scholarly journals A Note on Program Specialization. What Syntactical Properties of Residual Programs Can Reveal?

10.29007/cj27 ◽  
2018 ◽  
Author(s):  
Alexei Lisitsa ◽  
Andrei Nemytykh
Keyword(s):  
Solution Set ◽  
Program Specialization ◽  
Word Equation

The paper presents two examples of non-traditional using of program specialization by Turchin's supercompilation method.In both cases we are interested in syntactical properties of residual programs produced by supercompilation.In the first example we apply supercompilation to a program encoding a word equation and as a result we obtain a program representing a graphdescribing the solution set of the word equation.The idea of the second example belongs to Alexandr V. Korlyukov. He considered an interpreter simulating the dynamic of the well known missionaries-cannibals puzzle. Supercompilation of the interpreter allows us to solve the puzzle.The interpreter may also be seen as an encoding of a non-deterministic protocol.

scholarly journals Solutions to twisted word equations and equations in virtually free groups

2020 ◽  
Vol 30 (04) ◽  
pp. 731-819
Author(s):  
Volker Diekert ◽  
Murray Elder
Keyword(s):  
Free Groups ◽  
Expressive Power ◽  
Solution Set ◽  
Complexity Bound ◽  
Free Monoids ◽  
Word Equation ◽  
All Solutions ◽  
Standard Normal ◽  
Word Equations ◽  
Solving Equations

It is well known that the problem solving equations in virtually free groups can be reduced to the problem of solving twisted word equations with regular constraints over free monoids with involution. In this paper, we prove that the set of all solutions of a twisted word equation is an EDT0L language whose specification can be computed in PSPACE . Within the same complexity bound we can decide whether the solution set is empty, finite, or infinite. In the second part of the paper we apply the results for twisted equations to obtain in PSPACE an EDT0L description of the solution set of equations with rational constraints for finitely generated virtually free groups in standard normal forms with respect to a natural set of generators. If the rational constraints are given by a homomorphism into a fixed (or “small enough”) finite monoid, then our algorithms can be implemented in [Formula: see text], that is, in quasi-quadratic nondeterministic space. Our results generalize the work by Lohrey and Sénizergues (ICALP 2006) and Dahmani and Guirardel (J. of Topology 2010) with respect to both complexity and expressive power. Neither paper gave any concrete complexity bound and the results in these papers are stated for subsets of solutions only, whereas our results concern all solutions.

WORD EQUATIONS WITH ONE UNKNOWN

2011 ◽  
Vol 22 (02) ◽  
pp. 345-375 ◽  
Author(s):  
MARKKU LAINE ◽  
WOJCIECH PLANDOWSKI
Keyword(s):  
Infinite Number ◽  
Linear Time ◽  
Time Algorithm ◽  
Solution Set ◽  
Infinitely Many Solutions ◽  
Linear Time Algorithm ◽  
Number Of Solutions ◽  
Word Equation ◽  
Word Equations

We consider properties of the solution set of a word equation with one unknown. We prove that the solution set of a word equation possessing infinite number of solutions is of the form (pq)*p where pq is primitive. Next, we prove that a word equation with at most four occurrences of the unknown possesses either infinitely many solutions or at most two solutions. We show that there are equations with at most four occurrences of the unknown possessing exactly two solutions. Finally, we prove that a word equation with at most 2k occurrences of the unknown possesses either infinitely many solutions or at most 8 log k + O(1) solutions. Hence, if we consider a class εk of equations with at most 2k occurrences of the unknown, then each equation in this class possesses either infinitely many solutions or O( log k) number of solutions. Our considerations allow to construct the first alphabet independent linear time algorithm for computing the solution set of an equation in a nontrivial class of equations.

