Recursive Set Research Articles

On matrix-model approach to simplified Khovanov–Rozansky calculus

Wilson-loop averages in Chern–Simons theory (HOMFLY polynomials) can be evaluated in different ways – the most difficult, but most interesting of them is the hypercube calculus, the only one applicable to virtual knots and used also for categorification (higher-dimensional extension) of the theory. We continue the study of quantum dimensions, associated with hypercube vertices, in the drastically simplified version of this approach to knot polynomials. At q=1 the problem is reformulated in terms of fat (ribbon) graphs, where Seifert cycles play the role of vertices. Ward identities in associated matrix model provide a set of recursions between classical dimensions. For q≠1 most of these relations are broken (i.e. deformed in a still uncontrollable way), and only few are protected by Reidemeister invariance of Chern–Simons theory. Still they are helpful for systematic evaluation of entire series of quantum dimensions, including negative ones, which are relevant for virtual link diagrams. To illustrate the effectiveness of developed formalism we derive explicit expressions for the 2-cabled HOMFLY of virtual trefoil and virtual 3.2 knot, which involve respectively 12 and 14 intersections – far beyond any dreams with alternative methods. As a more conceptual application, we describe a relation between the genus of fat graph and Turaev genus of original link diagram, which is currently the most effective tool for the search of thin knots.

SAFE RECURSIVE SET FUNCTIONS

AbstractWe introduce the safe recursive set functions based on a Bellantoni–Cook style subclass of the primitive recursive set functions. We show that the functions computed by safe recursive set functions under a list encoding of finite strings by hereditarily finite sets are exactly the polynomial growth rate functions computed by alternating exponential time Turing machines with polynomially many alternations. We also show that the functions computed by safe recursive set functions under a more efficient binary tree encoding of finite strings by hereditarily finite sets are exactly the quasipolynomial growth rate functions computed by alternating quasipolynomial time Turing machines with polylogarithmic many alternations.We characterize the safe recursive set functions on arbitrary sets in definability-theoretic terms. In its strongest form, we show that a function on arbitrary sets is safe recursive if and only if it is uniformly definable in some polynomial level of a refinement of Jensen's J-hierarchy, relativized to the transitive closure of the function's arguments.We observe that safe recursive set functions on infinite binary strings are equivalent to functions computed by infinite-time Turing machines in time less than ωω. We also give a machine model for safe recursive set functions which is based on set-indexed parallel processors and the natural bound on running times.

On language equations with concatenation and various sets of Boolean operations

Systems of equations of the form X i = ϕ i (X 1 , ... , X n ), for 1 ⩽ i ⩽ n , in which the unknowns X i are formal languages, and the right-hand sides ϕ i may contain concatenation and union, are known for representing context-free grammars. If, instead of union only, another set of Boolean operations is used, the expressive power of such equations may change: for example, using both union and intersection leads to conjunctive grammars [A. Okhotin, J. Automata, Languages and Combinatorics 6 (2001) 519–535], whereas using all Boolean operations allows all recursive sets to be expressed by unique solutions [A. Okhotin, Decision problems for language equations with Boolean operations, Automata, Languages and Programming, ICALP 2003, Eindhoven, The Netherlands, 239–251]. This paper investigates the expressive power of such equations with any possible set of Boolean operations. It is determined that different sets of Boolean operations give rise to exactly seven families of formal languages: the recursive languages, the conjunctive languages, the context-free languages, a certain family incomparable with the context-free languages, as well as three subregular families.

Attribute Set Based Signature Secure in the Standard Model

We introduce attribute set based signature (ASBS), a new cryptographic primitive which organizes user attributes into a recursive set based structure such that dynamic constraints can be imposed on how those attributes may be combined to satisfy a signing policy. Compared with attribute based signature (ABS), ASBS is more flexible and efficient in managing user attributes and specifying signing policies. We present a practical construction of ASBS and prove its security in the standard model under three subgroup decision related assumptions. Its efficiency is comparable to that of the most efficient ABS scheme.

How much is a word? Predicting ease of articulation planning from apraxic speech error patterns

