Partial tests, universal tests and decomposability

For a property P and a sub-property P', we say that P is P'-partially testable with q queries} if there exists an algorithm that distinguishes, with high probability, inputs in P' from inputs ε-far from P, using q queries. Some natural properties require many queries to test, but can be partitioned into a small number of subsets for which they are partially testable with very few queries, sometimes even a number independent of the input size. For properties over {0,1}, the notion of being thus partitionable ties in closely with Merlin-Arthur proofs of Proximity (MAPs) as defined independently in [14] a partition into r partially-testable properties is the same as a Merlin-Arthur system where the proof consists of the identity of one of the r partially-testable properties, giving a 2-way translation to an O(log r) size proof. Our main result is that for some low complexity properties a partition as above cannot exist, and moreover that for each of our properties there does not exist even a single sub-property featuring both a large size and a query-efficient partial test, in particular improving the lower bound set in [14]. For this we use neither the traditional Yao-type arguments nor the more recent communication complexity method, but open up a new approach for proving lower bounds. First, we use entropy analysis, which allows us to apply our arguments directly to 2-sided tests, thus avoiding the cost of the conversion in [14] from 2-sided to 1-sided tests. Broadly speaking we use "distinguishing instances" of a supposed test to show that a uniformly random choice of a member of the sub-property has "low entropy areas", ultimately leading to it having a low total entropy and hence having a small base set. Additionally, to have our arguments apply to adaptive tests, we use a mechanism of "rearranging" the input bits (through a decision tree that adaptively reads the entire input) to expose the low entropy that would otherwise not be apparent. We also explore the possibility of a connection in the other direction, namely whether the existence of a good partition (or MAP) can lead to a relatively query-efficient standard property test. We provide some preliminary results concerning this question, including a simple lower bound on the possible trade-off. Our second major result is a positive trade-off result for the restricted framework of 1-sided proximity oblivious tests. This is achieved through the construction of a "universal tester" that works the same for all properties admitting the restricted test. Our tester is very related to the notion of sample-based testing (for a non-constant number of queries) as defined by Goldreich and Ron in [13]. In particular it partially resolves an open problem raised by [13]

Classical and quantum sublinear algorithms

This thesis investigates the capabilities of classical and quantum sublinear algorithms through the lens of complexity theory. The formal classification of problems between “tractable” (by constructing efficient algorithms that solve them) and “intractable” (by proving no efficient algorithm can) is among the most fruitful lines of work in theoretical computer science, which includes, amongst an abundance of fundamental results and open problems, the notorious P vs. NP question. This particular incarnation of the decision-versus-verification question stems from a choice of computational model: polynomial-time Turing machines. It is far from the only model worthy of investigation, however; indeed, measuring time up to polynomial factors is often too “coarse” for practical applications. We focus on quantum computation, a more complete model of physically realisable computation where quantum mechanical phenomena (such as interference and entanglement) may be used as computational resources; and sublinear algorithms, a formalisation of ultra-fast computation where merely reading or storing the entire input is impractical, e.g., when processing massive datasets such as social networks or large databases. We begin our investigation by studying structural properties of local algorithms, a large class of sublinear algorithms that includes property testers and is characterised by the inability to even see most of the input. We prove that, in this setting, queries – the main complexity measure – can be replaced with random samples. Applying this transformation yields, among other results, the state-of-the-art query lower bound for relaxed local decoders. Focusing our attention onto property testers, we begin to chart the complexity�theoretic landscape arising from the classical vs. quantum and decision vs. verification questions in testing. We show that quantum hardware and communication with a powerful but untrusted prover are “orthogonal” resources, so that one cannot be substituted for the other. This implies all of the possible separations among the analogues of QMA, MA and BQP in the property-testing setting. We conclude with a study of zero-knowledge for (classical) streaming algorithms, which receive one-pass access to the entirety of their input but only have sublinear space. Inspired by cryptographic tools, we construct commitment protocols that are unconditionally secure in the streaming model and can be leveraged to obtain zero-knowledge streaming interactive proofs – and, in particular, show that zero-knowledge is achievable in this model