57 research outputs found
Cutting corners
We define a class of subshifts defined by a family of allowed patterns of the same shape where, for any contents of the shape minus a corner, the number of ways to fill in the corner is the same. For such a subshift, a locally legal pattern of convex shape is globally legal, and there is a measure that samples uniformly on convex sets. We show by example that these subshifts need not admit a group structure by shift-commuting continuous operations. Our approach to convexity is axiomatic, and only requires an abstract convex geometry that is “midpointed with respect to the shape”. We construct such convex geometries on several groups, in particular strongly polycyclic groups and free groups. We also show some other methods for sampling finite patterns, and show a link to conjectures of Gottshalk and Kaplansky.</p
The Minimum Generating Set Problem
Let be a finite group. In order to determine the smallest cardinality
of a generating set of and a generating set with this cardinality,
one should repeat many times the test whether a subset of of small
cardinality generates . We prove that if a chief series of is known,
then the numbers of these generating tests can be drastically reduced. At most
subsets must be tested. This implies that the minimum generating
set problem for a finite group can be solved in polynomial time
On the Complexity Dichotomy for the Satisfiability of Systems of Term Equations over Finite Algebras
For a fixed finite algebra ?, we consider the decision problem SysTerm(?): does a given system of term equations have a solution in ?? This is equivalent to a constraint satisfaction problem (CSP) for a relational structure whose relations are the graphs of the basic operations of ?. From the complexity dichotomy for CSP over fixed finite templates due to Bulatov [Bulatov, 2017] and Zhuk [Zhuk, 2017], it follows that SysTerm(?) for a finite algebra ? is in P if ? has a not necessarily idempotent Taylor polymorphism and is NP-complete otherwise. More explicitly, we show that for a finite algebra ? in a congruence modular variety (e.g. for a quasigroup), SysTerm(?) is in P if the core of ? is abelian and is NP-complete otherwise. Given ? by the graphs of its basic operations, we show that this condition for tractability can be decided in quasi-polynomial time
New Techniques for Static Symmetry Breaking in Many-Sorted Finite Model Finding
© 2023 IEEE. J. Poremba, N. A. Day and A. Vakili, "New Techniques for Static Symmetry Breaking in Many-Sorted Finite Model Finding," in IEEE Transactions on Software Engineering, vol. 49, no. 6, pp. 3487-3503, 1 June 2023, doi: 10.1109/TSE.2023.3256939.Symmetry in finite model finding problems of many-sorted first-order logic (MSFOL) can be exploited to reduce the number of interpretations considered during search, thereby improving solver performance for tools such as the Alloy Analyzer. We present a framework to soundly compose static symmetry breaking schemes for MSFOL: 1) one for functions with distinct sorts in the domain and range; 2) one for functions where the range sort appears in the domain; and 3) one for predicates. We provide a novel presentation of sort inference in the context of symmetry breaking that yields a new mathematical link between sorts and symmetries. We empirically investigate how our symmetry breaking approaches affect solving performance.This work was supported in part by the Natural Sciences and Engineering Research Council of Canada (NSERC)
Selected Topics in Gravity, Field Theory and Quantum Mechanics
Quantum field theory has achieved some extraordinary successes over the past sixty years; however, it retains a set of challenging problems. It is not yet able to describe gravity in a mathematically consistent manner. CP violation remains unexplained. Grand unified theories have been eliminated by experiment, and a viable unification model has yet to replace them. Even the highly successful quantum chromodynamics, despite significant computational achievements, struggles to provide theoretical insight into the low-energy regime of quark physics, where the nature and structure of hadrons are determined. The only proposal for resolving the fine-tuning problem, low-energy supersymmetry, has been eliminated by results from the LHC. Since mathematics is the true and proper language for quantitative physical models, we expect new mathematical constructions to provide insight into physical phenomena and fresh approaches for building physical theories
A Geometric Approach to the Projective Tensor Norm
The main focus of this thesis is on the projective norm on finite-dimensional real or complex tensor products. There are various mathematical subjects with relations to the projective norm. For instance, it appears in the context of operator algebras or in quantum physics.
The projective norm on multipartite tensor products is considered to be less accessible. So we use a method from convex algebraic geometry to approximate the projective unit ball by convex supersets, so-called theta bodies. For real multipartite tensor products we obtain theta bodies which are close to the projective unit ball, leading to a generalisation of the Schmidt decomposition. In a second step the method is applied to complex tensor products, in a third step to separable states.
In a more general context, the projective norm can be related to binomial ideals, especially to so-called Hibi relations. In this respect, we also focus on a generalisation of the projective unit ball, here called Hibi body, and its theta bodies. It turns out that many statements also hold in this general context
Dynamically weakened constraints in bounded search for constraint optimisation problems
Combinatorial optimisation problems, where the goal is to an optimal solution from the set of solutions of a problem involving resources, constraints on how these resources can be used, and a ranking of solutions are of both theoretical and practical interest. Many real world problems (such as routing vehicles or planning timetables) can be modelled as constraint optimisation problems, and solved via a variety of solver technologies which rely on differing algorithms for search and inference. The starting point for the work presented in this thesis is two existing approaches to solving constraint optimisation problems: constraint programming and decision diagram branch and bound search. Constraint programming models problems using variables which have domains of values and valid value assignments to variables are restricted by constraints. Constraint programming is a mature approach to solving optimisation problems, and typically relies on backtracking search algorithms combined with constraint propagators (which infer from incomplete solutions which values can be removed from the domains of variables which are yet to be assigned a value). Decision diagram branch and bound search is a less mature approach which solves problems modelled as dynamic programming models using width restricted decision diagrams to provide bounds during search. The main contribution of this thesis is adapting decision diagram branch and bound to be the search scheme in a general purpose constraint solver. To achieve this we propose a method in which we introduce a new algorithm for each constraint that we wish to include in our solver and these new algorithms weaken individual constraints, so that they respect the problem relaxations introduced while using decision diagram branch and bound as the search algorithm in our solver. Constraints are weakened during search based on the problem relaxations imposed by the search algorithm: before search begins there is no way of telling which relaxations will be introduced. We attempt to provide weakening algorithms which require little to no changes to existing propagation algorithms. We provide weakening algorithms for a number of built-in constraints in the Flatzinc specifi- cation, as well as for global constraints and symmetry reduction constraints. We implement a solver in Go and empirically verify the competitiveness of our approach. We show that our solver can be parallelised using Goroutines and channels and that our approach scales well. Finally, we also provide an implementation of our approach in a solver which is tailored towards solving extremal graph problems. We use the forbidden subgraph problem to show that our approach of using decision diagram branch and bound as a search scheme in a constraint solver can be paired with canonical search. Canonical search is a technique for graph search which ensures that no two isomorphic graphs are returned during search. We pair our solver with the Nauty graph isomorphism algorithm to achieve this, and explore the relationship between branch and bound and canonical search
Proceedings of the 22nd Conference on Formal Methods in Computer-Aided Design – FMCAD 2022
The Conference on Formal Methods in Computer-Aided Design (FMCAD) is an annual conference on the theory and applications of formal methods in hardware and system verification. FMCAD provides a leading forum to researchers in academia and industry for presenting and discussing groundbreaking methods, technologies, theoretical results, and tools for reasoning formally about computing systems. FMCAD covers formal aspects of computer-aided system design including verification, specification, synthesis, and testing
Coding Theory
This book explores the latest developments, methods, approaches, and applications of coding theory in a wide variety of fields and endeavors. It consists of seven chapters that address such topics as applications of coding theory in networking and cryptography, wireless sensor nodes in wireless body area networks, the construction of linear codes, and more
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