BackgroundAccording to intuitive concepts, ‘ease of articulation’ is influenced by factors like word length or the presence of consonant clusters in an utterance. Imaging studies of speech motor control use these factors to systematically tax the speech motor system. Evidence from apraxia of speech, a disorder supposed to result from speech motor planning impairment after lesions to speech motor centers in the left hemisphere, supports the relevance of these and other factors in disordered speech planning and the genesis of apraxic speech errors. Yet, there is no unified account of the structural properties rendering a word easy or difficult to pronounce. AimTo model the motor planning demands of word articulation by a nonlinear regression model trained to predict the likelihood of accurate word production in apraxia of speech. MethodWe used a tree-structure model in which vocal tract gestures are embedded in hierarchically nested prosodic domains to derive a recursive set of terms for the computation of the likelihood of accurate word production. The model was trained with accuracy data from a set of 136 words averaged over 66 samples from apraxic speakers. In a second step, the model coefficients were used to predict a test dataset of accuracy values for 96 new words, averaged over 120 samples produced by a different group of apraxic speakers. ResultsAccurate modeling of the first dataset was achieved in the training study (R2adj = .71). In the cross-validation, the test dataset was predicted with a high accuracy as well (R2adj = .67). The model shape, as reflected by the coefficient estimates, was consistent with current phonetic theories and with clinical evidence. In accordance with phonetic and psycholinguistic work, a strong influence of word stress on articulation errors was found. ConclusionsThe proposed model provides a unified and transparent account of the motor planning requirements of word articulation.

Resource-bounded martingales and computable Dowd-type generic sets

Martin-Löf defined algorithmic randomness, the use of martingales in this definition and several variants were explored by Schnorr, and Lutz introduced resource-bounded randomness. Ambos-Spies et al. have studied resource-bounded randomness under time-bounds and have shown that resource-bounded randomness implies resource-bounded genericity. While the genericity of Ambos-Spies is based on time-bound on finite-extension strategy, the genericity of Dowd, the main topic of this paper, is based on an analogy of the forcing theorem. For a positive integer r, an oracle D is r-generic in the sense of Dowd (r-Dowd) if the following holds: If a certain formula F on an exponential-sized portion of D is a tautology then a polynomial-sized subfunction of D forces F to be a tautology. Here, r is the number of occurrences of query symbols in F. The relationship between resource-bounded randomness and Dowd-type genericity has been not clear so far. We show that there exists a primitive recursive function t(n) with the following property: Every t(n)-random set is r-Dowd for each positive integer r. A proof is done by means of constructing resource-bounded martingales. In our former paper, we left an open problem whether there exists a primitive recursive set which is r-Dowd for every positive integer r. In our recent work, we give an affirmative answer to the problem. The main theorem of the present paper gives an alternative proof of the affirmative answer.

RECURSIVELY ENUMERABLE SETS AND WELL-ORDERING OF THEIR ENUMERATIONS

A b s t r a c t. We will introduce the special kind of the order relations into recursively enumerable sets and prove that they can be used to distinguish (albeit in a non-constructive way) between recursive and non-recursive sets.

A fault detection scheme for linear discrete-time systems with an integrated online performance evaluation

This paper is concerned with the design of the fault detection systems, into which a residual generation, evaluation and threshold are integrated, for linear discrete time-varying processes over a finite horizon. In the proposed design scheme, the residual generation is realised in the context of H∞ fault estimation with a prescribed attenuation level. This attenuation level is minimised by using the Krein-space linear estimation theory and, subsequently, an H∞ fault estimator with the minimum attenuation level is designed in terms of the solution to a set of Riccati-like recursions. For the residual evaluation and decision making purpose, the false alarm rate and fault detection rate indicators are introduced in the norm-based framework, which is integrated into the decision making procedure. For the online computations of the false alarm rate and fault detection rate indicators, further estimates delivered by the H∞ fault estimator are applied without additional (online) computations. By means of checking the change in the false alarm rate and fault detection rate indicators, a decision is then made. In this way, the fault detection performance can be significantly improved. Finally, one application example is exploited to demonstrate the application of the proposed integrated fault detection and performance evaluation schemes.

Computational completeness of equations over sets of natural numbers

Systems of finitely many equations of the form φ(X1,…,Xn)=ψ(X1,…,Xn) are considered, in which the unknowns Xi are sets of natural numbers, while the expressions φ,ψ may contain singleton constants and the operations of union and pairwise addition S+T={m+n|m∈S,n∈T}. It is shown that the family of sets representable by unique (least, greatest) solutions of such systems is exactly the family of recursive (r.e., co-r.e., respectively) sets of numbers. Basic decision problems for these systems are located in the arithmetical hierarchy. The same results are established for equations with addition and intersection.

A reducibility related to being hyperimmune-free

The main topic of the present work is the relation that a set X is strongly hyperimmune-free relative to Y. Here X is strongly hyperimmune-free relative to Y if and only if for every partial X-recursive function p there is a partial Y-recursive function q such that every a in the domain of p is also in the domain of q and satisfies p(a)<p(a), that is, p is majorised by q. For X being hyperimmune-free relative to Y, one also says that X is pmaj-reducible to Y(X≤pmajY). It is shown that between degrees not above the Halting problem the pmaj-reducibility coincides with the Turing reducibility and that therefore only recursive sets have a strongly hyperimmune-free Turing degree while those sets Turing above the Halting problem bound uncountably many sets with respect to the pmaj-relation. So pmaj-reduction is identical with Turing reduction on a class of sets of measure 1. Moreover, every pmaj-degree coincides with the corresponding Turing degree. The pmaj-degree of the Halting problem is definable with the pmaj-relation.

