47,169 research outputs found
Using ACL2 to Verify Loop Pipelining in Behavioral Synthesis
Behavioral synthesis involves compiling an Electronic System-Level (ESL)
design into its Register-Transfer Level (RTL) implementation. Loop pipelining
is one of the most critical and complex transformations employed in behavioral
synthesis. Certifying the loop pipelining algorithm is challenging because
there is a huge semantic gap between the input sequential design and the output
pipelined implementation making it infeasible to verify their equivalence with
automated sequential equivalence checking techniques. We discuss our ongoing
effort using ACL2 to certify loop pipelining transformation. The completion of
the proof is work in progress. However, some of the insights developed so far
may already be of value to the ACL2 community. In particular, we discuss the
key invariant we formalized, which is very different from that used in most
pipeline proofs. We discuss the needs for this invariant, its formalization in
ACL2, and our envisioned proof using the invariant. We also discuss some
trade-offs, challenges, and insights developed in course of the project.Comment: In Proceedings ACL2 2014, arXiv:1406.123
Relating decision and search algorithms for rational points on curves of higher genus
For affine plane curves defined over the rationals of genus at least two, we
show that a decision algorithm for the existence of solutions also yields a
search algorithm for all solutions.Comment: to be published in Archive for Mathematical Logi
Phase Transition for Glauber Dynamics for Independent Sets on Regular Trees
We study the effect of boundary conditions on the relaxation time of the
Glauber dynamics for the hard-core model on the tree. The hard-core model is
defined on the set of independent sets weighted by a parameter ,
called the activity. The Glauber dynamics is the Markov chain that updates a
randomly chosen vertex in each step. On the infinite tree with branching factor
, the hard-core model can be equivalently defined as a broadcasting process
with a parameter which is the positive solution to
, and vertices are occupied with probability
when their parent is unoccupied. This broadcasting process
undergoes a phase transition between the so-called reconstruction and
non-reconstruction regions at . Reconstruction has
been of considerable interest recently since it appears to be intimately
connected to the efficiency of local algorithms on locally tree-like graphs,
such as sparse random graphs. In this paper we show that the relaxation time of
the Glauber dynamics on regular -ary trees of height and
vertices, undergoes a phase transition around the reconstruction threshold. In
particular, we construct a boundary condition for which the relaxation time
slows down at the reconstruction threshold. More precisely, for any , for with any boundary condition, the relaxation time is
and . In contrast, above the reconstruction
threshold we show that for every , for ,
the relaxation time on with any boundary condition is , and we construct a boundary condition where the relaxation time is
Graph Sparsification by Edge-Connectivity and Random Spanning Trees
We present new approaches to constructing graph sparsifiers --- weighted
subgraphs for which every cut has the same value as the original graph, up to a
factor of . Our first approach independently samples each
edge with probability inversely proportional to the edge-connectivity
between and . The fact that this approach produces a sparsifier resolves
a question posed by Bencz\'ur and Karger (2002). Concurrent work of Hariharan
and Panigrahi also resolves this question. Our second approach constructs a
sparsifier by forming the union of several uniformly random spanning trees.
Both of our approaches produce sparsifiers with
edges. Our proofs are based on extensions of Karger's contraction algorithm,
which may be of independent interest
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