Fundamentals and applications of order dependencies

Business-intelligence queries often involve SQL functions and algebraic expressions. There can be clear semantic relationships between a column's values and the values of a function over that column. A common property is monotonicity: as the column's values ascend, so do the function's values (or the other column's values). This we call an order dependency (OD). Queries can be evaluated more efficiently when the query optimizer uses order dependencies. They can be run even faster when the optimizer can also reason over known ODs to infer new ones. Order dependencies can be declared as integrity constraints, and they can be detected automatically for many types of SQL functions and algebraic expressions. We present optimization techniques using ODs for queries that involve join, order by, group by, partition by, and distinct. Essentially, ODs can further exploit interesting orders to eliminate or simplify potentially expensive sorts in the query plan. We evaluate these techniques over our prototype implementation in IBM® DB2® using the TPC-DS® benchmark schema and some customer inspired queries. Our experimental results demonstrate a significant performance gain. Dependencies have played an important role in database theory. We study the theoretical aspects of order dependencies-and unidirectional order dependencies (UODs), a proper sub-class of ODs-which describe the relationships among lexicographical orderings of sets of tuples. We investigate the inference problem for order dependencies. We establish the following: (i) a sound and complete axiomatization for UODs which is sound for ODs; (ii) a hierarchy of order dependency classes; (iii) a proof of co-NP-completeness of the inference problem for ODs and for the subclass of UODs; (iv) a proof of co-NP-completeness of the inference problem of functional dependencies (FDs) from ODs in general, but demonstrate linear time complexity for the inference of FDs from UODs; (v) a sound and complete elimination procedure for testing logical implication over ODs; and (vi) a sound and complete polynomial inference algorithm for sets of UODs over natural domains

Constant-Soundness Interactive Proofs for Local Hamiltonians

$\newcommand{\Xlin}{\mathcal{X}} \newcommand{\Zlin}{\mathcal{Z}} \newcommand{\C}{\mathbb{C}}$We give a quantum multiprover interactive proof system for the local Hamiltonian problem in which there is a constant number of provers, questions are classical of length polynomial in the number of qubits, and answers are of constant length. The main novelty of our protocol is that the gap between completeness and soundness is directly proportional to the promise gap on the (normalized) ground state energy of the Hamiltonian. This result can be interpreted as a concrete step towards a quantum PCP theorem giving entangled-prover interactive proof systems for QMA-complete problems. The key ingredient is a quantum version of the classical linearity test of Blum, Luby, and Rubinfeld, where the function $f:\{0,1\}^n\to\{0,1\}$ is replaced by a pair of functions \Xlin, \Zlin:\{0,1\}^n\to \text{Obs}_d(\C), the set of $d$-dimensional Hermitian matrices that square to identity. The test enforces that (i) each function is exactly linear, \Xlin(a)\Xlin(b)=\Xlin(a+b) and \Zlin(a) \Zlin(b)=\Zlin(a+b), and (ii) the two functions are approximately complementary, \Xlin(a)\Zlin(b)\approx (-1)^{a\cdot b} \Zlin(b)\Xlin(a).Comment: 33 page

Predictable arguments of knowledge

We initiate a formal investigation on the power of predictability for argument of knowledge systems for NP. Specifically, we consider private-coin argument systems where the answer of the prover can be predicted, given the private randomness of the verifier; we call such protocols Predictable Arguments of Knowledge (PAoK). Our study encompasses a full characterization of PAoK, showing that such arguments can be made extremely laconic, with the prover sending a single bit, and assumed to have only one round (i.e., two messages) of communication without loss of generality. We additionally explore PAoK satisfying additional properties (including zero-knowledge and the possibility of re-using the same challenge across multiple executions with the prover), present several constructions of PAoK relying on different cryptographic tools, and discuss applications to cryptography