Using the “Chandrasekhar Recursions” for Likelihood Evaluation of DSGE Models

In likelihood-based estimation of linearized Dynamic Stochastic General Equilibrium (DSGE) models, the evaluation of the Kalman Filter dominates the running time of the entire algorithm. In this paper, we revisit a set of simple recursions known as the “Chandrasekhar Recursions” developed by Morf (Fast Algorithms for Multivariate Systems, Ph.D. thesis, Stanford University, 1974) and Morf et al. (IEEE Trans Autom Control 19:315–323, 1974) for evaluating the likelihood of a Linear Gaussian State Space System. We show that DSGE models are ideally suited for the use of these recursions, which work best when the number of states is much greater than the number of observables. In several examples, we show that there are substantial benefits to using the recursions, with likelihood evaluation up to five times faster. This gain is especially pronounced in light of the trivial implementation costs—no model modification is required. Moreover, the algorithm is complementary with other approaches.

The Complexity of Recursive Splittings of Random Sets

It is investigated how much information of a random set can be preserved if one splits the random set into two halves or, more generally, cuts out an infinite portion with an infinite recursive set. The two main results are the following ones: 1. Every high Turing degree contains a Schnorr random set such that T Z R for every infinite recursive set R. 2. For each set X there is a Martin-Lrandom set T X such that for all recursive sets R, either X T Z R or X T Z R.

A constraint selection technique for recursive set membership identification

Recursive approximation of the set of feasible parameters is a key problem in set membership identification. In this paper a new technique is presented, aimed at recursively computing an outer orthotopic approximation of polytopic feasible parameter sets. The main idea is to exploit the concept of binding constraints in linear programs, in order to select a limited number of constraints providing a good approximation of the exact feasible set. Numerical tests demonstrate that the proposed technique outperforms existing recursive approximation algorithms, with a limited increase of the required computational burden.

On a closed-form solution of the point kinetics equations with reactivity feedback of temperature

An analytical solution of the point kinetics equations to calculate time-dependent reactivity by the decomposition method has recently appeared in the literature. In this paper, we consider the neutron point kinetics equations together with temperature feedback effects. To this end, point kinetics is perturbed by a temperature equation that depends on the neutron density, obtaining a second-order non-linear ordinary differential equation. This equation is then solved by the decomposition method by expanding the neutron density in a series and expressing the non-linear terms by Adomian polynomials. Upon substituting these expansions into the non-linear ordinary equation, we construct a recursive set of linear problems that can be solved and resulting in an exact analytical representation for the solution. We also report numerical simulations and comparison against literature results.

A recursive set of invariants of the magnetotelluric impedance tensor

We present a recursive relation that produces pairs of rotation invariants of the magnetotelluric impedance tensor starting from the series and parallel impedances (S-P), which are readily computed from field data. Subsequent iterations combine the two impedances again as series and parallel impedances until, as the number of iterations tends to infinity, they converge to a single quantity. This single quantity is the well known impedance derived from the determinant of the magnetotelluric tensor. This particular way of splitting the determinant impedance into pairs of invariants, with progressive dilution of information, behaves as a natural regularizer when inverting the data. To tackle inversion in 2D we develop corresponding formulas for the partial derivatives with respect to the ground resistivities. We illustrate the performance of this set of invariant responses with synthetic data produced by a 3D model as well as with real soundings from the BC87 data set. In both cases we present 2D inversions of the S-P responses as they compare with the determinant and a couple of intermediate recursions. The models are necessarily related, with decreasing resolution as the iterations increase.

Accelerating calculations of RNA secondary structure partition functions using GPUs

BackgroundRNA performs many diverse functions in the cell in addition to its role as a messenger of genetic information. These functions depend on its ability to fold to a unique three-dimensional structure determined by the sequence. The conformation of RNA is in part determined by its secondary structure, or the particular set of contacts between pairs of complementary bases. Prediction of the secondary structure of RNA from its sequence is therefore of great interest, but can be computationally expensive. In this work we accelerate computations of base-pair probababilities using parallel graphics processing units (GPUs).ResultsCalculation of the probabilities of base pairs in RNA secondary structures using nearest-neighbor standard free energy change parameters has been implemented using CUDA to run on hardware with multiprocessor GPUs. A modified set of recursions was introduced, which reduces memory usage by about 25%. GPUs are fastest in single precision, and for some hardware, restricted to single precision. This may introduce significant roundoff error. However, deviations in base-pair probabilities calculated using single precision were found to be negligible compared to those resulting from shifting the nearest-neighbor parameters by a random amount of magnitude similar to their experimental uncertainties. For large sequences running on our particular hardware, the GPU implementation reduces execution time by a factor of close to 60 compared with an optimized serial implementation, and by a factor of 116 compared with the original code.ConclusionsUsing GPUs can greatly accelerate computation of RNA secondary structure partition functions, allowing calculation of base-pair probabilities for large sequences in a reasonable amount of time, with a negligible compromise in accuracy due to working in single precision. The source code is integrated into the RNAstructure software package and available for download at http://rna.urmc.rochester.edu.

