153 research outputs found
A SAT+CAS Approach to Finding Good Matrices: New Examples and Counterexamples
We enumerate all circulant good matrices with odd orders divisible by 3 up to
order 70. As a consequence of this we find a previously overlooked set of good
matrices of order 27 and a new set of good matrices of order 57. We also find
that circulant good matrices do not exist in the orders 51, 63, and 69, thereby
finding three new counterexamples to the conjecture that such matrices exist in
all odd orders. Additionally, we prove a new relationship between the entries
of good matrices and exploit this relationship in our enumeration algorithm.
Our method applies the SAT+CAS paradigm of combining computer algebra
functionality with modern SAT solvers to efficiently search large spaces which
are specified by both algebraic and logical constraints
Nonexistence Certificates for Ovals in a Projective Plane of Order Ten
In 1983, a computer search was performed for ovals in a projective plane of
order ten. The search was exhaustive and negative, implying that such ovals do
not exist. However, no nonexistence certificates were produced by this search,
and to the best of our knowledge the search has never been independently
verified. In this paper, we rerun the search for ovals in a projective plane of
order ten and produce a collection of nonexistence certificates that, when
taken together, imply that such ovals do not exist. Our search program uses the
cube-and-conquer paradigm from the field of satisfiability (SAT) checking,
coupled with a programmatic SAT solver and the nauty symbolic computation
library for removing symmetries from the search.Comment: Appears in the Proceedings of the 31st International Workshop on
Combinatorial Algorithms (IWOCA 2020
Computational Methods for Combinatorial and Number Theoretic Problems
Computational methods have become a valuable tool for studying mathematical problems and for constructing large combinatorial objects. In fact, it is often not possible to find large combinatorial objects using human reasoning alone and the only known way of accessing such objects is to use computational methods. These methods require deriving mathematical properties which the object in question must necessarily satisfy, translating those properties into a format that a computer can process, and then running a search through a space which contains the objects which satisfy those properties.
In this thesis, we solve some combinatorial and number theoretic problems which fit into the above framework and present computational strategies which can be used to perform the search and preprocessing. In particular, one strategy we examine uses state-of-the-art tools from the symbolic computation and SAT/SMT solving communities to execute a search more efficiently than would be the case using the techniques from either community in isolation. To this end, we developed the tool MathCheck2, which combines the sophisticated domain-specific knowledge of a computer algebra system (CAS) with the powerful general-purpose search routines of a SAT solver. This fits into the recently proposed SAT+CAS paradigm which is based on the insight that modern SAT solvers (some of the best general-purpose search tools ever developed) do not perform well in all applications but can be made more efficient if supplied with appropriate domain-specific knowledge. To our knowledge, this is the first PhD thesis which studies the SAT+CAS paradigm which we believe has potential to be used in many problems for a long time to come.
As case studies for the methods we examine, we study the problem of computing Williamson matrices, the problem of computing complex Golay sequences, and the problem of computing minimal primes. In each case, we provide results which are competitive with or improve on the best known results prior to our work. In the first case study, we provide for the first time an enumeration of all Williamson matrices up to order 45 and show that 35 is the smallest order for which Williamson matrices do not exist. These results were previously known under the restriction that the order was odd but our work also considers even orders, as Williamson did when he defined such matrices in 1944. In the second case study, we provide an independent verification of the 2002 conjecture that complex Golay sequences do not exist in order 23 and enumerate all complex Golay sequences up to order 25. In the third case study, we compute the set of minimal primes for all bases up to 16 as well for all bases up to 30 with possibly a small number of missing elements
A SAT Solver and Computer Algebra Attack on the Minimum Kochen-Specker Problem
One of the foundational results in quantum mechanics is the Kochen-Specker
(KS) theorem, which states that any theory whose predictions agree with quantum
mechanics must be contextual, i.e., a quantum observation cannot be understood
as revealing a pre-existing value. The theorem hinges on the existence of a
mathematical object called a KS vector system. While many KS vector systems are
known to exist, the problem of finding the minimum KS vector system has
remained stubbornly open for over 55 years, despite significant attempts by
leading scientists and mathematicians. In this paper, we present a new method
based on a combination of a SAT solver and a computer algebra system (CAS) to
address this problem. Our approach improves the lower bound on the minimum
number of vectors in a KS system from 22 to 24, and is about 35,000 times more
efficient compared to the previous best computational methods. The increase in
efficiency derives from the fact we are able to exploit the powerful
combinatorial search-with-learning capabilities of a SAT solver together with
the isomorph-free exhaustive generation methods of a CAS. The quest for the
minimum KS vector system is motivated by myriad applications such as
simplifying experimental tests of contextuality, zero-error classical
communication, dimension witnessing, and the security of certain quantum
cryptographic protocols. To the best of our knowledge, this is the first
application of a novel SAT+CAS system to a problem in the realm of quantum
foundations
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