The Isomorphism Problem for FST Injection Structures

We present a flexible rule compiler developed for a text-to-speech (TTS) system. The compiler converts a set of rules into a finite-state transducer (FST). The input and output of the FST are subject to parameterization, so that the system can be applied to strings and sequences of feature-structures. The resulting transducer is guaranteed to realize a function (as opposed to a relation), and therefore can be implemented as a deterministic device (either a deterministic FST or a bimachine).

Regular languages are languages recognised by finite automata; automatic structures are a generalisation of regular languages where one also uses automatic relations (which are relations recognised by synchronous finite automata) and automatic functions (which are functions whose graph is an automatic relation). Functions and relations first-order definable from other automatic functions and relations are again automatic. Automatic functions coincide with the functions computed by position-faithful one-tape Turing machines in linear time. This survey addresses recent results and open questions on topics related to automatic structures: How difficult is the isomorphism problem for various types of automatic structures? Which groups are automatic? When are automatic groups Abelian or orderable? How can one overcome some of the limitations to represent rings and fields by weakening the automaticity requirements of a structure?

An automatic structure \(\mathcal{A}\) is one whose domain A and atomic relations are finite automaton (FA) recognisable. A structure isomorphic to \(\mathcal{A}\) is called automatically presentable. Suppose R is an FA recognisable relation on A. This paper concerns questions of the following type. For which automatic presentations of \(\mathcal{A}\) is (the image of) R also FA recognisable? To this end we say that a relation R is intrinsically regular in a structure \(\mathcal{A}\) if it is FA recognisable in every automatic presentation of the structure. For example, in every automatic structure all relations definable in first order logic are intrinsically regular. We characterise the intrinsically regular relations of some automatic fragments of arithmetic in the first order logic extended with quantifiers ∃ ∞ interpreted as ‘there exists infinitely many’, and ∃ (i) interpreted as ‘there exists a multiple of i many’.

It is well known that reversible finite automata do not accept all regular languages, that reversible pushdown automata do not accept all deterministic context-free languages, and that reversible queue automata are less powerful than deterministic real-time queue automata. It is of significant interest from both a practical and theoretical point of view to close these gaps. We here extend these reversible models by a preprocessing unit which is basically a reversible injective and length-preserving finite state transducer. It turns out that preprocessing the input using such weak devices increases the computational power of reversible deterministic finite automata to the acceptance of all regular languages, whereas for reversible pushdown automata the accepted family of languages lies strictly in between the reversible deterministic context-free languages and the real-time deterministic context-free languages. For reversible queue automata the preprocessing of the input leads to machines that are stronger than real-time reversible queue automata, but less powerful than real-time deterministic (irreversible) queue automata. Moreover, it is shown that the computational power of all three types of machines is not changed by allowing the preprocessing finite state transducer to work irreversibly. Finally, we examine the closure properties of the family of languages accepted by such machines.

Normality and automata

Tree regular model checking: A simulation-based approach

The application of statistical methods to natural language processing has been remarkably successful over the past two decades. The wide availability of text and speech corpora has played a critical role in their success since, as for all learning techniques, these methods rely heavily on data. Many of the components of complex natural language processing systems, for example, text normalizers, morphological or phonological analyzers, part-of-speech taggers, grammars or language models, pronunciation models, context-dependency models, acoustic Hidden-Markov Models (HMMs), are statistical models derived from large data sets using modern learning techniques. These models are often given as weighted automata or weighted finite-state transducers either directly or as a result of the approximation of more complex models.

“Regular model checking” is the name of a family of techniques for analyzing infinite-state systems in which states are represented by words or trees, sets of states by finite automata on these objects, and transitions by finite automata operating on pairs of state encodings, i.e. finite-state transducers. In this context, the central problem is then to compute the iterative closure of a finite-state transducer. This paper addresses the use of regular model-checking like techniques for systems whose states are represented by infinite (omega) words. Its main motivation is to show the feasibility and usefulness of this approach through a combination of the necessary theoretical developments, implementation, and experimentation. The iteration technique that is used is adapted from recent work of the authors on the iteration of finite-word transducers. It proceeds by comparing successive elements of a sequence of approximations of the iteration, detecting an “increment” that is added to move from one approximation to the next, and extrapolating the sequence by allowing arbitrary repetitions of this increment. By restricting oneself to weak deterministic Buchi automata, and using a number of implementation optimizations, examples of significant size can be handled. The proposed transducer iteration technique can just as well be exploited to compute the closure of a given set of states by the transducer iteration, which has proven to be a very effective way of using the technique. Examples such as a leaking gas burner in which time is modeled by real variables have been handled completely within the automata-theoretic setting.

Regular sets over extended tree structures

This paper proposes a layered Finite State Transducer (FST) framework integrating hierarchical supra-lexical linguistic knowledge into speech recognition based on shallow parsing. The shallow parsing grammar is derived directly from the full fledged grammar for natural language understanding, and augmented with top-level n-gram probabilities and phrase-level context-dependent probabilities, which is beyond the standard context-free grammar (CFG) formalism. Such a shallow parsing approach can help balance sufficient grammar coverage and tight structure constraints. The context-dependent probabilistic shallow parsing model is represented by layered FSTs, which can be integrated with speech recognition seamlessly to impose early phrase-level structural constraints consistent with natural language understanding. It is shown that in the JUPITER [1] weather information domain, the shallow parsing model achieves lower recognition word error rates, compared to a regular class n-gram model with the same order. However, we find that, with a higher order top-level n-gram model, pre-composition and optimization of the FSTs are highly restricted by the computational resources available. Given the potential of such models, it may be worth pursing an incremental approximation strategy [2], which includes part of the linguistic model FST in early optimization, while introducing the complete model through dynamic composition.

