1,404 research outputs found
Adaptive Monte Carlo Search for Conjecture Refutation in Graph Theory
Graph theory is an interdisciplinary field of study that has various
applications in mathematical modeling and computer science. Research in graph
theory depends on the creation of not only theorems but also conjectures.
Conjecture-refuting algorithms attempt to refute conjectures by searching for
counterexamples to those conjectures, often by maximizing certain score
functions on graphs. This study proposes a novel conjecture-refuting algorithm,
referred to as the adaptive Monte Carlo search (AMCS) algorithm, obtained by
modifying the Monte Carlo tree search algorithm. Evaluated based on its success
in finding counterexamples to several graph theory conjectures, AMCS
outperforms existing conjecture-refuting algorithms. The algorithm is further
utilized to refute six open conjectures, two of which were chemical graph
theory conjectures formulated by Liu et al. in 2021 and four of which were
formulated by the AutoGraphiX computer system in 2006. Finally, four of the
open conjectures are strongly refuted by generalizing the counterexamples
obtained by AMCS to produce a family of counterexamples. It is expected that
the algorithm can help researchers test graph-theoretic conjectures more
effectively.Comment: 27 pages, 11 figures, 3 tables; Milo Roucairol pointed out that both
of our papers used an incorrect formula for the harmonic of a graph. The
revised Conjecture 4 was able to be refuted. This paper and the GitHub
repository have been updated accordingl
Learning-Based Synthesis of Safety Controllers
We propose a machine learning framework to synthesize reactive controllers
for systems whose interactions with their adversarial environment are modeled
by infinite-duration, two-player games over (potentially) infinite graphs. Our
framework targets safety games with infinitely many vertices, but it is also
applicable to safety games over finite graphs whose size is too prohibitive for
conventional synthesis techniques. The learning takes place in a feedback loop
between a teacher component, which can reason symbolically about the safety
game, and a learning algorithm, which successively learns an overapproximation
of the winning region from various kinds of examples provided by the teacher.
We develop a novel decision tree learning algorithm for this setting and show
that our algorithm is guaranteed to converge to a reactive safety controller if
a suitable overapproximation of the winning region can be expressed as a
decision tree. Finally, we empirically compare the performance of a prototype
implementation to existing approaches, which are based on constraint solving
and automata learning, respectively
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