Coverability, Termination, and Finiteness in Recursive Petri Nets

In the early two-thousands, Recursive Petri nets have been introduced in order to model distributed planning of multi-agent systems for which counters and recursivity were necessary. Although Recursive Petri nets strictly extend Petri nets and context-free grammars, most of the usual problems (reachability, coverability, finiteness, boundedness and termination) were known to be solvable by using non-primitive recursive algorithms. For almost all other extended Petri nets models containing a stack, the complexity of coverability and termination are unknown or strictly larger than EXPSPACE. In contrast, we establish here that for Recursive Petri nets, the coverability, termination, boundedness and finiteness problems are EXPSPACE-complete as for Petri nets. From an expressiveness point of view, we show that coverability languages of Recursive Petri nets strictly include the union of coverability languages of Petri nets and context-free languages. Thus we get a more powerful model than Petri net for free.

Permutations Avoiding Certain Partially-Ordered Patterns

A permutation $\pi$ contains a pattern $\sigma$ if and only if there is a subsequence in $\pi$ with its letters in the same relative order as those in $\sigma$. Partially ordered patterns (POPs) provide a convenient way to denote patterns in which the relative order of some of the letters does not matter. This paper elucidates connections between the avoidance sets of a few POPs with other combinatorial objects, directly answering five open questions posed by Gao and Kitaev in 2019. This was done by thoroughly analysing the avoidance sets and developing recursive algorithms to derive these sets and their corresponding combinatorial objects in parallel, which yielded natural bijections. We also analysed an avoidance set whose simple permutations are enumerated by the Fibonacci numbers and derived an algorithm to obtain them recursively.

Patterns of the Expanding City: An Algorithmic Interpretation of Otto Wagner’s Work

Central Europe witnessed an urban boom at the beginning of the 20th century. By that time, the leading state of the area was Austria-Hungary, with Vienna as its capital. Before the First World War, even larger expansion of the cities was predictable. Otto Wagner, a leading architect of the empire and an expert in urban planning and architectural theory, published his vision about the future of the evolution of cities in 1911. In this book, he formulates clear rules about how a city should sustainably expand in a controlled manner. In this article, these rules of the inherited patterns are systematised and turned into recursive algorithms to simulate the urban growth controlled by them and the resulting patterns. The algorithms are tested on 1911 Vienna and, as comparison, on 2021 Miskolc, a medium-sized city in Hungary with different geographic surroundings. In the article, the resulting patterns are presented in 2D and 3D.

Recursive Algorithm and its Practice in C Language Online Course Teaching

The recursive algorithm has two core issues which are the design of recursive parameter lists and exit condition. We summarized the types and characteristics of recursive algorithms, and extracted four types of representative recursive algorithms with different levels of difficulty. Pseudocodes of these algorithms are given and the core issues complexity of these algorithms is compared. The relatively complex and representative recursive algorithm of the maze path finding is described in detail. The description includes that the maze is expressed mathematically, using numbers from 0 to 9 to represent the path cost, using * to represent walls or obstacles, and abstracting the maze problem as a square maze represented by numbers and *. The maze path finding recursive algorithm has five steps which are the storage structure of the maze numbers and *, the consideration of the two core problems of the recursive algorithm, the recursive call in the four directions of the middle point and the printing of the maze path. The screenshot of the running result in C language is showed. Students are required to implement the maze path finding recursive algorithm in C language. Because the maze path finding recursive algorithm is interesting and challenging, it can stimulate students' enthusiasm and initiative in learning. In the current situation of online course teaching during the epidemic, considering the demand for C language programming ability, combining with the characteristics of higher vocational students and the difficulty of the maze path finding recursive algorithm, we designed and practiced the C Language online course teaching mode led by the recursive algorithms. The every step of the teaching mode is described in detail. From the feedback of students’ evaluation of teaching and teacher’s evaluation of learning, this teaching mode is praised by teachers and students.

Recursive Algorithms of Maximum Entropy Thresholding on Circular Histogram

Circular histogram thresholding is a novel color image segmentation method, which makes full use of the hue component color information of the image, so that the desired target can be better separated from the background. Maximum entropy thresholding on circular histogram is one of the exist circular histogram thresholding methods. However, this method needs to search for a pair of optimal thresholds on the circular histogram of two-class thresholding in an exhaustive way, and its running time is even longer than that of the existing circular histogram thresholding based on the Otsu criteria, so the segmentation efficiency is extremely low, and the real-time application cannot be realized. In order to solve this problem, a recursive algorithm of maximum entropy thresholding on circular histogram is proposed. Moreover, the recursive algorithm is extended to the case of multiclass thresholding. A large number of experimental results show that the proposed recursive algorithms are more efficient than brute force and the existing circular histogram thresholding based on the Otsu criteria.

Algorithms for finding the maximum clique based on continuous time quantum walks

In this work, the application of continuous time quantum walks (CTQW) to the Maximum Clique (MC) problem was studied. Performing CTQW on graphs can generate distinct periodic probability amplitudes for different vertices. We found that the intensities of the probability amplitudes at some frequencies imply the clique structure of special kinds of graphs. Recursive algorithms with time complexity O(N^6) in classical computers were proposed to determine the maximum clique. We have experimented on random graphs where each edge exists with different probabilities. Although counter examples were not found for random graphs, whether these algorithms are universal is beyond the scope of this work.

A Behavioural Theory of Recursive Algorithms

“What is an algorithm?” is a fundamental question of computer science. Gurevich’s behavioural theory of sequential algorithms (aka the sequential ASM thesis) gives a partial answer by defining (non-deterministic) sequential algorithms axiomatically, without referring to a particular machine model or programming language, and showing that they are captured by (nondeterministic) sequential Abstract State Machines (nd-seq ASMs). However, recursive algorithms such as mergesort are not covered by this theory, as has been pointed out by Moschovakis, who had independently developed a different framework to mathematically characterize the concept of (in particular recursive) algorithm. In this article we propose an axiomatic definition of the notion of sequential recursive algorithm which extends Gurevich’s axioms for sequential algorithms by a Recursion Postulate and allows us to prove that sequential recursive algorithms are captured by recursive Abstract State Machines, an extension of nd-seq ASMs by a CALL rule. Applying this recursive ASM thesis yields a characterization of sequential recursive algorithms as finitely composed concurrent algorithms all of whose concurrent runs are partial-order runs.