On PSPACE generation of a solution set of a word equation and its applications

2019 ◽  
Vol 792 ◽  
pp. 20-61
Author(s):  
Wojciech Plandowski
Keyword(s):  
Solution Set ◽  
Word Equation

scholarly journals Multi-Objective Whale Optimization Algorithm for Computation Offloading Optimization in Mobile Edge Computing

Sensors ◽  
2021 ◽  
Vol 21 (8) ◽  
pp. 2628
Author(s):  
Mengxing Huang ◽  
Qianhao Zhai ◽  
Yinjie Chen ◽  
Siling Feng ◽  
Feng Shu
Keyword(s):  
Optimization Algorithm ◽  
Edge Computing ◽  
Solution Set ◽  
Whale Optimization Algorithm ◽  
Computation Offloading ◽  
Mobile Edge Computing ◽  
Multi Objective ◽  
Whale Optimization ◽  
Artificial Immune System Algorithm

Computation offloading is one of the most important problems in edge computing. Devices can transmit computation tasks to servers to be executed through computation offloading. However, not all the computation tasks can be offloaded to servers with the limitation of network conditions. Therefore, it is very important to decide quickly how many tasks should be executed on servers and how many should be executed locally. Only computation tasks that are properly offloaded can improve the Quality of Service (QoS). Some existing methods only focus on a single objection, and of the others some have high computational complexity. There still have no method that could balance the targets and complexity for universal application. In this study, a Multi-Objective Whale Optimization Algorithm (MOWOA) based on time and energy consumption is proposed to solve the optimal offloading mechanism of computation offloading in mobile edge computing. It is the first time that MOWOA has been applied in this area. For improving the quality of the solution set, crowding degrees are introduced and all solutions are sorted by crowding degrees. Additionally, an improved MOWOA (MOWOA2) by using the gravity reference point method is proposed to obtain better diversity of the solution set. Compared with some typical approaches, such as the Grid-Based Evolutionary Algorithm (GrEA), Cluster-Gradient-based Artificial Immune System Algorithm (CGbAIS), Non-dominated Sorting Genetic Algorithm III (NSGA-III), etc., the MOWOA2 performs better in terms of the quality of the final solutions.

Nonlocal fractional semilinear differential inclusions with noninstantaneous impulses and of order α ∈ (1, 2)

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
JinRong Wang ◽  
Ahmed G. Ibrahim ◽  
Donal O’Regan ◽  
Adel A. Elmandouh
Keyword(s):  
Differential Inclusions ◽  
Mild Solutions ◽  
Multivalued Function ◽  
Linear Operators ◽  
Solution Set ◽  
Cosine Family ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Noninstantaneous Impulses

AbstractIn this paper, we establish the existence of mild solutions for nonlocal fractional semilinear differential inclusions with noninstantaneous impulses of order α ∈ (1,2) and generated by a cosine family of bounded linear operators. Moreover, we show the compactness of the solution set. We consider both the case when the values of the multivalued function are convex and nonconvex. Examples are given to illustrate the theory.

Rule-based Word Equation Solving

2020 ◽  
Author(s):  
Joel D. Day ◽  
Mitja Kulczynski ◽  
Florin Manea ◽  
Dirk Nowotka ◽  
Danny Bøgsted Poulsen
Keyword(s):  
Rule Based ◽  
Equation Solving ◽  
Word Equation

scholarly journals Structure of Pareto Solutions of Generalized Polyhedral-Valued Vector Optimization Problems in Banach Spaces

2013 ◽  
Vol 2013 ◽  
pp. 1-10
Author(s):  
Qinghai He ◽  
Weili Kong
Keyword(s):  
Banach Spaces ◽  
Vector Optimization ◽  
Optimization Problems ◽  
Vector Optimization Problem ◽  
Solution Set ◽  
Pareto Optimal ◽  
Pareto Solution ◽  
Value Set ◽  
Set Valued Mapping ◽  
Optimal Value

In general Banach spaces, we consider a vector optimization problem (SVOP) in which the objective is a set-valued mapping whose graph is the union of finitely many polyhedra or the union of finitely many generalized polyhedra. Dropping the compactness assumption, we establish some results on structure of the weak Pareto solution set, Pareto solution set, weak Pareto optimal value set, and Pareto optimal value set of (SVOP) and on connectedness of Pareto solution set and Pareto optimal value set of (SVOP). In particular, we improved and generalize, Arrow, Barankin, and Blackwell’s classical results in Euclidean spaces and Zheng and Yang’s results in general Banach spaces.

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