuit of size s we require a specificatio of s gates, each of which ca be oe of B 2 = 16 tpes, each havig two iputs. These iputs could come from the s 1 gates that have positive outdegree, or from the origial iputs to the fuctio f. Thus there are at most s + 1 total choices for a iput to a gate. Therefore there are at most 16(s + 1) 2 descriptios for a sigle gate. We also ote that the order of the gates does ot matter, so log as the iputs are well-defied. So the umber of differet possible circuits of size s is bouded b ( 16(s + 1) 2 ) s F (s, ) s! = 16s (s + 1) 2s s! 16s (s + 1) 2s s s /e s for some costat c. = (16e)s s 2s ( s (64e) s s s = c s s s Now let s = 2. The the above gives ( ) ( ) 2 2 F, c 2 2 ( c ) 2 = (2 ) 2 ( c = )2 ( ) 2 2 s s = o ( 2 2 ). ) 2s

6 6 SCRIBE: CHRISTIAN WOODS But 2 2 is the umber of -ar Boolea fuctios. This proves Shao s Theorem. A similar theorem for formulae was proved earlier b Riorda ad Shao, which we will discuss et lecture: Theorem (Riorda-Shao, 1942). Let be sufficietl large ad fi δ < 1. The δ2 most -ar Boolea fuctios have L B2 -size at least log. Eercise: Prove that the Haltig Problem does ot have polomial-size circuits. (Here the Haltig Problem is defied as the set of Turig machies that halt o blak iput, usig a efficiet ecodig of Turig machies b Gödel umbers.) Future Topics: All -ar fuctios have {,, }-circuits of -size 2 (1 + o(1)) All -ar fuctios have {,, }-formulae of -size 2+1 (1 + o(1)) log