Recursive Codes Research Articles

Recursion vs. iteration: An empirical study of comprehension

An extensive study involving three test groups over a period of three different years was performed to determine differences between comprehension of recursive and iterative code constructs. This study involved computer science students enrolled in six different data structure classes using PASCAL and obtained 275 responses, which were statistically analyzed. Using a χ 2 test, one task—searching a linked list for a given value—showed a statistically significant ( p = 0.006) difference in comprehension favoring recursion. Another task—making a copy of a linked list—favored iteration, but not with statistical significance. Among responses with correct comprehension, a comparison of means test was performed on the comprehension times among the latter two groups in the study. There was a faster rate of comprehension of the recursive code for both of the tasks ( p = 0.096 for the search and p = 0.045 for the copy). A two-way analysis of variance suggested a tendency toward an interaction effect between task and construct in terms of comprehension time.

The expressiveness of locally stratified programs

This paper completes an investigation of the logical expressibility of finite, locally stratified, general logic programs. We show that every hyperarithmetic set can be defined by a suitably chosen locally stratified logic program (as a set of values of a predicate over its perfect model). This is an optimal result, since the perfect model of a locally stratified program is itself an implicitly definable hyperarithmetic set (under a recursive coding of the Herbrand base); hence, to obtain all hyperarithmetic sets requires something new, in this case selecting one predicate from the model. We find that the expressive power of programs does not increase when one considers the programs which have a unique stable model or a total well-founded model. This shows that all these classes of structures (perfect models of logically stratified logic programs, well-founded models which turn out to be total, and stable models of programs possessing a unique stable model) are all closely connected with Kleene's hyperarithmetical hierarchy. Thus, for general logic programming, negation with respect to two-valued logic is related to the hyperarithmetic hierarchy in the same way as Horn logic is to the class of recursively enumerable sets. In particular, a set is definable in the well-founded semantics by a programP whose well-founded partial model is total iff it is hyperarithmetic.

Role of recursive convolutional codes in turbocodes

Recently proposed 'turbo codes' yield very good performance in combination with simple iterative decoding strategies. They can be thought of as parallel concatenated coding schemes employing systematic convolutional codes as constituent codes linked by an interleaver. A distinctive feature of turbo codes is that they use, as constituent codes. Recursive convolutional codes. Through the application of simple analytical considerations reinforced by a complete example, we show that the presence of a recursive code is crucial to the performance of the concatenated code.

The Stable Models of a Predicate Logic Program

We study the family of stable models of finite and recursive predicate logic programs. We show that the family of stable models of a recursive predicate logic program is, up to a recursive coding, a ∏ 1 0 class (i.e. an effectively closed set) and, vice versa, that each ∏ 1 0 class is, up to a recursive coding, the family of stable models of a finite predicate logic program. Since the structure of the Turing degrees of elements of ∏ 1 0 classes has been extensively studied, these coding results automatically imply many results about the degrees of stable models of finite predicate logic programs. For example, there exists a finite predicate logic program which has a stable model but which has no stable model which is hyperarithmetic and the existence problem for stable models of finite predicate logic programs is a ∑ 1 1 -complete problem.

Does rule recursion make melodies easier to reproduce? If not, what does?

Two experiments examined the effects of variations in melodic rule structure and rhythm upon the ability of musically sophisticated listeners to reproduce auditory patterns. These experiments were designed to evaluate different theoretical perspectives on auditory pattern perception and the role of rule structure in perceiving and remembering. Predictions of a coding model, which emphasizes the impact of rule recursion, were contrasted with those of an accent model, which emphasizes the relative timing of melodic and temporal accents. Effects of recursive (hierarchical) and nonrecursive (linear) rule arrangements were studied in contexts where pattern contour differences were controlled for. Rhythm was also varied. Measured pauses were inserted between certain tones to make a rhythm compatible or incompatible with melodic rule phrases. Experiment 1 showed that pattern simplicity was determined not by rule recursive codes, but by the number and timing of contour changes and melodic rule breaks. Experiment 2 examined additional effects of rhythm on listeners' response to rule recursion and melodic phrasing in melodies of equivalent contour. Although modest effects of rule recursion appeared, these effects were again outweighed by large performance differences due to the relative timing of changes in contour and melodic rule structure. Implications of the accent model for dynamic attending are discussed in the context of a new proposal involving temporal phasing of accents.

