The Complexity of Finite Functions Ravi B Boppana Department of Computer Science New York University New York NY Michael Sipser Mathematics Department Mas sachusetts Institute of Technology Cambridge MA August Abstract Thi s pap er surveys the recent re sults on the complexity of Bo olean functions in terms of Bo olean circuits formulas and branching programs The pr imary aim i s to give acce s s ible pro ofs of the more dicult theorems proving lower b ounds on the complexity of sp ecic functions in re str icted computational mo dels Thes e include b ounded depth circuits monotone circuits and b ounded width branching programs Application to other are as are descr ib e d including Turing machine complexity relativization and rst order denability Keywords Alternation Approximation Bo olean Circuit Bo olean Formula Bo olean Function Branching Program Clique Function Connectivity Function Depth Fanin Fanout First Order Denability Gate Log Time Hierarchy Ma jor ity Function Monotone Circuit Monotone Function Nonconstructive Pro of Non determinism Oracle Parity Function Polynomial Time Hierarchy Relativization Re str iction Size Slice Function Space Sunower Symmetric Function Thre sh old Function Time Turing Machine Supported by an NSF Mathematical Science s Postdoctoral Fellowship and NSF Grant CCR Supported by NSF Grant DCR and Air Force Contract AFOSR Introduction The clas s ication of problems according to computational diculty compr i s e s two subdi sciplines of dierent character One the theory of algor ithms gives up p er b ounds on the amount of computational re source needed to solve particular problems Thi s endeavor has enjoyed much succes s in recent years with a large number of strong re sults and f ruitful connections with other branches of mathe matics and engineering The other calle d the theory of computational complexity i s an attempt to show that certain problems cannot b e solved eciently by e s tablishing lower b ounds on their inherent computational diculty Thi s has not b een as succes sful Except in a few sp ecial circumstances we have b een unable to demonstrate that particular problems are computationally dicult even though there are many which appe ar to b e so The mo st fundamental que stions such as the f amous P versus NP que stion which s imply asks whether it i s harder to nd a pro of than to check one remain f ar b eyond our pre s ent abilities Thi s chapter sur veys the pre s ent state of understanding on this and related que stions in complexity theory The diculty in proving that problems have high complexity s eems to lie in the nature of the adversary the algor ithm Fast algor ithms may work in a counterintuitive f ashion us ing deep devious and endishly clever ideas How do e s one prove that there i s no clever way to quickly solve a problem Thi s i s the i s sue conf ronting the complexity theori st One way to make some progre s s on this i s to limit the capabilities of the computational mo del thereby limiting the clas s of p otential algor ithms In this way it has b een p o s s ible to achieve some interesting re sults Perhaps thes e may le ad the way to lower b ounds for more p owerful computational mo dels There i s by now an extensive literature on b ounds for various mo dels of com putation We have excluded some of thes e f rom this survey us ing the following cr iteria First we will concentrate on recent mo stly combinatorial b ounds for Bo olean circuits formulas and branching programs We will not include the older work on lower b ounds via the diagonalization method Though imp ortant e arly progre s s was made in that way the relativization re sults of Baker Gill and Solo vay show that such methods f rom recurs ive function theory are inadequate for the remaining interesting que stions Second we will mo stly fo cus on b ounds that dierentiate p olynomial f rom nonp olynomial rather than lower level b ounds Thi s reects our feeling that this type of re sult i s clo s er to the interesting unsolved que stions Finally in s everal cas e s we ref rain f rom giving the tightest re sult known to keep the exp o s ition as cle ar as p o s s ible Br ief Hi story Circuit complexity theory dates f rom Shannons s eminal pap er There he propo s e d the s ize of the smallest circuit computing a function as a measure of its complexity Hi s motivation was s imply to minimize the hardware neces sary for computation He proved an upper b ound on the complexity of all n input functions and us e d a counting argument to show that for mo st functions this b ound i s not too f ar o The s saw the introduction of the algor ithm as a way of measur ing the complexity of functions Edmonds gave a p olynomial time algor ithm for the matching problem and fore