3,540 research outputs found
Beta Reduction is Invariant, Indeed (Long Version)
Slot and van Emde Boas' weak invariance thesis states that reasonable
machines can simulate each other within a polynomially overhead in time. Is
-calculus a reasonable machine? Is there a way to measure the
computational complexity of a -term? This paper presents the first
complete positive answer to this long-standing problem. Moreover, our answer is
completely machine-independent and based over a standard notion in the theory
of -calculus: the length of a leftmost-outermost derivation to normal
form is an invariant cost model. Such a theorem cannot be proved by directly
relating -calculus with Turing machines or random access machines,
because of the size explosion problem: there are terms that in a linear number
of steps produce an exponentially long output. The first step towards the
solution is to shift to a notion of evaluation for which the length and the
size of the output are linearly related. This is done by adopting the linear
substitution calculus (LSC), a calculus of explicit substitutions modelled
after linear logic and proof-nets and admitting a decomposition of
leftmost-outermost derivations with the desired property. Thus, the LSC is
invariant with respect to, say, random access machines. The second step is to
show that LSC is invariant with respect to the -calculus. The size
explosion problem seems to imply that this is not possible: having the same
notions of normal form, evaluation in the LSC is exponentially longer than in
the -calculus. We solve such an impasse by introducing a new form of
shared normal form and shared reduction, deemed useful. Useful evaluation
avoids those steps that only unshare the output without contributing to
-redexes, i.e., the steps that cause the blow-up in size.Comment: 29 page
(Leftmost-Outermost) Beta Reduction is Invariant, Indeed
Slot and van Emde Boas' weak invariance thesis states that reasonable
machines can simulate each other within a polynomially overhead in time. Is
lambda-calculus a reasonable machine? Is there a way to measure the
computational complexity of a lambda-term? This paper presents the first
complete positive answer to this long-standing problem. Moreover, our answer is
completely machine-independent and based over a standard notion in the theory
of lambda-calculus: the length of a leftmost-outermost derivation to normal
form is an invariant cost model. Such a theorem cannot be proved by directly
relating lambda-calculus with Turing machines or random access machines,
because of the size explosion problem: there are terms that in a linear number
of steps produce an exponentially long output. The first step towards the
solution is to shift to a notion of evaluation for which the length and the
size of the output are linearly related. This is done by adopting the linear
substitution calculus (LSC), a calculus of explicit substitutions modeled after
linear logic proof nets and admitting a decomposition of leftmost-outermost
derivations with the desired property. Thus, the LSC is invariant with respect
to, say, random access machines. The second step is to show that LSC is
invariant with respect to the lambda-calculus. The size explosion problem seems
to imply that this is not possible: having the same notions of normal form,
evaluation in the LSC is exponentially longer than in the lambda-calculus. We
solve such an impasse by introducing a new form of shared normal form and
shared reduction, deemed useful. Useful evaluation avoids those steps that only
unshare the output without contributing to beta-redexes, i.e. the steps that
cause the blow-up in size. The main technical contribution of the paper is
indeed the definition of useful reductions and the thorough analysis of their
properties.Comment: arXiv admin note: substantial text overlap with arXiv:1405.331
12th International Workshop on Termination (WST 2012) : WST 2012, February 19–23, 2012, Obergurgl, Austria / ed. by Georg Moser
This volume contains the proceedings of the 12th International Workshop on Termination (WST 2012), to be held February 19–23, 2012 in Obergurgl, Austria. The goal of the Workshop on Termination is to be a venue for presentation and discussion of all topics in and around termination. In this way, the workshop tries to bridge the gaps between different communities interested and active in research in and around termination. The 12th International Workshop on Termination in Obergurgl continues the successful workshops held in St. Andrews (1993), La Bresse (1995), Ede (1997), Dagstuhl (1999), Utrecht (2001), Valencia (2003), Aachen (2004), Seattle (2006), Paris (2007), Leipzig (2009), and Edinburgh (2010). The 12th International Workshop on Termination did welcome contributions on all aspects of termination and complexity analysis. Contributions from the imperative, constraint, functional, and logic programming communities, and papers investigating applications of complexity or termination (for example in program transformation or theorem proving) were particularly welcome. We did receive 18 submissions which all were accepted. Each paper was assigned two reviewers. In addition to these 18 contributed talks, WST 2012, hosts three invited talks by Alexander Krauss, Martin Hofmann, and Fausto Spoto
Generalized Unitary Coupled Cluster Wavefunctions for Quantum Computation
We introduce a unitary coupled-cluster (UCC) ansatz termed -UpCCGSD that
is based on a family of sparse generalized doubles (D) operators which provides
an affordable and systematically improvable unitary coupled-cluster
wavefunction suitable for implementation on a near-term quantum computer.
-UpCCGSD employs products of the exponential of pair coupled-cluster
double excitation operators (pCCD), together with generalized single (S)
excitation operators. We compare its performance in both efficiency of
implementation and accuracy with that of the generalized UCC ansatz employing
the full generalized SD excitation operators (UCCGSD), as well as with the
standard ansatz employing only SD excitations (UCCSD). -UpCCGSD is found to
show the best scaling for quantum computing applications, requiring a circuit
depth of , compared with for UCCGSD and
for UCCSD where is the number of spin
orbitals and is the number of electrons. We analyzed the accuracy of
these three ans\"atze by making classical benchmark calculations on the ground
state and the first excited state of H (STO-3G, 6-31G), HO (STO-3G),
and N (STO-3G), making additional comparisons to conventional coupled
cluster methods. The results for ground states show that -UpCCGSD offers a
good tradeoff between accuracy and cost, achieving chemical accuracy for lower
cost of implementation on quantum computers than both UCCGSD and UCCSD. Excited
states are calculated with an orthogonally constrained variational quantum
eigensolver approach. This is seen to generally yield less accurate energies
than for the corresponding ground states. We demonstrate that using a
specialized multi-determinantal reference state constructed from classical
linear response calculations allows these excited state energetics to be
improved
On Sharing, Memoization, and Polynomial Time (Long Version)
We study how the adoption of an evaluation mechanism with sharing and
memoization impacts the class of functions which can be computed in polynomial
time. We first show how a natural cost model in which lookup for an already
computed value has no cost is indeed invariant. As a corollary, we then prove
that the most general notion of ramified recurrence is sound for polynomial
time, this way settling an open problem in implicit computational complexity
Projected Density Matrix Embedding Theory with Applications to the Two-Dimensional Hubbard Model
Density matrix embedding theory (DMET) is a quantum embedding theory for
strongly correlated systems. From a computational perspective, one bottleneck
in DMET is the optimization of the correlation potential to achieve
self-consistency, especially for heterogeneous systems of large size. We
propose a new method, called projected density matrix embedding theory
(p-DMET), which achieves self-consistency without needing to optimize a
correlation potential. We demonstrate the performance of p-DMET on the
two-dimensional Hubbard model.Comment: 25 pages, 8 figure
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