Universality in one-dimensional hierarchical coalescence processes

Motivated by several models introduced in the physics literature to study the nonequilibrium coarsening dynamics of one-dimensional systems, we consider a large class of "hierarchical coalescence processes" (HCP). An HCP consists of an infinite sequence of coalescence processes ${\xi^{(n)}(\cdot)}_{n\ge1}$: each process occurs in a different "epoch" (indexed by $n$) and evolves for an infinite time, while the evolution in subsequent epochs are linked in such a way that the initial distribution of $\xi^{(n+1)}$ coincides with the final distribution of $\xi^{(n)}$. Inside each epoch the process, described by a suitable simple point process representing the boundaries between adjacent intervals (domains), evolves as follows. Only intervals whose length belongs to a certain epoch-dependent finite range are active, that is, they can incorporate their left or right neighboring interval with quite general rates. Inactive intervals cannot incorporate their neighbors and can increase their length only if they are incorporated by active neighbors. The activity ranges are such that after a merging step the newly produced interval always becomes inactive for that epoch but active for some future epoch. Without making any mean-field assumption we show that: (i) if the initial distribution describes a renewal process, then such a property is preserved at all later times and all future epochs; (ii) the distribution of certain rescaled variables, for example, the domain length, has a well-defined and universal limiting behavior as $n\to \infty$ independent of the details of the process (merging rates, activity ranges$,...$). This last result explains the universality in the limiting behavior of several very different physical systems (e.g., the East model of glassy dynamics or the Paste-all model) which was observed in several simulations and analyzed in many physics papers. The main idea to obtain the asymptotic result is to first write down a recursive set of nonlinear identities for the Laplace transforms of the relevant quantities on different epochs and then to solve it by means of a transformation which in some sense linearizes the system.

Model Checking Vector Addition Systems with one zero-test

We design a variation of the Karp-Miller algorithm to compute, in a forward manner, a finite representation of the cover (i.e., the downward closure of the reachability set) of a vector addition system with one zero-test. This algorithm yields decision procedures for several problems for these systems, open until now, such as place-boundedness or LTL model-checking. The proof techniques to handle the zero-test are based on two new notions of cover: the refined and the filtered cover. The refined cover is a hybrid between the reachability set and the classical cover. It inherits properties of the reachability set: equality of two refined covers is undecidable, even for usual Vector Addition Systems (with no zero-test), but the refined cover of a Vector Addition System is a recursive set. The second notion of cover, called the filtered cover, is the central tool of our algorithms. It inherits properties of the classical cover, and in particular, one can effectively compute a finite representation of this set, even for Vector Addition Systems with one zero-test.

Using the "Chandrasekhar Recursions" for Likelihood Evaluation of DSGE Models

In likelihood-based estimation of linearized dynamic stochastic general equilibrium (DSGE) models, the evaluation of the Kalman filter dominates the running time of the entire algorithm. In this paper, we revisit a set of simple recursions known as the &amp;amp;#x27;Chandrasekhar Recursions&amp;amp;#x27; developed by Morf (1974) and Morf, Sidhu, and Kalaith (1974) for evaluating the likelihood of a linear Gaussian state space system. We show that DSGE models are ideally suited for the use of these recursions, which work best when the number of states is much greater than the number of observables. In several examples, we show that there are substantial benefits to using the recursions, with likelihood evaluation up to five times faster. This gain is especially pronounced in light of the trivial implementation costs -- no model modification is required. Moreover, the algorithm is complementary with other approaches.

Using the 'Chandrasekhar Recursions' for Likelihood Evaluation of DSGE Models

In likelihood-based estimation of linearized Dynamic Stochastic General Equilibrium (DSGE) models, the evaluation of the Kalman Filter dominates the running time of the entire algorithm. In this paper, we revisit a set of simple recursions known as the Chandrasekhar Recursions developed by Morf (1974) and Morf, Sidhu, and Kalaith (1974) for evaluating the likelihood of a Linear Gaussian State Space System. We show that DSGE models are ideally suited for the use of these recursions, which work best when the number of states is much greater than the number of observables. In several examples, we show that there are substantial benefits to using the recursions, with likelihood evaluation up to five times faster. This gain is especially pronounced in light of the trivial implementation costs--no model modification is required. Moreover, the algorithm is complementary with other approaches.