Recursive Block Coding--A New Approach to Transform Coding

The concept of fast KL transform coding introduced earlier [7], [8] for first-order Markov processes and certain random fields has been extended to higher order autoregressive (AR) sequences and practical images yielding what we call recursive block coding (RBC) algorithms. In general, the rate-distortion performance for these algorithms is significantly superior to that of the conventional block KL transform algorithm. Moreover, these algorithms permit the use of small size transforms, thereby removing the need for fast transforms and making the hardware implementation of such coders more appealing. This improved performance has been verified for practical image data and results in suppression of the block-boundary effect commonly observed in traditional transform coding techniques. This is illustrated by comparing RBC with cosine transform coding using both one- and twodimensional algorithms. Examples of RBC encoded images at various rates are given.

A recursion theoretic analysis of the clopen Ramsey theorem

AbstractSolovay has shown that if F: [ω]ω → 2 is a clopen partition with recursive code, then there is an infinite homogeneous hyperarithmetic set for the partition (a basis result). Simpson has shown that for every 0α, where α is a recursive ordinal, there is a clopen partition F: [ω]ω → 2 such that every infinite homogeneous set is Turing above 0α (an anti-basis result). Here we refine these results, by associating the “order type” of a clopen set with the Turing complexity of the infinite homogeneous sets. We also consider the Nash-Williams barrier theorem and its relation to the clopen Ramsey theorem.

Partial degrees and the density problem. Part 2: The enumeration degrees of the Σ2 sets are dense

As in Rogers [3], we treat the partial degrees as notational variants of the enumeration degrees (that is, the partial degree of a function is identified with the enumeration degree of its graph). We showed in [1] that there are no minimal partial degrees. The purpose of this paper is to show that the partial degrees below 0′ (that is, the partial degrees of the Σ2 partial functions) are dense. From this we see that the Σ2 sets play an analagous role within the enumeration degrees to that played by the recursively enumerable sets within the Turing degrees. The techniques, of course, are very different to those required to prove the Sacks Density Theorem (see [4, p. 20]) for the recursively enumerable Turing degrees.Notation and terminology are similar to those of [1]. In particular, We, Dx, 〈m, n〉, ψe are, respectively, notations for the e th r.e. set in a given standard listing of the r.e. sets, the finite set whose canonical index is x, the recursive code for (m, n) and the e th enumeration operator (derived from We). Recursive approximations etc. are also defined as in [1].Theorem 1. If B and C are Σ2sets of numbers, and B ≰e C, then there is an e-operator Θ withProof. We enumerate an e-operator Θ so as to satisfy the list of conditions:Let {Bs ∣ s ≥ 0}, {Cs ∣ s ≥ 0} be recursive sequences of approximations to B, C respectively, for which, for each х, х ∈ B ⇔ (∃s*)(∀s ≥ s*)(х ∈ Bs) and х ∈ C ⇔ (∃s*)(∀s ≥ s*)(х ∈ Cs).

Recursive code generation based on m -sequence

Families of binary sequences are recursively defined, with the first family consisting of a single reference m-sequence. The crosscorrelation function of any of these sequences with the reference m-sequence has a unique pattern which determines a code word corresponding to the given sequence. One suggested application of the technique is to a data transmission system using modulation of an m-sequence carrier.

A note on the failure of the relativized enumeration theorem in recursive function theory

where [u] is the uth finite set in some recursive coding of finite sets. Since A is 21 (1) shows that any 4 partial recursive in 6A has 21 graph; furthermore since A is many-one complete for 21 sets we have that all b with 21 graphs are partial recursive in AA. Since a total function has :1 graph if it is hyperarithmetic this proves (i). (ii) follows from (i) since there is no highest degree among hyperarithmetic functions. To show (iii) we first show