saw the i s sue of p olynomial versus exp onential com plexity Hartmanis and Stear ns formalized this measure as time on a Turing machine Savage e stablished a clo s e relationship b etween the time require d to compute a function on a Turing machine and its circuit complexity At this p oint the imp ortance of proving go o d lower b ounds on circuit s ize was apparent but it was also b ecoming cle ar that this was going to b e dicult to accomplish See Harp er and Savage for a di scus s ion of this By the end of the s e s s entially the only re sults known were the linear lower b ound of Paul and the nearly quadratic lower b ound of Neciporuk for the sp ecial cas e of formula s ize A new direction in the s brought on a burst of activity By placing suciently strong re str ictions on the clas s of circuits it b ecame p o s s ible to prove strong lower b ounds The rst re sults of this kind were independently obtained by Furst Saxe and Sip s er and Ajtai for the b ounded depth circuit mo del Razb orov a and subs equently Andreev gave strong lower b ounds for the monotone circuit mo del Numerous pap ers strengthening thes e re sults and giving others in the same vein have s ince appe are d Our chapter emphasize s this latter work giving only a sketchy overview of the pro cee ding p er io d For a more comprehensive di scus s ion of the e arlier work s ee the survey pap er of Paterson and the b o ok of Savage Much of the recent work also i s covered in the b o oks of Wegener and Dunne Contents Introduction General Circuits Bo olean Circuits and Turing Machines Nonexplicit Lower Bounds Explicit Lower Bounds Bounde d Depth Circuits Denitions Re str ictions Hastad Switching Lemma Lower Bound for the Parity Function Depth Hierarchy Monotone Bounded Depth Circuits Probabilistic Bounded Depth Circuits Razb orovSmolensky Lower Bound for Circuits with MOD Gates p Applications Relativization of the Polynomial Time Hierarchy Log time Hierarchy First Order Denability on Finite Structures Monotone Circuits Background Razb orov Lower Bound for Monotone Circuit Size Lower Bound for the Clique Function Polynomial Lower Bounds Formulas Karchmer and Wigderson Lower Bound for Monotone Formula Size Lower Bound for the Connectivity Function Nonmonotone Formulas Symmetric Functions Branching Programs Relationship with Space Complexity Bounds on Size Conclus ion General Circuits In an e arly pap er Shannon cons idered the s ize of Bo olean circuits as a measure of computational diculty Circuits are an attractive mo del for proving lower b ounds for s everal re asons They are clo s ely related in computational p ower to Turing machines so that a go o d lower b ound on circuit s ize directly gives a lower b ound on time complexity Among computational mo dels the circuit mo del has an e sp ecially s imple denition and so may b e more amenable to combinatorial analysi s Even so we are currently able to prove only very weak lower b ounds on circuit s ize A Boolean circuit i s a directed acyclic graph The no des of indegree are calle d inputs and are labele d with a variable x or with a constant or The i no des of indegree k are calle d gates and are labele d with a Bo olean function on k inputs We refer to the indegree of a no de as its fanin and its outdegree as its fanout Unle s s otherwis e sp ecie d we re str ict to the Bo olean functions AND OR and NOT One of the no des i s des ignated the output no de The size i s the number of gates and the depth i s the maximum di stance f rom an input to the output A Boolean formula i s a sp ecial kind of circuit whos e underlying graph i s a tree A Bo olean circuit repre s ents a Bo olean function in a natural way Let N n denote the natural numbers f g the s et of binary str ings of length n and f g n the s et of all nite binary str ings Let f f g f g Then C f i s the s ize of the smallest circuit repre s enting f Let g f g f g and h N N Say g has circuit complexity h if for all n C g hn where g i s g re str icted to n n n f g A language i s a subs et of f g The circuit complexity of a language i s that of its characteri stic function Bo olean Circuits and Turing Machines In this subs ection we e stablish a relationship b etween the circuit complexity of a problem and the amount of time that i s require d to solve it First we must s elect a mo del of computation on which to measure time The Turing machine Tm mo del was propo s e d by Alan Turing as a means of formalizing the notion of eective pro ce dure Thi s intuitively appe aling mo del s erves as a convenient foundation for many re sults in complexity theory The choice i s arbitrary among the many p olynomially equivalent mo dels The complexity of Turing machine computations was rst cons idered by Hartmanis