Decomposing finite-valued two-way finite transducers

Sequential reactive systems include programs and devices that work with two streams of data and convert input streams of data into output streams. Such information processing systems include controllers, device drivers, computer interpreters. The result of the operation of such computing systems are infinite sequences of pairs of events of the request-response type, and, therefore, finite transducers are most often used as formal models for them. The behavior of transducers is represented by binary relations on infinite sequences, and so, traditional applied temporal logics (like HML, LTL, CTL, mu-calculus) are poorly suited as specification languages, since omega-languages, not binary relations on omega-words are used for interpretation of their formulae. To provide temporal logics with the ability to define properties of transformations that characterize the behavior ofreactive systems, we introduced new extensions ofthese logics, which have two distinctive features: 1) temporal operators are parameterized, and languages in the input alphabet oftransducers are used as parameters; 2) languages in the output alphabet oftransducers are used as basic predicates. Previously, we studied the expressive power ofnew extensions Reg-LTL and Reg-CTL ofthe well-known temporal logics oflinear and branching time LTL and CTL, in which it was allowed to use only regular languages for parameterization of temporal operators and basic predicates. We discovered that such a parameterization increases the expressive capabilities oftemporal logic, but preserves the decidability of the model checking problem. For the logics mentioned above, we have developed algorithms for the verification of finite transducers. At the next stage of our research on the new extensions of temporal logic designed for the specification and verification of sequential reactive systems, we studied the verification problem for these systems using the temporal logic Reg-CTL*, which is an extension ofthe Generalized Computational Tree Logics CTL*. In this paper we present an algorithm for checking the satisfiability of Reg-CTL* formulae on models of finite state transducers and show that this problem belongs to the complexity class ExpSpace.

The generalized theory of resonance scattering (GTRS) by an elastic spherical target in acoustics is extended to describe the arbitrary scattering of a finite beam using the addition theorem for the spherical wave functions of the first kind under a translation of the coordinate origin. The advantage of the proposed method over the standard discrete spherical harmonics transform previously used in the GTRS formalism is the computation of the off-axial beam-shape coefficients (BSCs) stemming from a closed-form partial-wave series expansion representing the axial BSCs in spherical coordinates. With this general method, the arbitrary acoustical scattering can be evaluated for any particle shape and size, whether the particle is partially or completely illuminated by the incident beam. Numerical examples for the axial and off-axial resonance scattering from an elastic sphere placed arbitrarily in the field of a finite circular piston transducer with uniform vibration are provided. Moreover, the 3-D resonance directivity patterns illustrate the theory and reveal some properties of the scattering. Numerous applications involving the scattering phenomenon in imaging, particle manipulation, and the characterization of multiphase flows can benefit from the present analysis because all physically realizable beams radiate acoustical waves from finite transducers as opposed to waves of infinite extent.

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.

In this paper finite-valued finite transducers are investigated in connection with their inner structure. The following results are shown: A finite-valued nondeterministic generalized sequential machine (NGSM) M can be effectively decomposed into finitely many single-valued NGSMs $M_1 , \ldots ,M_N $ such that the transduction realized by M is the union of the transductions realized by $M_1 , \ldots ,M_N $. Using this decomposition, the equivalence of finite-valued NGSMs is decidable in deterministic double exponential time. By reduction, both results can be generalized to normalized finite transducers.

The notion of a grammar as a relation between sets of strings is quite ancient. In the 4th century BCE, Pāṇini—Pynini’s eponym—proposed a sophisticated computational system (see Penn and Kiparsky 2012) in which Sanskrit words and phrases are constructed by repeated application of rewrite rules. These rules consist of a target (for example, in Aṣṭdhyāyā 6.1.77, [i, i:, u:,ṛ, ṛ.:]), a replacement (the corresponding semivowels), and a context in which it is applied (before another vowel). Priscian, writing in the 6th century CE, proposes a system of rewrite rules generating inflectional variants of Latin verbs (Matthews 1972). In the 19th century, historical linguists—particularly those now known as the Neogrammarians—employed rewrite rules to describe language change. For instance, Grimm’s Law correctly predicts that the initial *p in the Proto-Indo-European word meaning ‘father’ (cf. Latin pater) surfaces as [f] in Germanic (e.g., Old High German fater). In the 20th century, linguists adapted rewrite rules—some of which closely resemble historical sound changes—to sketch the morphophonemics of languages such as Menomini (Bloomfield 1939), Russian (Jakobson 1948), and Modern Hebrew (Chomsky 1951), and began to formalize rewrite systems. One particularly influential formal approach is the phonological rule notation popularized by Chomsky and Halle (1968). Subsequent work, reviewed below, establishes strong connections between the notation used by Chomsky and Halle and the regular languages,1 provides algorithms to compile rewrite rules into finite-state transducers, and employs them for speech and language technology applications. This chapter formalizes rewrite rules and shows how they are compiled into finite transducers, combined into rule cascades, and applied to input strings. Their use is then illustrated with applications in speech and language technology, including grapheme-to-phoneme conversion, morphological generation, and text normalization; the latter two applications are discussed in greater detail in later chapters.

Ciliates are protozoic organisms having two types of nuclei: micronuclei (that store the DNA) and macronuclei (that provide the RNA). Another feature that makes them distinct is the phenomenon of transformation of micronuclear genes into macronuclear genes during their sexual reproduction, called the Gene Assembly Process in Ciliates (GAPC). Parallel Communicating Finite Transducer Systems (PCFTS) are translating devices composed of several finite transducers that work in parallel in a synchronized manner. They communicate with each other by states and by the output tapes. In this paper we present a simulation of the molecular operations performed during GAPC with PCFTS.

