8,911 research outputs found
The -operator and Invariant Subtraction Games
We study 2-player impartial games, so called \emph{invariant subtraction
games}, of the type, given a set of allowed moves the players take turn in
moving one single piece on a large Chess board towards the position
. Here, invariance means that each allowed move is available
inside the whole board. Then we define a new game, of the old game, by
taking the -positions, except , as moves in the new game. One
such game is \W^\star= (Wythoff Nim), where the moves are defined by
complementary Beatty sequences with irrational moduli. Here we give a
polynomial time algorithm for infinitely many -positions of \W^\star. A
repeated application of turns out to give especially nice properties
for a certain subfamily of the invariant subtraction games, the
\emph{permutation games}, which we introduce here. We also introduce the family
of \emph{ornament games}, whose -positions define complementary Beatty
sequences with rational moduli---hence related to A. S. Fraenkel's `variant'
Rat- and Mouse games---and give closed forms for the moves of such games. We
also prove that (-pile Nim) = -pile Nim.Comment: 30 pages, 5 figure
Number sense : the underpinning understanding for early quantitative literacy
The fundamental meaning of Quantitative Literacy (QL) as the application of quantitative knowledge or reasoning in new/unfamiliar contexts is problematic because how we acquire knowledge, and transfer it to new situations, is not straightforward. This article argues that in the early development of QL, there is a specific corpus of numerical knowledge which learners need to integrate into their thinking, and to which teachers should attend. The paper is a rebuttal to historically prevalent (and simplistic) views that the terrain of early numerical understanding is little more than simple counting devoid of cognitive complexity. Rather, the knowledge upon which early QL develops comprises interdependent dimensions: Number Knowledge, Counting Skills and Principles, Nonverbal Calculation, Number Combinations and Story Problems - summarised as Number Sense. In order to derive the findings for this manuscript, a realist synthesis of recent Education and Psychology literature was conducted. The findings are of use not only when teaching very young children, but also when teaching learners who are experiencing learning difficulties through the absence of prerequisite numerical knowledge. As well distilling fundamental quantitative knowledge for teachers to integrate into practice, the review emphasises that improved pedagogy is less a function of literal applications of reported interventions, on the grounds of perceived efficacy elsewhere, but based in refinements of teachers' understandings. Because teachers need to adapt instructional sequences to the actual thinking and learning of learners in their charge, they need knowledge that allows them to develop their own theoretical understanding rather than didactic exhortations
Quantum-over-classical Advantage in Solving Multiplayer Games
We study the applicability of quantum algorithms in computational game theory
and generalize some results related to Subtraction games, which are sometimes
referred to as one-heap Nim games.
In quantum game theory, a subset of Subtraction games became the first
explicitly defined class of zero-sum combinatorial games with provable
separation between quantum and classical complexity of solving them. For a
narrower subset of Subtraction games, an exact quantum sublinear algorithm is
known that surpasses all deterministic algorithms for finding solutions with
probability .
Typically, both Nim and Subtraction games are defined for only two players.
We extend some known results to games for three or more players, while
maintaining the same classical and quantum complexities:
and respectively
Decomposable Theories
We present in this paper a general algorithm for solving first-order formulas
in particular theories called "decomposable theories". First of all, using
special quantifiers, we give a formal characterization of decomposable theories
and show some of their properties. Then, we present a general algorithm for
solving first-order formulas in any decomposable theory "T". The algorithm is
given in the form of five rewriting rules. It transforms a first-order formula
"P", which can possibly contain free variables, into a conjunction "Q" of
solved formulas easily transformable into a Boolean combination of
existentially quantified conjunctions of atomic formulas. In particular, if "P"
has no free variables then "Q" is either the formula "true" or "false". The
correctness of our algorithm proves the completeness of the decomposable
theories.
Finally, we show that the theory "Tr" of finite or infinite trees is a
decomposable theory and give some benchmarks realized by an implementation of
our algorithm, solving formulas on two-partner games in "Tr" with more than 160
nested alternated quantifiers
The efficacy of working memory training in improving crystallized intelligence
Crystallized intelligence (Gc) is thought to reflect skills acquired through knowledge and experience and is related to verbal ability, language development^1^ and academic success^2^. Gc, together with fluid intelligence (Gf), are constructs of general intelligence^3^. While Gc involves learning, knowledge and skills, Gf refers to our ability in tests of problem-solving, pattern matching, and reasoning. Although there is evidence that Gf can be improved through memory training in adults^4^, the efficacy of memory training in improving acquired skills, such as Gc and academic attainment, has yet to be established. Furthermore, evidence of transfer effects from gains made in the trained tasks is sparse^5^. Here we demonstrate improvements in Gc and academic attainment using working memory training. Participants in the Training group displayed superior performance in all measures of cognitive assessments post-training compared to the Control group, who received knowledge-based training. While previous studies have indicated that gains in intelligence are due to improvements in test-taking skills^6^, this study demonstrates that it is possible to improve crystallized skills through working memory training. Considering the fundamental importance of Gc in acquiring and using knowledge and its predictive power for a large variety of intellectual tasks, these findings may be highly relevant to improving educational outcomes in those who are struggling
Exact and Approximate Determinization of Discounted-Sum Automata
A discounted-sum automaton (NDA) is a nondeterministic finite automaton with
edge weights, valuing a run by the discounted sum of visited edge weights. More
precisely, the weight in the i-th position of the run is divided by
, where the discount factor is a fixed rational number
greater than 1. The value of a word is the minimal value of the automaton runs
on it. Discounted summation is a common and useful measuring scheme, especially
for infinite sequences, reflecting the assumption that earlier weights are more
important than later weights. Unfortunately, determinization of NDAs, which is
often essential in formal verification, is, in general, not possible. We
provide positive news, showing that every NDA with an integral discount factor
is determinizable. We complete the picture by proving that the integers
characterize exactly the discount factors that guarantee determinizability: for
every nonintegral rational discount factor , there is a
nondeterminizable -NDA. We also prove that the class of NDAs with
integral discount factors enjoys closure under the algebraic operations min,
max, addition, and subtraction, which is not the case for general NDAs nor for
deterministic NDAs. For general NDAs, we look into approximate determinization,
which is always possible as the influence of a word's suffix decays. We show
that the naive approach, of unfolding the automaton computations up to a
sufficient level, is doubly exponential in the discount factor. We provide an
alternative construction for approximate determinization, which is singly
exponential in the discount factor, in the precision, and in the number of
states. We also prove matching lower bounds, showing that the exponential
dependency on each of these three parameters cannot be avoided. All our results
hold equally for automata over finite words and for automata over infinite
words
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