Digraph Complexity Measures and Applications in Formal Language Theory

We investigate structural complexity measures on digraphs, in particular the cycle rank. This concept is intimately related to a classical topic in formal language theory, namely the star height of regular languages. We explore this connection, and obtain several new algorithmic insights regarding both cycle rank and star height. Among other results, we show that computing the cycle rank is NP-complete, even for sparse digraphs of maximum outdegree 2. Notwithstanding, we provide both a polynomial-time approximation algorithm and an exponential-time exact algorithm for this problem. The former algorithm yields an O((log n)^(3/2))- approximation in polynomial time, whereas the latter yields the optimum solution, and runs in time and space O*(1.9129^n) on digraphs of maximum outdegree at most two. Regarding the star height problem, we identify a subclass of the regular languages for which we can precisely determine the computational complexity of the star height problem. Namely, the star height problem for bideterministic languages is NP-complete, and this holds already for binary alphabets. Then we translate the algorithmic results concerning cycle rank to the bideterministic star height problem, thus giving a polynomial-time approximation as well as a reasonably fast exact exponential algorithm for bideterministic star height.Comment: 19 pages, 1 figur

Topological Optimization of the Evaluation of Finite Element Matrices

We present a topological framework for finding low-flop algorithms for evaluating element stiffness matrices associated with multilinear forms for finite element methods posed over straight-sided affine domains. This framework relies on phrasing the computation on each element as the contraction of each collection of reference element tensors with an element-specific geometric tensor. We then present a new concept of complexity-reducing relations that serve as distance relations between these reference element tensors. This notion sets up a graph-theoretic context in which we may find an optimized algorithm by computing a minimum spanning tree. We present experimental results for some common multilinear forms showing significant reductions in operation count and also discuss some efficient algorithms for building the graph we use for the optimization

From Finite Automata to Regular Expressions and Back--A Summary on Descriptional Complexity

The equivalence of finite automata and regular expressions dates back to the seminal paper of Kleene on events in nerve nets and finite automata from 1956. In the present paper we tour a fragment of the literature and summarize results on upper and lower bounds on the conversion of finite automata to regular expressions and vice versa. We also briefly recall the known bounds for the removal of spontaneous transitions (epsilon-transitions) on non-epsilon-free nondeterministic devices. Moreover, we report on recent results on the average case descriptional complexity bounds for the conversion of regular expressions to finite automata and brand new developments on the state elimination algorithm that converts finite automata to regular expressions.Comment: In Proceedings AFL 2014, arXiv:1405.527

Ackermann Encoding, Bisimulations, and OBDDs

We propose an alternative way to represent graphs via OBDDs based on the observation that a partition of the graph nodes allows sharing among the employed OBDDs. In the second part of the paper we present a method to compute at the same time the quotient w.r.t. the maximum bisimulation and the OBDD representation of a given graph. The proposed computation is based on an OBDD-rewriting of the notion of Ackermann encoding of hereditarily finite sets into natural numbers.Comment: To appear on 'Theory and Practice of Logic Programming

A topological approach to neural complexity

Considerable efforts in modern statistical physics is devoted to the study of networked systems. One of the most important example of them is the brain, which creates and continuously develops complex networks of correlated dynamics. An important quantity which captures fundamental aspects of brain network organization is the neural complexity C(X)introduced by Tononi et al. This work addresses the dependence of this measure on the topological features of a network in the case of gaussian stationary process. Both anlytical and numerical results show that the degree of complexity has a clear and simple meaning from a topological point of view. Moreover the analytical result offers a straightforward algorithm to compute the complexity than the standard one.Comment: 6 pages, 4 figure

Simulating quantum computation by contracting tensor networks

The treewidth of a graph is a useful combinatorial measure of how close the graph is to a tree. We prove that a quantum circuit with $T$ gates whose underlying graph has treewidth $d$ can be simulated deterministically in $T^{O(1)}\exp[O(d)]$ time, which, in particular, is polynomial in $T$ if $d=O(\log T)$. Among many implications, we show efficient simulations for log-depth circuits whose gates apply to nearby qubits only, a natural constraint satisfied by most physical implementations. We also show that one-way quantum computation of Raussendorf and Briegel (Physical Review Letters, 86:5188--5191, 2001), a universal quantum computation scheme with promising physical implementations, can be efficiently simulated by a randomized algorithm if its quantum resource is derived from a small-treewidth graph.Comment: 7 figure