Quantum, Stochastic, and Pseudo Stochastic Languages with Few States

Stochastic languages are the languages recognized by probabilistic finite automata (PFAs) with cutpoint over the field of real numbers. More general computational models over the same field such as generalized finite automata (GFAs) and quantum finite automata (QFAs) define the same class. In 1963, Rabin proved the set of stochastic languages to be uncountable presenting a single 2-state PFA over the binary alphabet recognizing uncountably many languages depending on the cutpoint. In this paper, we show the same result for unary stochastic languages. Namely, we exhibit a 2-state unary GFA, a 2-state unary QFA, and a family of 3-state unary PFAs recognizing uncountably many languages; all these numbers of states are optimal. After this, we completely characterize the class of languages recognized by 1-state GFAs, which is the only nontrivial class of languages recognized by 1-state automata. Finally, we consider the variations of PFAs, QFAs, and GFAs based on the notion of inclusive/exclusive cutpoint, and present some results on their expressive power.Comment: A new version with new results. Previous version: Arseny M. Shur, Abuzer Yakaryilmaz: Quantum, Stochastic, and Pseudo Stochastic Languages with Few States. UCNC 2014: 327-33

The Formal Dynamism of Categories: Stops vs. Fricatives, Primitivity vs. Simplicity

Minimalist Phonology (MP; Pöchtrager 2006) constructs its theory based on the phonological epistemological principle (Kaye 2001) and exposes the arbitrary nature of standard Government Phonology (sGP) and strict-CV (sCV), particularly with reference to their confusion of melody and structure. For Pöchtrager, these are crucially different, concluding that place of articulation is melodic (expressed with elements), while manner of articulation is structural. In this model, the heads (xN and xO) can license and incorporate the length of the other into their own interpretation, that is xN influences xO projections as well as its own and vice versa. This dynamism is an aspect of the whole framework and this paper in particular will show that stops and fricatives evidence a plasticity of category and that, although fricatives are simpler in structure, stops are the more primitive of the two. This will be achieved phonologically through simply unifying the environment of application of the licensing forces within Pöchtrager's otherwise sound onset structure. In doing so, we automatically make several predictions about language acquisition and typology and show how lenition in Qiang (Sino-Tibetan) can be more elegantly explained

Robust Grammatical Analysis for Spoken Dialogue Systems

We argue that grammatical analysis is a viable alternative to concept spotting for processing spoken input in a practical spoken dialogue system. We discuss the structure of the grammar, and a model for robust parsing which combines linguistic sources of information and statistical sources of information. We discuss test results suggesting that grammatical processing allows fast and accurate processing of spoken input.Comment: Accepted for JNL

On equations over sets of integers

Systems of equations with sets of integers as unknowns are considered. It is shown that the class of sets representable by unique solutions of equations using the operations of union and addition S+T=\makeset{m+n}{m \in S, \: n \in T} and with ultimately periodic constants is exactly the class of hyper-arithmetical sets. Equations using addition only can represent every hyper-arithmetical set under a simple encoding. All hyper-arithmetical sets can also be represented by equations over sets of natural numbers equipped with union, addition and subtraction S \dotminus T=\makeset{m-n}{m \in S, \: n \in T, \: m \geqslant n}. Testing whether a given system has a solution is $\Sigma^1_1$-complete for each model. These results, in particular, settle the expressive power of the most general types of language equations, as well as equations over subsets of free groups.Comment: 12 apges, 0 figure

On Varieties of Ordered Automata

The Eilenberg correspondence relates varieties of regular languages to pseudovarieties of finite monoids. Various modifications of this correspondence have been found with more general classes of regular languages on one hand and classes of more complex algebraic structures on the other hand. It is also possible to consider classes of automata instead of algebraic structures as a natural counterpart of classes of languages. Here we deal with the correspondence relating positive $\mathcal C$-varieties of languages to positive $\mathcal C$-varieties of ordered automata and we present various specific instances of this correspondence. These bring certain well-known results from a new perspective and also some new observations. Moreover, complexity aspects of the membership problem are discussed both in the particular examples and in a general setting

Syntactic Complexity of Prefix-, Suffix-, Bifix-, and Factor-Free Regular Languages

The syntactic complexity of a regular language is the cardinality of its syntactic semigroup. The syntactic complexity of a subclass of the class of regular languages is the maximal syntactic complexity of languages in that class, taken as a function of the state complexity $n$ of these languages. We study the syntactic complexity of prefix-, suffix-, bifix-, and factor-free regular languages. We prove that $n^{n-2}$ is a tight upper bound for prefix-free regular languages. We present properties of the syntactic semigroups of suffix-, bifix-, and factor-free regular languages, conjecture tight upper bounds on their size to be $(n-1)^{n-2}+(n-2)$, $(n-1)^{n-3} + (n-2)^{n-3} + (n-3)2^{n-3}$, and $(n-1)^{n-3} + (n-3)2^{n-3} + 1$, respectively, and exhibit languages with these syntactic complexities.Comment: 28 pages, 6 figures, 3 tables. An earlier version of this paper was presented in: M. Holzer, M. Kutrib, G. Pighizzini, eds., 13th Int. Workshop on Descriptional Complexity of Formal Systems, DCFS 2011, Vol. 6808 of LNCS, Springer, 2011, pp. 93-106. The current version contains improved bounds for suffix-free languages, new results about factor-free languages, and new results about reversa