The set ℤ β of β-integers is a Meyer set when β is a Pisot number, and thus there exists a finite set F such that ℤ β − ℤ β ⊂ ℤ β + F. We give finite automata describing the expansions of the elements of ℤ β and of ℤ β − ℤ β . We present a construction of such a finite set F, and a method to minimize the size of F. We obtain in this way a finite transducer that performs the decomposition of the elements of ℤ β − ℤ β as a sum belonging to ℤ β + F.

In this paper, we investigate the synthesis problem of terminating reactive systems from quantitative specifications. Such systems are modeled as finite transducers whose executions are represented as finite words in (I × O)^*, where I, O are finite sets of input and output symbols, respectively. A weighted specification S assigns a rational value (or -∞) to words in (I × O)^*, and we consider three kinds of objectives for synthesis, namely threshold objectives where the system’s executions are required to be above some given threshold, best-value and approximate objectives where the system is required to perform as best as it can by providing output symbols that yield the best value and e-best value respectively w.r.t. S. We establish a landscape of decidability results for these three objectives and weighted specifications with partial domain over finite words given by deterministic weighted automata equipped with sum, discounted-sum and average measures. The resulting objectives are not regular in general and we develop an infinite game framework to solve the corresponding synthesis problems, namely the class of (weighted) critical prefix games.

Transductions are binary relations of finite words. For rational transductions, i.e., transductions defined by finite transducers, the inclusion, equivalence and sequential uniformisation problems are known to be undecidable. In this paper, we investigate stronger variants of inclusion, equivalence and sequential uniformisation, based on a general notion of transducer resynchronisation, and show their decidability. We also investigate the classes of finite-valued rational transductions and deterministic rational transductions, which are known to have a decidable equivalence problem. We show that sequential uniformisation is also decidable for them.

The acousto-optic interaction in the Raman-Nath regime was used to study sound waves propagating in the principal planes of piezoelectric lithium niobate. Complete cuts of the inverse phase velocity surfaces were obtained for surface waves on reflection and for three bulk waves and one surface wave on transmission through the surface acoustic wave (SAW) propagation surface. It is also shown how simple diffraction can be used to measure the group velocity direction and the electro-mechanical coupling coefficient for surface waves. A modified schlieren technique was used to visualize bulk waves generated by a finite transducer and to observe shear waves reflecting and mode converting to longitudinal waves at an interface.

A transducer is infinite-valued if the maximal number of different outputs for an input string is not bounded by any constant. For one-way finite transducers, Weber gave sufficient and necessary conditions in terms of the structure of a transducer to characterize whether the transducer is infinite-valued or not. As crossing sequences in two-way automata often play similar roles as states in their one-way counterparts, we derive in this paper analogous criteria in the setting of crossing sequences to characterize the infinite-valuedness of two-way finite transducers.

When a sound beam in water is incident on a sediment at a sufficiently small grazing angle, the resulting wave field in the sediment is complicated, even for the case of flat, fluidlike sediments. The wave field in the sediment for a sound beam from a simple, unshaded, finite transducer has an evanescent component and diffractive components. These components can interfere to produce a series of nulls outside the spatial region dominated by the evanescent wave field. This situation has been experimentally simulated by using a combination of previously described immiscible liquids [Osterhoudt et al., J. Acoust. Soc. Am. 117, 2483 (2005)]. The spacing between the observed nulls is similar to that seen in a wave-number-integration-based synthesis (using OASES) for a related problem. An analysis of a dephasing distance for evanescent and algebraically decaying components [T .J. Matula and P. L. Marston, J. Acoust. Soc. Am. 97, 1389–1398 (1995)] explains the spacing of the nulls. [Work supported by ONR.]

A mixed prefix-postfix notation for representations of the constituent structures of the expressions of natural languages is proposed, which are of limited degree of center embedding if the original expressions are noncenter-embedding. The method of constructing these representations is applicable to expressions with center embedding, and results in representations which seem to reflect the ways in which people actually parse those expressions. Both the representations and their interpretations can be computed from the expressions from left to right by finite-state devices.The class of acceptable expressions of a natural language L all manifest no more than a small, fixed, finite degree n of center embedding. From this observation, it follows that the ability of human beings to parse the expressions of L can be modeled by a finite transducer that associates with the acceptable expressions of L representations of the structural descriptions of those expressions. This paper considers some initial steps in the construction of such a model. The first step is to determine a method of representating the class of constituent structures of the expressions of L without center embedding in such a way that the members of that class themselves have no more than a small fixed finite degree of center embedding. Given a grammar that directly generates that class of constituent structures, it is not difficult to construct a deterministic finite-state transducer (parser) that assigns the appropriate members of that class to the noncenter-embedded expressions of L from left to right. The second step is to extend the method so that it is capable of representing the class of constituent structures of expressions of L with no more than degree n of center embedding in a manner which appears to accord with the way in which human beings actually parse those sentences. Given certain reasonable assumptions about the character of the rules of grammar of natural languages, we show how this step can also be taken.

We obtain a sharp upper bound for the nondeterministic state complexity of the application of a finite transducer to a regular language.

The interaction of ultrasound waves with human tissues is complex, and it is inevitable that simplifying physical and mathematical modelling must be invoked to aid our understanding of the processes involved. For pulse-echo methods, finite (bounded) transducers are used to generate three-dimensional (3-D) pulses which travel into the 3-D inhomogeneous medium that is tissue. For soft tissues, it is reasonable to disregard shear wave propagation, but attenuation of the ultrasound pulse is marked, and must be included. Moreover, since attenuation rises in a near-linear fashion with frequency, the shape of the pulse changes as it traverses tissue. Scattered waves are generated by tissue inhomogeneities (density, elasticity and absorption), and are coherently detected.