In this paper, we present an encoding of the λ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\lambda $$\\end{document}-calculus in a multiset rewriting system and provide a few applications of the construction. For this purpose, we choose the calculus named String MultiSet Rewriting, which was introduced in Barbuti et al. (Electron Notes Theor Comput Sci 194:19–34, 2008) by Barbuti et al. With the help of our encoding, we give alternative proofs for the standardization and the finiteness of developments theorems in the λ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\lambda $$\\end{document}-calculus.

Simon's problem is one of the most important problems demonstrating the power of quantum algorithms, as it greatly inspired the proposal of Shor's algorithm. The generalized Simon's problem is a natural extension of Simon's problem, and also a special hidden subgroup problem: Given a function f : {0, 1} n → {0, 1} m , with the property that for any x, y ∈ {0, 1} n , there is some unknown The goal of generalized Simon's problem is to find the hidden subgroup S. In this paper, we present two key contributions. Firstly, we characterize the structure of the generalized Simon's problem in distributed scenario and introduce a corresponding distributed quantum algorithm. Secondly, we refine the algorithm to ensure exactness due to the application of quantum amplitude amplification technique. Our algorithm offers exponential acceleration compared to the distributed classical algorithm. When contrasted with the centralized quantum algorithm for the generalized Simon's problem, our algorithm's oracle requires fewer qubits, thus making it easier to be physically implemented. Particulary, the exact distributed quantum algorithm we develop for the generalized Simon's problem outperforms the best previously proposed distributed quantum algorithm for Simon's problem in terms of generalizability and exactness.

This paper considers a situation where each process has a set of operations Op and executes each operation in Op infinitely often in distributed systems. Then, let Oe ⊂ Op be a subset of operations, which cannot be executed by a process while its closed neighborhood execute operations in Op \ Oe. For such a situation, this new concept of Neighborhood Mutual Remainder (NMR) is defined. A distributed algorithm that satisfies the NMR requirement should satisfy the following two properties: (1) Liveness is satisfied if a process executes each operation in Op infinitely often, and (2) safety is satisfied if, when each process executes operations in Oe, no process in its closed neighborhood executes operations in Op \ Oe. * Preliminary brief announcement versions (one of five pages and the other of six pages) of this detailed complete paper (of twenty-two pages) can be found in [1, 2] . Neighborhood Mutual Remainder We formalize the concept of NMR and give a simple self-stabilizing algorithm to demonstrate the design paradigm to achieve NMR. We present an application of NMR to a LOOK-COMPUTE-MOVE (LCM) robot system for implementing a move-atomic property, where robots possess an independent clock that is advanced at the same speed. It is the first self-stabilizing implementations of the LCM synchronization for environments where each robot can have limited visibility and lights.