18,759 research outputs found
Logic Meets Algebra: the Case of Regular Languages
The study of finite automata and regular languages is a privileged meeting
point of algebra and logic. Since the work of Buchi, regular languages have
been classified according to their descriptive complexity, i.e. the type of
logical formalism required to define them. The algebraic point of view on
automata is an essential complement of this classification: by providing
alternative, algebraic characterizations for the classes, it often yields the
only opportunity for the design of algorithms that decide expressibility in
some logical fragment.
We survey the existing results relating the expressibility of regular
languages in logical fragments of MSO[S] with algebraic properties of their
minimal automata. In particular, we show that many of the best known results in
this area share the same underlying mechanics and rely on a very strong
relation between logical substitutions and block-products of pseudovarieties of
monoid. We also explain the impact of these connections on circuit complexity
theory.Comment: 37 page
Redundant Logic Insertion and Fault Tolerance Improvement in Combinational Circuits
This paper presents a novel method to identify and insert redundant logic
into a combinational circuit to improve its fault tolerance without having to
replicate the entire circuit as is the case with conventional redundancy
techniques. In this context, it is discussed how to estimate the fault masking
capability of a combinational circuit using the truth-cum-fault enumeration
table, and then it is shown how to identify the logic that can introduced to
add redundancy into the original circuit without affecting its native
functionality and with the aim of improving its fault tolerance though this
would involve some trade-off in the design metrics. However, care should be
taken while introducing redundant logic since redundant logic insertion may
give rise to new internal nodes and faults on those may impact the fault
tolerance of the resulting circuit. The combinational circuit that is
considered and its redundant counterparts are all implemented in semi-custom
design style using a 32/28nm CMOS digital cell library and their respective
design metrics and fault tolerances are compared
Statistical Power Supply Dynamic Noise Prediction in Hierarchical Power Grid and Package Networks
One of the most crucial high performance systems-on-chip design challenge is to front their power supply noise sufferance due to high frequencies, huge number of functional blocks and technology scaling down. Marking a difference from traditional post physical-design static voltage drop analysis, /a priori dynamic voltage drop/evaluation is the focus of this work. It takes into account transient currents and on-chip and package /RLC/ parasitics while exploring the power grid design solution space: Design countermeasures can be thus early defined and long post physical-design verification cycles can be shortened. As shown by an extensive set of results, a carefully extracted and modular grid library assures realistic evaluation of parasitics impact on noise and facilitates the power network construction; furthermore statistical analysis guarantees a correct current envelope evaluation and Spice simulations endorse reliable result
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
Lower bounds on the non-Clifford resources for quantum computations
We establish lower-bounds on the number of resource states, also known as
magic states, needed to perform various quantum computing tasks, treating
stabilizer operations as free. Our bounds apply to adaptive computations using
measurements and an arbitrary number of stabilizer ancillas. We consider (1)
resource state conversion, (2) single-qubit unitary synthesis, and (3)
computational tasks.
To prove our resource conversion bounds we introduce two new monotones, the
stabilizer nullity and the dyadic monotone, and make use of the already-known
stabilizer extent. We consider conversions that borrow resource states, known
as catalyst states, and return them at the end of the algorithm. We show that
catalysis is necessary for many conversions and introduce new catalytic
conversions, some of which are close to optimal.
By finding a canonical form for post-selected stabilizer computations, we
show that approximating a single-qubit unitary to within diamond-norm precision
requires at least
-states on average. This is the first lower bound that applies to synthesis
protocols using fall-back, mixing techniques, and where the number of ancillas
used can depend on .
Up to multiplicative factors, we optimally lower bound the number of or
states needed to implement the ubiquitous modular adder and
multiply-controlled- operations. When the probability of Pauli measurement
outcomes is 1/2, some of our bounds become tight to within a small additive
constant.Comment: 62 page
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