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

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

On Knot Polynomials of Annular Surfaces and their Boundary Links

Stoimenow and Kidwell asked the following question: Let $K$ be a non-trivial knot, and let $W(K)$ be a Whitehead double of $K$. Let $F(a,z)$ be the Kauffman polynomial and $P(v,z)$ the skein polynomial. Is then always $\max\deg_z P_{W(K)} - 1 = 2 \max\deg_z F_K$? Here this question is rephrased in more general terms as a conjectured relation between the maximum $z$-degrees of the Kauffman polynomial of an annular surface $A$ on the one hand, and the Rudolph polynomial on the other hand, the latter being defined as a certain M\"obius transform of the skein polynomial of the boundary link $\partial A$. That relation is shown to hold for algebraic alternating links, thus simultaneously solving the conjecture by Kidwell and Stoimenow and a related conjecture by Tripp for this class of links. Also, in spite of the heavyweight definition of the Rudolph polynomial $\{K\}$ of a link $K$, the remarkably simple formula \{\bigcirc\}\{L#M\}=\{L\}\{M\} for link composition is established. This last result can be used to reduce the conjecture in question to the case of prime links.Comment: Version 4: revision as of October 10, 2008. Fixed several errors and inaccuracies. 11 pages, 1 figure. To appear in Mathematical Proceedings of the Cambridge Philosophical Societ

Optimal Regular Expressions for Palindromes of Given Length

The language P_n (P?_n, respectively) consists of all words that are palindromes of length 2n (2n-1, respectively) over a fixed binary alphabet. We construct a regular expression that specifies P_n (P?_n, respectively) of alphabetic width 4? 2?-4 (3? 2?-4, respectively) and show that this is optimal, that is, the expression has minimum alphabetic width among all expressions that describe P_n (P?_n, respectively). To this end we give optimal expressions for the first k palindromes in lexicographic order of odd and even length, proving that the optimal bound is 2n+4(k-1)-2 S?(k-1) in case of odd length and 2n+3(k-1)-2 S?(k-1)-1 for even length, respectively. Here S?(n) refers to the Hamming weight function, which denotes the number of ones in the binary expansion of the number n

Elementary steps of the catalytic NO_x reduction with NH₃: Cluster studies on reaction paths and energetics at vanadium oxide substrate

We consider different reaction scenarios of the selective catalytic reduction (SCR) of NO in the presence of ammonia at perfect as well as reduced vanadium oxide surfaces modeled by V2O5(010) without and with oxygen vacancies. Geometric and energetic details as well as reaction paths are evaluated using extended cluster models together with density-functional theory. Based on earlier work of adsorption, diffusion, and reaction of the different surface species participating in the SCR we confirm that at Brønsted acid sites (i.e., OH groups) of the perfect oxide surface nitrosamide, NH2NO, forms a stable intermediate. Here adsorption of NH3 results in NH4 surface species which reacts with gas phase NO to produce the intermediate. Nitrosamide is also found as intermediate of the SCR near Lewis acid sites of the reduced oxide surface (i.e., near oxygen vacancies). However, here the adsorbed NH3 species is dehydrogenated to surface NH2 before it reacts with gas phase NO to produce the intermediate. The calculations suggest that reaction barriers for the SCR are overall higher near Brønsted acid sites of the perfect surface compared with Lewis acid sites of the reduced surface, examined for the first time in this work. The theoretical results are consistent with experimental findings and confirm the importance of surface reduction for the SCR proces