Two Lower Bounds for Circuits over the Basis (&, V, -)

A @choral upprox~laat~on tec]%n~que £o gel 1o~er bounds /or tJ%e comp~ex~tt; o/ comb~ncxt~onaL c~rcu~ts ouez" a;% urb~tr=r~ u~gebrus o/ operations ~s presented. The technique @enera1~=es recent meZhods )cot monotone c~rcu~s end y~elds some ne%~ res~Its, rh~s report contu~ns un expCf)C[o@=nDD toy;or bound for the complexity of rea1~zuZ~on o~ non-~%onotone Boo~eun /unctions by c~rcu~ts o~)e~" ~l%e b=s~s C~,V,-%5 co~apu~n@ su/)C~c~ent1~ laan~ pr~ae ~lap~cc~n~s , und o/ thPee-~=~ued /unctions b~; c~l-cu~ts o~;er solne ~ncomp~efe ~%ree-~cz[ued extensions o/ C&,V,"~. INTRODUCTION The general idea of approximation technique in the theory of lower bounds for Boolean circuits is to approximate the circuits by more restricted ones. Various refinements of such an approach have already been used in a great many of lower bounds proofs. At present we have three main refinements. These are : probub~L~st~c approximations, by Furst,Saxe and Sipser [5], Ajtal [I], Ha~tad [T], Yao [17], H~jnal et al. [6] , etc. /unct£onal approxirnut~ons, by Andreev [3,4], Razborov [13-15], Alon and Boppana [~], P~terson [le], Smolensky [18]. Ugol'nikov Jig], etc. ; tOpO~O@~CaL approx~mcz~ons, [8-10]. The aim of this report is to develope the functional approximation technique in order to obtain lower bounds for circuits over an arbitrary algebras of operations. The technique generalizes the methods of [~-4,1R-1B] and yields some new results. The first result concerns Boolean circuits over the basis (~,V,-D with ~ -gates on the top of circuit. Any such circuit S computes some Boolean function fS and also some disjunctive normal form CDNF for short3 D E of fS Csee Section 3 for detailsg. A circuit S is called to be a 6-c~rcu~t CO S 6 S 13 iff [ DE ~ ImpCfs 3 [ > ~ impCfs 3 [6 _ I . where ImpCf9 denotes the set of all prime implicants of f of minimal length ; S is ~-c~rcu~£ if D S = ImpEfs3 For 6 E [0,1] 0 (W) and a Boolean function f , let C6Cf3 denote the minimum