On Testing Computability by small Width OBDDs Oded Goldreich Weizmann Institute of Science

Gothic cathedral ? Property Testing: an illustration

Property Testing: informal definition A relaxation of a decision problem: For a fixed property P and any object O, determine whether O has property P or is far from having property P ( i.e., O is far from any other object having P ). Focus: sub-linear time algorithms – performing the task by inspecting the object at few locations. ?? ? ? ? Objects viewed as functions. Inspecting = querying the function/oracle.

Property Testing applied to complexity classes Objects are viewed as functions; Properties are sets or classes of functions; So what’s more natural than considering complexity classes ( I mean traditional complexity classes ). Study of width 2 OBDDs [Ron+Tsur, 2009]: An O(log n) tester for the class of functions from n bits to 1 bit that can be computed by a width 2 OBDD. Learning requests  (n) queries. We ask: Does this extend to width 3? Or 4? *) Trad. Complex. Classes ≠ Juntas, low degree polynomials, or constant-term DNFs. **) Don’t confuse this with the property of being accepted by a fixed computing device.

Ordered Binary Decision Diagrams A non-uniform model of computation; A restricted class of read-once Branching Programs: Variables read in a fixed (canonical) order. THM: An  (n 1/2 ) lower bound for testing the class of functions (from n bits to 1 bit) that can be computed by a width 4 OBDD. Learning possible in  (n) queries. CONJ: An  (n) lower bound. X i X i+1 X i+2 X i+3 X i+4 =1 =0 +X i set(X i+1 ) ig(X i+2 )  X i+3 set(1) Recall that testing the class of width 2 OBDDs can be done with O(log n) queries

Hardness of testing of natural subclasses (of width 2 OBDDs, e.g., of linear functions) Property testing is not monotone w.r.t subset/superset. E.g., testing may become harder when restricting to a subclass (trivial examples). We are interested in natural examples. Linear functions over GF(p) are computable by width p OBDDs. THM: An  (n 1/2 ) lower bound for testing for the class of linear functions (from n bits to 1 bit) that depend on n/2 variables. CONJ: An  (n) lower bound. Ditto for linear functions over GF(3) with 0-1 coefficients.

On the techniques A triviality we all know: The values of a uniformly selected random function over GF(p) at t (adaptively selected inputs) are uniformly distributed over GF(p) t. LEM: For p=2, the deviation from uniform is at most O(t 2 /n), and for non-adaptive queries it is at most O(t/n). CONJ: The stronger bound holds also for adaptive queries. Question: What if the function is selected uniformly among all linear functions that depend on exactly (n±1)/2 variables? For random linear functions over GF(3) with 0-1 coefficients versus such with a single exceptional coefficient, we have a O(t 2 /n) bound for general queries, and conjecture O(t/n).

Open problems Improve all  (n 1/2 ) lower bounds to  (n) : This includes the lower bounds on testing the following classes The class of functions implemented by width 4 OBDDs. The class of GF(2)-linear function that depend on n/2 vars. The class of GF(3)-linear functions with 0-1 coefficients. Seems to call for better analysis of the output distribution of linear functions selected from various subclasses ( as above ). What about the class of functions that are implemented by width 3 OBDDs?

Not the End The slides of this talk are available at The paper itself is available at Other papers are available at

Testing Graph Blow-Up (with Lidor Avigad) A project proposed in “Algorithmic Aspects of Testing Grpah Properties in the Dense Graph Model” (w. Dana Ron): Characterize the class of graph properties that are testable (in the dense graph model) within a number of queries that is inversely (linearly) related to the proximity parameter (i.e., complexity O(1  ). Ditto for non-adaptive testing. Blow-Up Properties: For a fixed graph H, the set of graphs G that are obtained as a blow-up of H; that is, G consists of “clouds” that are connected as in H. [GR] a non-adaptive O(1  query tester for H  c-clique. New: Ditto for any H.

Proximity Oblivious Testing and the Role of Invariances (with Tali Kaufman) Proximity Oblivious Testing: Tester makes a O(1) queries, independent of the proximity parameter (not given to it), and reject with probability related to the actual distance. Invariant local conditions: The property is characterized by a set of local conditions that is generated by actions of a group defined over the function’s domain. Conjecture: A property has a POT if and only if it has an invariant local characterization. The conjecture holds in two graph testing models, but each direction fails in some other models.

The End The slides of this talk are available at The paper itself is available at Other papers are available at