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

Syntactic-Semantic Form of Mizar Articles

Mizar Mathematical Library is most appreciated for the wealth of mathematical knowledge it contains. However, accessing this publicly available huge corpus of formalized data is not straightforward due to the complexity of the underlying Mizar language, which has been designed to resemble informal mathematical papers. For this reason, most systems exploring the library are based on an internal XML representation format used by semantic modules of Mizar. This representation is easily accessible, but it lacks certain syntactic information available only in the original human-readable Mizar source files. In this paper we propose a new XML-based format which combines both syntactic and semantic data. It is intended to facilitate various applications of the Mizar library requiring fullest possible information to be retrieved from the formalization files

ProofWatch: Watchlist Guidance for Large Theories in E

Watchlist (also hint list) is a mechanism that allows related proofs to guide a proof search for a new conjecture. This mechanism has been used with the Otter and Prover9 theorem provers, both for interactive formalizations and for human-assisted proving of open conjectures in small theories. In this work we explore the use of watchlists in large theories coming from first-order translations of large ITP libraries, aiming at improving hammer-style automation by smarter internal guidance of the ATP systems. In particular, we (i) design watchlist-based clause evaluation heuristics inside the E ATP system, and (ii) develop new proof guiding algorithms that load many previous proofs inside the ATP and focus the proof search using a dynamically updated notion of proof matching. The methods are evaluated on a large set of problems coming from the Mizar library, showing significant improvement of E's standard portfolio of strategies, and also of the previous best set of strategies invented for Mizar by evolutionary methods.Comment: 19 pages, 10 tables, submitted to ITP 2018 at FLO

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