Finite state automata are of fundamental importance in theory as well as applications in computer science. This dissertation presents several parallel algorithms for processing of them, including minimizing the number of states for a given finite state automaton, finding homing sequences for a given finite state automaton, converting a regular expression into an $\\varepsilon$-free nondeterministic finite automaton. Implementation of some algorithms, as well as their experimental results, are also given. For minimization of deterministic finite automata, we present a parallel algorithm and discuss its implementation on a practical parallel machine using data parallel and message passing paradigms. Its time complexity on a p processor EREW PRAM $(p\\le n)$ for inputs of size n is $O({n\\log\\sp2n\\over p}+\\log n\\ \\log p)$ uniformly on almost all instances. The work done by our algorithm is thus within a factor of $O(\\log n)$ of the best known sequential algorithm. The space used by our algorithm is linear in the input size. The actual resource requirements of our implementations are consistent with these estimates. For finding homing sequences of deterministic finite automata, we present both deterministic and randomized algorithms. Two randomized algorithms of time complexity $O(log\\sp2n)$ are given. One of the randomized algorithms produces a homing sequence of length $O(n\\ log\\sp2n)$ for almost all DFA's with n states. The deterministic algorithm is based on a well known sequential algorithm. The implementation of the deterministic parallel algorithm shows that it can handle randomly generated instances with millions of states, and all DFA's with thousands of states using currently available parallel machines. Finally, we present an efficient parallel algorithm for converting a regular expression into an $\\varepsilon$-free nondeterministic finite automaton. The resulting automaton is natural representation of a regular expression and has some important and practical uses in the document processing area. The time complexity of the algorithm is $O(\\log n)$ and the number of processors used is $O(n\\sp2).$

Finite automata and finite transducers are used in a wide range of applications in software engineering, from regular expressions to specification languages. We extend these classic objects with symbolic alphabets represented as parametric theories. Admitting potentially infinite alphabets makes this representation strictly more general and succinct than classical finite transducers and automata over strings. Despite this, the main operations, including composition, checking that a transducer is single-valued, and equivalence checking for single-valued symbolic finite transducers are effective given a decision procedure for the background theory. We provide novel algorithms for these operations and extend composition to symbolic transducers augmented with registers. Our base algorithms are unusual in that they are nonconstructive, therefore, we also supply a separate model generation algorithm that can quickly find counterexamples in the case two symbolic finite transducers are not equivalent. The algorithms give rise to a complete decidable algebra of symbolic transducers. Unlike previous work, we do not need any syntactic restriction of the formulas on the transitions, only a decision procedure. In practice we leverage recent advances in satisfiability modulo theory (SMT) solvers. We demonstrate our techniques on four case studies, covering a wide range of applications. Our techniques can synthesize string pre-images in excess of 8,000 bytes in roughly a minute, and we find that our new encodings significantly outperform previous techniques in succinctness and speed of analysis.

We consider infinite graphs that are generated by ground tree (or term) rewriting systems. the vertices of these graphs are trees. thus, with a finite tree automaton one can represent a regular set of vertices. It is shown that for a regular set T of vertices the set of vertices from where one can reach (respectively, infinitely often reach) the set T is again regular. Furthermore it is shown that the problems, given a tree t and a regular set T, whether all paths starting in t eventually (respectively, infinitely often) reach T, are undecidable. We then define a logic which is in some sense a maximal fragment of temporal logic with a decidable model-checking problem for the class of ground tree rewriting graphs.

India is a country with a diversity of languages. Malayalam belongs to the Dravidian family of languages with literary tradition. In 2013, it is designated as one of the classical languages in India and is the official language of Kerala, the south-western state in India. In Kerala, people give a high priority for education. In most of the schools run by private trusts and individuals in Kerala, English is the medium of instruction. But the State Government schools use both English and Malayalam as language of instruction. The structure of spoken English has influenced the structure of spoken Malayalam in certain aspects. Due to this influence, Malayalam has assumed certain new dimensions on account of the impact of English grammar. It is also seen that Malayalam grammar is taught using the linguistic terms in English. For example, the words noun and verb are more commonly used instead of the corresponding Malayalam words and tafto even while teaching Malayalam grammar. Most students do not like grammar. This has resulted in the elimination of grammar from today's classrooms. In this paper, we have presented the details of how Finite State Transducer is utilized to implement a Malayalam morphological generator. The software helps the students to discover the rules of Malayalam grammar. The children of keralites, who are settled abroad can learn the rules of their mother tongue and know how native speakers speak in real Malayalam conversation. They can study structures of word, like morphemes. Morph generator provides different types of inflections to the root word according to the provided morpholexical information. In the present version of the software, two modules are included. In learning module, the user can study the grammar. Exercise module, conducts test to know the level of knowledge the user learned.

Multiple-entry finite automata