Search results for " Complexity"
showing 10 items of 623 documents
Understanding Quantum Algorithms via Query Complexity
2017
Query complexity is a model of computation in which we have to compute a function $f(x_1, \ldots, x_N)$ of variables $x_i$ which can be accessed via queries. The complexity of an algorithm is measured by the number of queries that it makes. Query complexity is widely used for studying quantum algorithms, for two reasons. First, it includes many of the known quantum algorithms (including Grover's quantum search and a key subroutine of Shor's factoring algorithm). Second, one can prove lower bounds on the query complexity, bounding the possible quantum advantage. In the last few years, there have been major advances on several longstanding problems in the query complexity. In this talk, we su…
On Physical Problems that are Slightly More Difficult than QMA
2013
We study the complexity of computational problems from quantum physics. Typically, they are studied using the complexity class QMA (quantum counterpart of NP) but some natural computational problems appear to be slightly harder than QMA. We introduce new complexity classes consisting of problems that are solvable with a small number of queries to a QMA oracle and use these complexity classes to quantify the complexity of several natural computational problems (for example, the complexity of estimating the spectral gap of a Hamiltonian).
The Alternating BWT: an algorithmic perspective
2020
Abstract The Burrows-Wheeler Transform (BWT) is a word transformation introduced in 1994 for Data Compression. It has become a fundamental tool for designing self-indexing data structures, with important applications in several areas in science and engineering. The Alternating Burrows-Wheeler Transform (ABWT) is another transformation recently introduced in Gessel et al. (2012) [21] and studied in the field of Combinatorics on Words. It is analogous to the BWT, except that it uses an alternating lexicographical order instead of the usual one. Building on results in Giancarlo et al. (2018) [23] , where we have shown that BWT and ABWT are part of a larger class of reversible transformations, …
Minimal forbidden words and symbolic dynamics
1996
We introduce a new complexity measure of a factorial formal language L: the growth rate of the set of minimal forbidden words. We prove some combinatorial properties of minimal forbidden words. As main result we prove that the growth rate of the set of minimal forbidden words for L is a topological invariant of the dynamical system defined by L.
Finite State Verifiers with Constant Randomness
2012
We give a new characterization of NL as the class of languages whose members have certificates that can be verified with small error in polynomial time by finite state machines that use a constant number of random bits, as opposed to its conventional description in terms of deterministic logarithmic-space verifiers. It turns out that allowing two-way interaction with the prover does not change the class of verifiable languages, and that no polynomially bounded amount of randomness is useful for constant-memory computers when used as language recognizers, or public-coin verifiers.
Team learning as a game
1997
A machine FIN-learning machine M receives successive values of the function f it is learning; at some point M outputs conjecture which should be a correct index of f. When n machines simultaneously learn the same function f and at least k of these machines outut correct indices of f, we have team learning [k,n]FIN. Papers [DKV92, DK96] show that sometimes a team or a robabilistic learner can simulate another one, if its probability p (or team success ratio k/n) is close enough. On the other hand, there are critical ratios which mae simulation o FIN(p2) by FIN(p1) imossible whenever p2 _< r < p1 or some critical ratio r. Accordingly to [DKV92] the critical ratio closest to 1/2 rom the let is…
On extremal cases of Hopcroft’s algorithm
2010
AbstractIn this paper we consider the problem of minimization of deterministic finite automata (DFA) with reference to Hopcroft’s algorithm. Hopcroft’s algorithm has several degrees of freedom, so there can exist different executions that can lead to different sequences of refinements of the set of the states up to the final partition. We find an infinite family of binary automata for which such a process is unique, whatever strategy is chosen. Some recent papers (cf. Berstel and Carton (2004) [3], Castiglione et al. (2008) [6] and Berstel et al. (2009) [1]) have been devoted to find families of automata for which Hopcroft’s algorithm has its worst execution time. They are unary automata as…
Counting with Probabilistic and Ultrametric Finite Automata
2014
We investigate the state complexity of probabilistic and ultrametric finite automata for the problem of counting, i.e. recognizing the one-word unary language \(C_n=\left\{ 1^n \right\} \). We also review the known results for other types of automata.
If P≠NP then some strongly noninvertible functions are invertible
2006
AbstractRabi, Rivest, and Sherman alter the standard notion of noninvertibility to a new notion they call strong noninvertibility, and show—via explicit cryptographic protocols for secret-key agreement (Rabi and Sherman attribute this protocol to Rivest and Sherman) and digital signatures (Rabi and Sherman)—that strongly noninvertible functions are very useful components in protocol design. Their definition of strong noninvertibility has a small twist (“respecting the argument given”) that is needed to ensure cryptographic usefulness. In this paper, we show that this small twist has a consequence: unless P=NP, some strongly noninvertible functions are invertible.
Two-Variable First-Order Logic with Equivalence Closure
2012
We consider the satisfiability and finite satisfiability problems for extensions of the two-variable fragment of first-order logic in which an equivalence closure operator can be applied to a fixed number of binary predicates. We show that the satisfiability problem for two-variable, first-order logic with equivalence closure applied to two binary predicates is in 2-NExpTime, and we obtain a matching lower bound by showing that the satisfiability problem for two-variable first-order logic in the presence of two equivalence relations is 2-NExpTime-hard. The logics in question lack the finite model property; however, we show that the same complexity bounds hold for the corresponding finite sa…