We establish a 1-1 correspondence between Valiant's character theory of matchgate/matchcircuit [14] and his signature theory of planar-matchgate/matchgrid [16], thus unifying the two theories in expressibility. In [5], we had established a complete characterization of general matchgates, in terms of a set of useful Grassmann-Plücker identities. With this correspondence, we give a corresponding set of identities which completely characterizes planar-matchgates and their signatures. Applying this characterization we prove some negative results for holographic algorithms. On the positive side, we also give a polynomial time algorithm for a simultaneous node-edge deletion problem, using holographic algorithms. Finally we give characterizations of symmetric signatures realizable in the Hadamard basis.

Valiant has proposed a new theory of algorithmic computation based on perfect matchings and the Pfaffian. We study the properties of matchgates-the basic building blocks in this new theory. We give a set of algebraic identities which completely characterize these objects in terms of the Grassmann-Plücker identities. In the important case of 4 by 4 matchgate matrices, which was used in Valiant's classical simulation of a fragment of quantum computations, we further realize a group action on the character matrix of a matchgate, and relate this information to its compound matrix. Then we use Jacobi's theorem to prove that in this case the invertible matchgate matrices form a multiplicative group. These results are useful in establishing limitations on the ultimate capabilities of Valiant's theory of matchgate computations and his closely related theory of Holographic Algorithms.

Automatic generation of Instruction Set Extensions (ISEs),to be executed on a custom processing unit or a coprocessor is an important step towards processor customization. A typical goal of a manual designer is to combine a large number of atomic instructions into an ISE satisfying microarchitectural constraints. However, memory operations pose a challenge for previous ISE approaches by limiting the size of the resulting instruction. In this paper, we introduce memory elements into custom units which result in ISEs closer to those sought after by the designers. We consider two kinds of memory elements for mapping to the specialized hardware: small hardware tables and architecturally-visible state registers. We devised a genetic algorithm to specifically exploit opportunities of introducing memory elements during ISE generation. Finally, we demonstrate the effectiveness of our approach by a detailed study of the variation in performance, area and energy in the presence of the generated ISEs, on a number of MediaBench, EEMBC and cryptographic applications. With the introduction of memory, the average speedup varied from 2.7X to 5X depending on the architectural configuration with a nominal area overhead. Moreover, we obtained an average energy reduction of 26% with respect to a 32-KB cache.

Valiant has proposed a new theory of algorithmic computation based on perfect matchings and the Pfaffian. We study the properties of matchgates-the basic building blocks in this new theory. We give a set of algebraic identities which completely characterize these objects in terms of the Grassmann-Plücker identities. In the important case of 4 by 4 matchgate matrices, which was used in Valiant's classical simulation of a fragment of quantum computations, we further realize a group action on the character matrix of a matchgate, and relate this information to its compound matrix. Then we use Jacobi's theorem to prove that in this case the invertible matchgate matrices form a multiplicative group. These results are useful in establishing limitations on the ultimate capabilities of Valiant's theory of matchgate computations and his closely related theory of Holographic Algorithms.