Finite model enumeration

• Main Author: Chow, Choiwah
• Publication Date: 2023
• Language: eng
• Source: Repositórios Científicos de Acesso Aberto de Portugal (RCAAP)
• Download full: http://hdl.handle.net/10400.2/14363
• topic: Álgebra; CREAM; Cubo-e-conquista; Enumeração de modelo finito; GAP; Monoide inerentemente não finito; Invariante; Isofilter; Isomorfismo; Magmaut; Mace4; Master Prover; Monomorfismo; Algoritmo paralelo; Algebra; Cube-and-conquer; Finite model enumeration; Inherently nonfinitely based monoid; Invariant; Isofilter; Isomorphism; Master Prover; Mmonomorphism; Parallel algorithm

To study and get intuition on different types of relational algebras (groups, semigroups, and their ordered versions, quasigroups, fields, rings, MV-algebras, lattices, etc.), mathematicians resort to libraries of all models, up to isomorphism, of order n (for small values of n) of the classes of algebras they are interested in. These libraries allow experimentation such as forming and/or testing conjectures etc., to gain insights into the classes of algebras in question. Since an isomorphism partitions the set of all models into equivalence classes, so, only one representative element in each equivalence class is needed. The rest of them are redundant and are removed to make it easy to discern the structure of the algebras. To construct these libraries, we need first to find all models up to isomorphism of the classes of algebras of interest. Enumerating all finite models from first-order formulas up to isomorphism is a hard problem, even for algebras of small orders. The current best practice of finite model enumeration is to divide the job into two steps: first generate models according to the laws laid down by the first-order formula defining the class of algebra, then in the post-processing step, find only one representative model of each equivalence class of isomorphic models. An algebra of finite order n defined by a first-order formula can be represented by operation tables (also known as multiplication tables) of finite sizes. Traditional finite model enumerators such as Mace4 basically perform combinatorial search on these operation tables to find instances that satisfy, or equivalently, not violate, the rules laid down by the first-order formula. This is a hard problem. For example, there are n n 2 possible binary operation tables of size n × n with n possible values in each cell. That is, even to enumerate all models of order 4 defined with just one binary operation in its first-order formula, we are facing the task of checking 4 4 2 ≈ 4.3 billion possibilities. Traditional finite model enumerators also produce a lot of isomorphic models to the extent that isomorphism models often constitute over 99% of the outputs. This is a characteristics of algebras definable by FOL: any permutation on the domain elements of a model is a model. Finding non-isomorphic models is very computationally intensive. If we follow a simple brute-force approach, then, given m models, it may require in the worst case m(m −1)/2 comparisons of models to check for isomorphism. Each comparison of models of order n may require in the worst case n! checks because there are n! different permutations on n symbols. Thus, both of these two steps (generating models from FOL and removing isomorphic models) are resource-intensive. They prompt us to find better algorithms that also make optimal use of the available computing resources. The matter is further complicated by the fact that while modern-day generalpurpose computers are predominantly multi-core, harnessing parallelism for combinatorial search is surprisingly difficult. In this thesis, we propose the isomorphic cubes removal algorithm and the parallel cubes search algorithm to improve the speed of the traditional finite model enumerator. Searching the operation tables can be seen as searching a search tree. Any partial branch from the root (called a cube) can be searched in parallel with other branches, making full use of the computing resources available. Furthermore, some (isomorphic) cubes produce isomorphic models and hence only one of them needs to be searched. Removing isomorphic cubes not only speeds up the search process, but also reduces the number of isomorphic models that need to be removed in the post-processing step. These parallel algorithms are very scalable: more cubes can be generated if more computing resources are available. They also cope with big problems well: longer cubes are possible with higher orders of algebra; with longer cubes, more isomorphic cubes can be removed. To speed up the isomorphic models removal process, we propose the parallel invariant-based isomorphic model removal algorithms. We note that models with different sets of invariant vectors (ordered list of invariants) cannot be isomorphic. We put models into blocks in a hash-table using invariant vectors as the keys. Models across different blocks thus have different sets of invariant vectors and hence cannot be isomorphic. These blocks of models can be checked for isomorphism separately in parallel. The improvement in speed comes from (1) smaller block size means fewer pairs of models to compare for isomorphism, (2) the blocks can be processed in parallel, making full use of the computing resources available. These algorithms also cope with big problems well: models of higher orders give more different invariant vectors to spread out the models. These algorithms together often give an improvement of orders of magnitude in the performance of the traditional finite model enumerators. Better yet, these algorithms can be incorporated in many common finite model enumerators with minimal change in their code base. Furthermore, the proposed algorithms are proved to be correct. In any mathematical research, humans are the most important factor to be given considerations. Not all mathematicians are computer experts, and not all of them want to invest their valuable time in the tools of mathematics. Thus, the tools must be designed to be user-friendly, without a steep learning curve for the user to be able to use the tools effectively. We therefore develop a GAP package generator to automatically generate GAP package for algebras of small orders. The generator hides the technical details of the generation process, but gives users the ability to control many of the non-technical aspects such as the name and the definition (in FOL) of the algebra to be explored. The GAP package generator makes extensive use of the facilities provided by the GAP system, and the generated package complies with the latest GAP standards. A separate, standalone GAP package, Magmaut, is developed to provide functions to manipulate isomorphism and automorphism for algebras of type (2 m, 1 n). It supplements the generated GAP package of algebra of small orders with isomorphism-related functionality. With the tools provided by this project, mathematicians can generate the GAP packages of algebras of type (2 m, 1 n) of their interest, without being bogged down by the complexity that often comes with the tools that generate such GAP packages. In this thesis, we propose many parallel algorithms to take advantage of modern-day multi-core computers to enumerate finite models. We show some important applications of these algorithms in developing GAP packages to calculate different morphisms such as isomorphism and monomorphism etc., in generating GAP packages of algebras, and in counting the number of algebras of small orders. We also develop a user-friendly tool to help mathematicians to generate GAP packages of algebras (of type (2 m, 1 n)) of small orders.