13,030 research outputs found
On the logical complexity of cyclic arithmetic
We study the logical complexity of proofs in cyclic arithmetic
(), as introduced in Simpson '17, in terms of quantifier
alternations of formulae occurring. Writing for (the logical
consequences of) cyclic proofs containing only formulae, our main
result is that and prove the same
theorems, for all . Furthermore, due to the 'uniformity' of our
method, we also show that and Peano Arithmetic ()
proofs of the same theorem differ only exponentially in size.
The inclusion is obtained by proof
theoretic techniques, relying on normal forms and structural manipulations of
proofs. It improves upon the natural result that is
contained in . The converse inclusion, , is obtained by calibrating the approach of Simpson '17 with
recent results on the reverse mathematics of B\"uchi's theorem in
Ko{\l}odziejczyk, Michalewski, Pradic & Skrzypczak '16 (KMPS'16), and
specialising to the case of cyclic proofs. These results improve upon the
bounds on proof complexity and logical complexity implicit in Simpson '17 and
also an alternative approach due to Berardi & Tatsuta '17.
The uniformity of our method also allows us to recover a metamathematical
account of fragments of ; in particular we show that, for , the consistency of is provable in but not
. As a result, we show that certain versions of McNaughton's
theorem (the determinisation of -word automata) are not provable in
, partially resolving an open problem from KMPS '16
Making proofs without Modus Ponens: An introduction to the combinatorics and complexity of cut elimination
This paper is intended to provide an introduction to cut elimination which is
accessible to a broad mathematical audience. Gentzen's cut elimination theorem
is not as well known as it deserves to be, and it is tied to a lot of
interesting mathematical structure. In particular we try to indicate some
dynamical and combinatorial aspects of cut elimination, as well as its
connections to complexity theory. We discuss two concrete examples where one
can see the structure of short proofs with cuts, one concerning feasible
numbers and the other concerning "bounded mean oscillation" from real analysis
Quantum resource estimates for computing elliptic curve discrete logarithms
We give precise quantum resource estimates for Shor's algorithm to compute
discrete logarithms on elliptic curves over prime fields. The estimates are
derived from a simulation of a Toffoli gate network for controlled elliptic
curve point addition, implemented within the framework of the quantum computing
software tool suite LIQ. We determine circuit implementations for
reversible modular arithmetic, including modular addition, multiplication and
inversion, as well as reversible elliptic curve point addition. We conclude
that elliptic curve discrete logarithms on an elliptic curve defined over an
-bit prime field can be computed on a quantum computer with at most qubits using a quantum circuit of at most Toffoli gates. We are able to classically simulate the
Toffoli networks corresponding to the controlled elliptic curve point addition
as the core piece of Shor's algorithm for the NIST standard curves P-192,
P-224, P-256, P-384 and P-521. Our approach allows gate-level comparisons to
recent resource estimates for Shor's factoring algorithm. The results also
support estimates given earlier by Proos and Zalka and indicate that, for
current parameters at comparable classical security levels, the number of
qubits required to tackle elliptic curves is less than for attacking RSA,
suggesting that indeed ECC is an easier target than RSA.Comment: 24 pages, 2 tables, 11 figures. v2: typos fixed and reference added.
ASIACRYPT 201
Universal lossless source coding with the Burrows Wheeler transform
The Burrows Wheeler transform (1994) is a reversible sequence transformation used in a variety of practical lossless source-coding algorithms. In each, the BWT is followed by a lossless source code that attempts to exploit the natural ordering of the BWT coefficients. BWT-based compression schemes are widely touted as low-complexity algorithms giving lossless coding rates better than those of the Ziv-Lempel codes (commonly known as LZ'77 and LZ'78) and almost as good as those achieved by prediction by partial matching (PPM) algorithms. To date, the coding performance claims have been made primarily on the basis of experimental results. This work gives a theoretical evaluation of BWT-based coding. The main results of this theoretical evaluation include: (1) statistical characterizations of the BWT output on both finite strings and sequences of length n â â, (2) a variety of very simple new techniques for BWT-based lossless source coding, and (3) proofs of the universality and bounds on the rates of convergence of both new and existing BWT-based codes for finite-memory and stationary ergodic sources. The end result is a theoretical justification and validation of the experimentally derived conclusions: BWT-based lossless source codes achieve universal lossless coding performance that converges to the optimal coding performance more quickly than the rate of convergence observed in Ziv-Lempel style codes and, for some BWT-based codes, within a constant factor of the optimal rate of convergence for finite-memory source
Enumerating Subgraph Instances Using Map-Reduce
The theme of this paper is how to find all instances of a given "sample"
graph in a larger "data graph," using a single round of map-reduce. For the
simplest sample graph, the triangle, we improve upon the best known such
algorithm. We then examine the general case, considering both the communication
cost between mappers and reducers and the total computation cost at the
reducers. To minimize communication cost, we exploit the techniques of (Afrati
and Ullman, TKDE 2011)for computing multiway joins (evaluating conjunctive
queries) in a single map-reduce round. Several methods are shown for
translating sample graphs into a union of conjunctive queries with as few
queries as possible. We also address the matter of optimizing computation cost.
Many serial algorithms are shown to be "convertible," in the sense that it is
possible to partition the data graph, explore each partition in a separate
reducer, and have the total computation cost at the reducers be of the same
order as the computation cost of the serial algorithm.Comment: 37 page
A Self-Repairing Execution Unit for Microprogrammed Processors
Describes a processor which dynamically reconfigures its internal microcode to execute each instruction using only fault-free blocks from the execution unit. Working without redundant or spare computational blocks, this self-repair approach permits a graceful performance degradatio
Bounded Reachability for Temporal Logic over Constraint Systems
We present CLTLB(D), an extension of PLTLB (PLTL with both past and future
operators) augmented with atomic formulae built over a constraint system D.
Even for decidable constraint systems, satisfiability and Model Checking
problem of such logic can be undecidable. We introduce suitable restrictions
and assumptions that are shown to make the satisfiability problem for the
extended logic decidable. Moreover for a large class of constraint systems we
propose an encoding that realize an effective decision procedure for the
Bounded Reachability problem
Improved Memoryless RNS Forward Converter Based on the Periodicity of Residues
The residue number system (RNS) is suitable for DSP architectures because of its ability to perform fast carry-free arithmetic. However, this advantage is over-shadowed by the complexity involved in the conversion of numbers between binary and RNS representations. Although the reverse conversion (RNS to binary) is more complex, the forward transformation is not simple either. Most forward converters make use of look-up tables (memory). Recently, a memoryless forward converter architecture for arbitrary moduli sets was proposed by Premkumar in 2002. In this paper, we present an extension to that architecture which results in 44% less hardware for parallel conversion and achieves 43% improvement in speed for serial conversions. It makes use of the periodicity properties of residues obtained using modular exponentiation
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