Jiamou Liu

The Isomorphism Problem on Classes of Automatic Structures

Several new undecidability results on isomorphism problems for automatic structures are shown: (i) The isomorphism problem for automatic equivalence relations is \pi^0_1-complete. (ii) The isomorphism problem for automatic trees of height

The isomorphism problem on classes of automatic structures with transitive relations

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Community Detection Based on Graph Dynamical Systems with Asynchronous Runs

??? 2 is \pi^0_2

Tree-Automatic Well-Founded Trees

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Unary automatic graphs: an algorithmic perspective

–3-complete. (iii) The isomorphism problem for automatic linear orders is not arithmetical.

Integrating Networks of Equipotent Nodes

Automatic structures are finitely presented structures where the universe and all relations can be recognized by finite automata. It is known that the isomorphism problem for automatic structures is complete for Σ 1 , the first existential level of the analytical hierarchy. Positive results on ordinals and on Boolean algebras raised hope that the isomorphism problem is simpler for transitive relations. We prove that this is not the case. More precisely, this paper shows: (i) The isomorphism problem for automatic equivalence relations is complete for Π 1 (first universal level of the arithmetical hierarchy). (ii) The isomorphism problem for automatic trees of height n ≥ 2 is Π 2n−3-complete. (iii) The isomorphism problem for well-founded automatic order trees is recursively equivalent to true first-order arithmetic. (iv) The isomorphism problem for automatic order trees is Σ 1 -complete. (v) The isomorphism problem for automatic linear orders is Σ 1 -complete. We also obtain Π 1 -completeness of the elementary equivalence problem for several classes of automatic structures and Σ 1 -completeness of the isomorphism problem for linear orders consisting of a deterministic context-free language together with the lexicographic order. This solves several open questions of Ésik, Khoussainov, Nerode, Rubin, and Stephan.

Finite automata over structures

A community in a network is a group of nodes that are densely connected internally but sparsely connected externally. We propose a novel approach for detecting communities in networks based on graph dynamical systems (GDS), which are computation models for networks of interacting entities. We introduce the Propose-Select-Adjust framework - a GDS-based computation model for solving network problems, and demonstrate how this model may be used in community detection. The advantage of this approach is that computation is distributed to each node which asynchronously computes its own solution. This makes the method suitable for decentralised and dynamic networks.

Deciding the isomorphism problem in classes of unary automatic structures

We investigate tree-automatic well-founded trees. Using Delhommes decomposition technique for tree-automatic structures, we show that the (ordinal) rank of a tree-automatic well-founded tree is strictly below omega^omega. Moreover, we make a step towards proving that the ranks of tree-automatic well-founded partial orders are bounded by omega^omega^omega: we prove this bound for what we call upwards linear partial orders. As an application of our result, we show that the isomorphism problem for tree-automatic well-founded trees is complete for level Delta^0_{omega^omega} of the hyperarithmetical hierarchy with respect to Turing-reductions.

Computable Categoricity of Graphs with Finite Components

This paper studies infinite graphs produced from a natural unfolding operation applied to finite graphs. Graphs produced via such operations are of finite degree and can be described by finite automata over the unary alphabet. We investigate algorithmic properties of such unfolded graphs given their finite presentations. In particular, we ask whether a given node belongs to an infinite component, whether two given nodes in the graph are reachable from one another, and whether the graph is connected. We give polynomial time algorithms for each of these questions. Hence, we improve on previous work, in which nonelementary or non-uniform algorithms were found.

Dynamic Relationship Building: Exploitation Versus Exploration on a Social Network

When two social groups merge, members of both groups should socialize effectively into the merged new entity. In other words, interpersonal ties should be established between the groups to give members appropriate access to resource and information. Viewing a social group as a network, we investigate such integration from a computational perspective. In particular, we assume that the networks have equipotent nodes, which refers to the situation when every member has equal privilege. We introduce the network integration problem: Given two networks, set up links between them so that the integrated network has diameter no more than a fixed value. We propose a few heuristics for solving this problem, study their computational complexity and compare their performance using experimental analysis. The results show that our approach is a feasible way to solve the network integration problem by establishing a small number of edges.