Representing Integer Sequences Using Piecewise-Affine Loops

A formal, high-level representation of programs is typically needed for static and dynamic analyses performed by compilers. However, the source code of target applications is not always available in an analyzable form, e.g., to protect intellectual property. To reason on such applications, it becomes necessary to build models from observations of its execution. This paper details an algebraic approach which, taking as input the trace of memory addresses accessed by a single memory reference, synthesizes an affine loop with a single perfectly nested reference that generates the original trace. This approach is extended to support the synthesis of unions of affine loops, useful for minimally modeling traces generated by automatic transformations of polyhedral programs, such as tiling. The resulting system is capable of processing hundreds of gigabytes of trace data in minutes, minimally reconstructing 100% of the static control parts in PolyBench/C applications and 99.99% in the Pluto-tiled versions of these benchmarks. As an application example of the trace modeling method, trace compression is explored. The affine representations built for the memory traces of PolyBench/C codes achieve compression factors of the order of 106 and 103 with respect to gzip for the original and tiled versions of the traces, respectively.

Drinfeld center of planar algebra

We introduce fusion, contragradient and braiding of Hilbert affine representations of a subfactor planar algebra P (not necessarily having finite depth). We prove that if N ⊂ M is a subfactor realization of P, then the Drinfeld center of the N–N-bimodule category generated byNL2(M)M, is equivalent to the category of Hilbert affine representations of P satisfying certain finiteness criterion. As a consequence, we prove Kevin Walker's conjecture for planar algebras.