5,642 research outputs found
L-Recursion and a new Logic for Logarithmic Space
We extend first-order logic with counting by a new operator that
allows it to formalise a limited form of recursion which can be
evaluated in logarithmic space. The resulting logic LREC has a
data complexity in LOGSPACE, and it defines LOGSPACE-complete
problems like deterministic reachability and Boolean formula
evaluation. We prove that LREC is strictly more expressive than
deterministic transitive closure logic with counting and
incomparable in expressive power with symmetric transitive closure
logic STC and transitive closure logic (with or without counting).
LREC is strictly contained in fixed-point logic with counting FPC.
We also study an extension LREC= of LREC that has nicer closure
properties and is more expressive than both LREC and STC, but is
still contained in FPC and has a data complexity in LOGSPACE.
Our main results are that LREC captures LOGSPACE on the class of
directed trees and that LREC= captures LOGSPACE on the class of
interval graphs
The parameterized space complexity of model-checking bounded variable first-order logic
The parameterized model-checking problem for a class of first-order sentences
(queries) asks to decide whether a given sentence from the class holds true in
a given relational structure (database); the parameter is the length of the
sentence. We study the parameterized space complexity of the model-checking
problem for queries with a bounded number of variables. For each bound on the
quantifier alternation rank the problem becomes complete for the corresponding
level of what we call the tree hierarchy, a hierarchy of parameterized
complexity classes defined via space bounded alternating machines between
parameterized logarithmic space and fixed-parameter tractable time. We observe
that a parameterized logarithmic space model-checker for existential bounded
variable queries would allow to improve Savitch's classical simulation of
nondeterministic logarithmic space in deterministic space .
Further, we define a highly space efficient model-checker for queries with a
bounded number of variables and bounded quantifier alternation rank. We study
its optimality under the assumption that Savitch's Theorem is optimal
Deciding regular grammar logics with converse through first-order logic
We provide a simple translation of the satisfiability problem for regular
grammar logics with converse into GF2, which is the intersection of the guarded
fragment and the 2-variable fragment of first-order logic. This translation is
theoretically interesting because it translates modal logics with certain frame
conditions into first-order logic, without explicitly expressing the frame
conditions.
A consequence of the translation is that the general satisfiability problem
for regular grammar logics with converse is in EXPTIME. This extends a previous
result of the first author for grammar logics without converse. Using the same
method, we show how some other modal logics can be naturally translated into
GF2, including nominal tense logics and intuitionistic logic.
In our view, the results in this paper show that the natural first-order
fragment corresponding to regular grammar logics is simply GF2 without extra
machinery such as fixed point-operators.Comment: 34 page
Efficient Quantum Algorithms for State Measurement and Linear Algebra Applications
We present an algorithm for measurement of -local operators in a quantum
state, which scales logarithmically both in the system size and the output
accuracy. The key ingredients of the algorithm are a digital representation of
the quantum state, and a decomposition of the measurement operator in a basis
of operators with known discrete spectra. We then show how this algorithm can
be combined with (a) Hamiltonian evolution to make quantum simulations
efficient, (b) the Newton-Raphson method based solution of matrix inverse to
efficiently solve linear simultaneous equations, and (c) Chebyshev expansion of
matrix exponentials to efficiently evaluate thermal expectation values. The
general strategy may be useful in solving many other linear algebra problems
efficiently.Comment: 17 pages, 3 figures (v2) Sections reorganised, several clarifications
added, results unchange
An in-between "implicit" and "explicit" complexity: Automata
Implicit Computational Complexity makes two aspects implicit, by manipulating
programming languages rather than models of com-putation, and by internalizing
the bounds rather than using external measure. We survey how automata theory
contributed to complexity with a machine-dependant with implicit bounds model
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