Proclaiming Dictators and Juntas or Testing Boolean Formulae Michal Parnas Dana Ron Alex Samorodnitsky

Testing Properties of Functions or: Testing membership in Function Classes f : {0,1} n -> {0,1} – Tested function; F – Class of functions (e.g. class of parity functions); dist(f,F) - min gF (Pr[f(x)g(x)]) f xf(x)f(x) If fF should accept with probability  2/3 If dist(f,F) >  should reject with probability  2/3

In this work: We consider “The most basic” function classes: Singletons: Monomials: DNF:

Motivation Use testing as preliminary step to learning. That is, to decide what hypothesis class to use. Gain new understanding about function class F through Property Testing perspective.

Previous Work on Property Testing Initially defined by Rubinfeld and Sudan in the context of Program Testing. Tested algebraic properties of functions: low-degree polynomials. Other work on testing algebraic properties: [BLR,R,EKKRV...]. Non-algebraic properties: Monotonicity [GGLRS,DGLRSS,B,FN]. Properties of functions: Properties of other objects: Main focus: Graph properties: [GGR,GR,AK,AFKS,BR,PR,CS...] Growing body of work deals with properties of strings [AKNS,N,PRR], sets of points [PR], geometric objects [CSZ], distributions [BFRW], and more.

Our Results Test whether f is a singleton using queries. Test whether f is a monomial using queries. Test whether f is a monotone DNF with at most t terms using queries. Common theme: no dependence in query complexity on size of input, n, and polynomial dependence on distance parameter, .

Learning Boolean Formulae Basic observation: (proper) learning implies testing. Can learn singletons and monomials under uniform distribution using queries [BEHW]. Can properly learn monotone DNF with t terms and r literals using queries [A+BJT]. Main difference in our work: no dependence on n (though worse dependence on  and t ), and different algorithmic approach.

Testing (Monotone) Singletons Singletons satisfy: (1) (2) Natural test: check, by sampling, that conditions hold (approximately). Can analyze natural test for case that distance between function and class of singletons is not too big (bounded from 1/2).

Testing Singletons II - Parity Testing Observation: Singletons are a special case of parity functions (i.e., functions of the form.) Claim: Let. If then Modified algorithm: (1) Test whether f is a parity function (with dist. par.  ) using algorithm of [BLR]. (2) Uniformly select constant number of pairs x,y and check whether any is a violating pair (i.e.: ).

Testing Singletons III - Self Correcting Use Self-Corrector of [BLR] to “fix” f into parity function (g), and then test violations on self-corrected version. This “almost works”: If f is singleton - always accepted. If f is  -far from parity - rejected w.h.p. But if f is  -close to parity function g, then cannot simply apply claim to argue that many violating pairs w.r.t. f. If we could only test violations w.r.t. g instead of f...

Testing Singletons IIII - The Algorithm Final Algorithm for Testing Singletons: (1) Test whether f is a parity function with dist. par.  using algorithm of [BLR]. (2) Uniformly select constant number of pairs x,y. Verify that Self-Cor(f,x)  Self-Cor(f,y) = Self-Cor(f,xy). (3) Verify that Self-Cor( ) = 1.

Testing (Monotone) Monomials Monomials satisfy: (1) for some int k (2) Here too use some additional structure: In addition to checking conditions (1) and (2), algorithm tests whether F 1 is an affine subspace. (How do we test affinity? Why does this help us? ) If f is a monomial, then F 1 ={x: f(x)=1} is an affine subspace (i.e., exist linear subspace V and y{0,1} n s.t. F 1 =Vy).

Testing Monomials II - Affinity Testing Fact: H is an affine subspace i.f.f. The Affinity Test: Repeat O(1/  2 ) times: Uniformly select x,yF 1, z {0,1} n, verify that ( implies )

Testing Monomials III - Affinity Lemmas Lemma2 (implications of test): If f is close to function g s.t. G 1 =[g(x)=1] is an affine subspace but g is not a monomial then is large. Lemma1 (correctness of affinity test): If f is monomial then affinity test always passes. If f is far from every function g s.t. G 1 =[g(x)=1] is an affine subspace, then test rejects w.h.p. What if f is far from every monomial (so should be rejected) but close to a function g s.t. G 1 =[g(x)=1] is an affine subspace (so may pass affinity test)?

Testing Monomials IIII - The Algorithm Algorithm for Testing Monomials: (1) Check that Pr[f(x)=1] is close to 2 -k for some integer k. (2) Test whether F 1 ={x: f(x)=1} is an affine subspace: Repeat O(1/ 2 ) times: Uniformly select x,yF 1, z {0,1} n, verify that (3) Repeat O(1/  ) times: Uniformly select xF 1, y{0,1} n check whether

Testing Monotone DNF If f is a t-term DNF, then f is the disjunction (“or”) of t monomials f 1,…,f t. High Level Description of Algorithm: Works in (at most t) iterations. In i’th iteration finds string x i that can be used to (implicitly) define function g i. Tests whether each g i is a monotone monomial. Basic idea: Reduce testing DNF to testing monomials.

Testing Monotone DNF II Can Show:  If f is monotone t-term DNF then, w.h.p. all g i ’s are monotone monomials.  If f is far from monotone t-term DNF then, at least one g i is far from monotone monomials (or don’t finish after t iterations). What are x i ’s and how are the g i ’s defined? If f is DNF then x i ’s are “single-term representatives”. That is, each satisfies a single term (monomial) f i in f. Define g i (y)=1 i.f.f. f(x)=1 and f(x i y)=1

Testing Monotone DNF III - Finding Single-Term Rep’s Basic Idea: Suppose f is DNF. For x in F 1 let S(x) denote subset of (indices of) terms satisfied by x. (Note that single-term-rep equiv to S(x).) Consider x,y in F 1. Then S(xy)=S(x)S(y). Furthermore, for fixed x, w.h.p. over random y in F 1, |S(xy)|(3/4)|S(x)|. On the other hand, can bound probability that |S(xy)|=0. Hence, if we start from some x in F 1 (s.t. S(x) contains term that is not yet represented) can obtain new single-term-rep (i.e., x’ s.t. S(x’)=1) w.h.p. in O(log t) steps.

Further Research Are parity/affinity tests necessary for singletons/monomials? Can (cubic) dependence on 1/  in monomial test be improved? Is polynomial (or even linear) dependence on t (num of terms in DNF) really necessary? Can algorithm for testing monotone DNF be extended to general DNF?