Dagstuhl News January - December 2001

"Dagstuhl News" is a publication edited especially for the members of the Foundation "Informatikzentrum Schloss Dagstuhl" to thank them for their support. The News give a summary of the scientific work being done in Dagstuhl. Each Dagstuhl Seminar is presented by a small abstract describing the contents and scientific highlights of the seminar as well as the perspectives or challenges of the research topic

Adaptivity Helps for Testing Juntas

We give a new lower bound on the query complexity of any non-adaptive algorithm for testing whether an unknown Boolean function is a k-junta versus epsilon-far from every k-junta. Our lower bound is that any non-adaptive algorithm must make Omega(( k * log*(k)) / ( epsilon^c * log(log(k)/epsilon^c))) queries for this testing problem, where c is any absolute constant <1. For suitable values of epsilon this is asymptotically larger than the O(k * log(k) + k/epsilon) query complexity of the best known adaptive algorithm [Blais,STOC\u2709] for testing juntas, and thus the new lower bound shows that adaptive algorithms are more powerful than non-adaptive algorithms for the junta testing problem

Partially Symmetric Functions are Efficiently Isomorphism-Testable

Given a function f: {0,1}^n \to {0,1}, the f-isomorphism testing problem requires a randomized algorithm to distinguish functions that are identical to f up to relabeling of the input variables from functions that are far from being so. An important open question in property testing is to determine for which functions f we can test f-isomorphism with a constant number of queries. Despite much recent attention to this question, essentially only two classes of functions were known to be efficiently isomorphism testable: symmetric functions and juntas. We unify and extend these results by showing that all partially symmetric functions---functions invariant to the reordering of all but a constant number of their variables---are efficiently isomorphism-testable. This class of functions, first introduced by Shannon, includes symmetric functions, juntas, and many other functions as well. We conjecture that these functions are essentially the only functions efficiently isomorphism-testable. To prove our main result, we also show that partial symmetry is efficiently testable. In turn, to prove this result we had to revisit the junta testing problem. We provide a new proof of correctness of the nearly-optimal junta tester. Our new proof replaces the Fourier machinery of the original proof with a purely combinatorial argument that exploits the connection between sets of variables with low influence and intersecting families. Another important ingredient in our proofs is a new notion of symmetric influence. We use this measure of influence to prove that partial symmetry is efficiently testable and also to construct an efficient sample extractor for partially symmetric functions. We then combine the sample extractor with the testing-by-implicit-learning approach to complete the proof that partially symmetric functions are efficiently isomorphism-testable.Comment: 22 page

Settling the Query Complexity of Non-Adaptive Junta Testing

We prove that any non-adaptive algorithm that tests whether an unknown Boolean function f is a k-junta or epsilon-far from every k-junta must make ~Omega(k^{3/2}/ epsilon) many queries for a wide range of parameters k and epsilon. Our result dramatically improves previous lower bounds from [BGSMdW13,STW15], and is essentially optimal given Blais\u27s non-adaptive junta tester from [Blais08], which makes ~O(k^{3/2})/epsilon queries. Combined with the adaptive tester of [Blais09] which makes O(k log k + k / epsilon) queries, our result shows that adaptivity enables polynomial savings in query complexity for junta testing

Dagstuhl News January - December 1999

"Dagstuhl News" is a publication edited especially for the members of the Foundation "Informatikzentrum Schloss Dagstuhl" to thank them for their support. The News give a summary of the scientific work being done in Dagstuhl. Each Dagstuhl Seminar is presented by a small abstract describing the contents and scientific highlights of the seminar as well as the perspectives or challenges of the research topic

Partially Symmetric Functions Are Efficiently Isomorphism Testable

Given a Boolean function f, the f-isomorphism testing problem requires a randomized algorithm to distinguish functions that are identical to f up to relabeling of the input variables from functions that are far from being so. An important open question in property testing is to determine for which functions f we can test f-isomorphism with a constant number of queries. Despite much recent attention to this question, essentially only two classes of functions were known to be efficiently isomorphism testable: symmetric functions and juntas. We unify and extend these results by showing that all partially symmetric functions---functions invariant to the reordering of all but a constant number of their variables---are efficiently isomorphism testable. This class of functions, first introduced by Shannon, includes symmetric functions, juntas, and many other functions as well. We conjecture that these functions are essentially the only functions efficiently isomorphism-testable. To prove our main result, we also show that partial symmetry is efficiently testable. In turn, to prove this result we had to revisit the junta testing problem. We provide a new proof of correctness of the nearly optimal junta tester. Our new proof replaces the Fourier machinery of the original proof with a purely combinatorial argument that exploits the connection between sets of variables with low influence and intersecting families. Another important ingredient in our proofs is a new notion of symmetric influence. We use this measure of influence to prove that partial symmetry is efficiently testable and also to construct an efficient sample extractor for partially symmetric functions. We then combine the sample extractor with the testing-by-implicit-learning approach to complete the proof that partially symmetric functions are efficiently isomorphism testable.Simons Foundation (Postdoctoral Fellowship

Inferring Symbolic Automata

We study the learnability of symbolic finite state automata (SFA), a model shown useful in many applications in software verification. The state-of-the-art literature on this topic follows the query learning paradigm, and so far all obtained results are positive. We provide a necessary condition for efficient learnability of SFAs in this paradigm, from which we obtain the first negative result. The main focus of our work lies in the learnability of SFAs under the paradigm of identification in the limit using polynomial time and data, and its strengthening efficient identifiability, which are concerned with the existence of a systematic set of characteristic samples from which a learner can correctly infer the target language. We provide a necessary condition for identification of SFAs in the limit using polynomial time and data, and a sufficient condition for efficient learnability of SFAs. From these conditions we derive a positive and a negative result. The performance of a learning algorithm is typically bounded as a function of the size of the representation of the target language. Since SFAs, in general, do not have a canonical form, and there are trade-offs between the complexity of the predicates on the transitions and the number of transitions, we start by defining size measures for SFAs. We revisit the complexity of procedures on SFAs and analyze them according to these measures, paying attention to the special forms of SFAs: normalized SFAs and neat SFAs, as well as to SFAs over a monotonic effective Boolean algebra. This is an extended version of the paper with the same title published in CSL'22

Distribution testing lower bounds via reductions from communication complexity

We present a new methodology for proving distribution testing lower bounds, establishing a connection between distribution testing and the simultaneous message passing (SMP) communication model. Extending the framework of Blais, Brody, and Matulef (Computational Complexity, 2012), we show a simple way to reduce (private-coin) SMP problems to distribution testing problems. This method allows us to prove new distribution testing lower bounds, as well as to provide simple proofs of known lower bounds. Our main result is concerned with testing identity to a specific distribution p, given as a parameter. In a recent and influential work, Valiant and Valiant (FOCS, 2014) showed that the sample complexity of the aforementioned problem is closely related to the 2/3-quasinorm of p. We obtain alternative bounds on the complexity of this problem in terms of an arguably more intuitive measure and using simpler proofs. More specifically, we prove that the sample complexity is essentially determined by a fundamental operator in the theory of interpolation of Banach spaces, known as Peetre's K-functional. We show that this quantity is closely related to the size of the effective support of p (loosely speaking, the number of supported elements that constitute the vast majority of the mass of p). This result, in turn, stems from an unexpected connection to functional analysis and refined concentration of measure inequalities, which arise naturally in our reduction