recurrence relations - How to get the characteristic equation? - Mathematics Stack Exchange
Modeling is about mapping entities and relationships of the world into the Recursive Relationships. • Recursive . Subclass = special case → Inheritance. Recursion, though, is a fairly elusive concept, often used in slightly different .. cursive relationships, and planning a dinner party may need care- ful attention to . might process it in this piecemeal way, as a succession of unrelated events. A relationship between two entities of similar entity type is called a recursive relationship. Here the same entity type participates more than once in a relationship.
In particular, we focus on a variant of the simplest recursive language, anbn, and find evidence that i participants trained on two levels of structure essentially ab and aabb generalize to the next higher level aaabbb more readily than participants trained on one level of structure ab combined with a filler sentence; nevertheless, they do not generalize immediately; ii participants trained up to three levels ab, aabb, aaabbb generalize more readily to four levels than participants trained on two levels generalize to three; iii when we present the levels in succession, starting with the lower levels and including more and more of the higher levels, participants show evidence of transitioning between the levels gradually, exhibiting intermediate patterns of behavior on which they were not trained; iv the intermediate patterns of behavior are associated with perturbations of an attractor in the sense of dynamical systems theory.
We argue that all of these behaviors indicate a theory of mental representation in which recursive systems lie on a continuum of grammar systems which are organized so that grammars that produce similar behaviors are near one another, and that people learning a recursive system are navigating progressively through the space of these grammars.
In natural language, it refers to morphological and syntactic patterns in which one phrase is embedded inside another of the same type e. Recursion is found in almost all natural languages, although it is still a matter of debate whether it is language universal Everett, ; Nevins et al. Center-embedding recursion [as in 1 below] refers to the case in which a phrase is embedded in the middle of a phrase of the same type. To process center-embedding recursion structures like 1-a and 1-bthe language system must keep track of each constituent that has been started and not finished, and in cases like 1-athe order in which these have occurred.
If one assumes that the recursion processing system is capable of processing center-embedding patterns to arbitrary levels of embedding, then an infinite state mechanism is necessary for recognizing or generating all and only the legal structures Chomsky, ; Hopcroft and Ullman, S] Cowper,quoted in Lewis, b. The processor of a counting recursion language e. The processor of a mirror recursion language e.
Much debate in the artificial language learning domain has centered around the question of whether animals or humans trained in a laboratory setting can learn a mirror recursive language, as opposed to a mere counting recursive language Fitch and Hauser, ; Perruchet and Rey, ; Bahlmann et al. Prior studies suggest that human learners can learn at least counting recursion in the lab and, according to Lai and Poletiekcan learn mirror recursion if they are exposed to lower levels of embedding earlier than higher levels of embedding.
In this study, we aim to elucidate the process of recursion learning. We therefore focus on counting recursion, with which it is relatively easy to get robust generalization behavior, suggesting abstraction to a recursive rule. We argue that new insight into the way the human mind reaches and encodes such abstraction can be achieved by looking in detail at the learning process.
Recursive Functions (Stanford Encyclopedia of Philosophy)
We agree that our participants, who certainly all learned how to count long before they participated in our experiment, may, in some cases, have eventually recognized the utility of using their counting ability to succeed with our task.
We claim that it is the process of getting to the point of that recognition that is worth examining in detail. Indeed, as we report below, the task is not trivial in that many participants fail to solve it, and these show a range of behaviors which suggest that their minds are developing toward a formulation of the counting principle before they actually reach it.
We also find that the work we have done on this counting recursion case establishes a foundation for analyses of mirror recursion cases, which we have explored in separate experiments Tabor et al.
What kind of novel understanding can be derived from this close examination of the learning process? There are three main points: A careful empirical examination of the intermediate states indicates that something additional is occurring: This phenomenon is reminiscent of attractors in nonlinear dynamical systems theory.
Helpfully, this theory offers a possible avenue out of the teleology. We go over these arguments carefully in General Discussion. Motivating dynamical systems models By dynamical systems, we mean formal systems that are characterized in terms of how they change. Typically, they are expressed as systems of differential equations or iterated maps Strogatz, Several language-learning connectionist networks of syntactic processing are iterated map dynamical systems Elman,; Tabor, For example, the Simple Recurrent Network SRN Elman,consists of a layer of input units which feeds forward to a recurrently connected layer of hidden units; this, in turn, feeds forward to a layer of output units.
In the context of language learning, the SRN is trained to receive a sequence of words one word at a time and predict the next word.
A prominent feature of such systems, which differentiates them from other models of cognitive processing, is their employment of continuous parameter and state spaces.
In the continuous parameter spaces e. Tabor explores such a model, called a Fractal Learning Neural Network of the learning of some mirror recursion languages. He finds that the model progresses through a series of stages: As training processes, the system masters two levels, then three, etc.
Moreover, between the mastery of each level, the system makes subtle changes in its encoding which shift it gradually from one level to the next see also Tabor et al. Inspired by these observations, we designed the experiments below to encourage participants to progress through successive levels and we sought evidence that they exhibit intermediate behavior when they transition between the levels. In their continuous state spaces e. An attractor is a subset of the state space with the following properties: The majority of research on dynamical systems has focused on autonomous systems: Here, we are interested in a non-autonomous case: Also in this case, it appears that the system's behavior can be organized around an attractor-like structure—i.
For example, in Tabor et al. Motivated by this observation, we sought evidence that humans who learned a recursive artificial language would gradually return to grammatical behavior after perturbation as they continued to receive grammatical input from the language. This prediction also hinges on the assumption that the system's states lie in at least an approximate continuum, because there need to be intermediate locations that it can occupy, as it returns gradually from the perturbed location back to the original trajectory.
Research hypothesis We hypothesize that people trained on a sample of sentences from a recursive artificial language are gradually adjusting the continuous parameters of a dynamical system which is capable of developing attractor-like structures; this system is a highly plastic learning device and it can mirror the distribution of the input well.
In relational, this is called a self-join. Network Recursion The many-to-many recursive relationship, sometimes called a bill of material BOMis more complex. For now, we will stick to the term BOM.
It is commonly used.
It is also very descriptive of an actual manufacturing bill of materials. Although it can be applied to many other recursive situations, an actual bill of material makes a good example for explaining the concept. Simplified View Of Bill Of Materials The Bill of Materials represents what component parts comprise a superior part and into what superior parts a component part goes. The ITEM represents the part out of context.
All items, regardless of their role or associations, are first represented in ITEM. Here is a typical bill of materials structure.
Why the multiple relationships between the two entities?
- Primitive recursive function
The combination of the two represents the relationship. There are multiple relationships between the two entities. Multiple relationships are valid when any of the following conditions exists: In this case, the occurrences would be different because one points to the parent occurrence, the other to the child occurrence.
It does not matter which attribute is chosen to play which role — parent or child. The parent does not have to be the first one and the child the second one.
However, once chosen, the role of the attribute is fixed. To break the item down further, the children of each parent may have child parts of their own. They are then used as parents with their own children to extend the structure, as is illustrated below for an F15 plane. Navigating The BOM If an assembly is navigated downward from any given parent, it is commonly called explosion. The quantity of parts needed to form an assembly can be identified.
We can ask, for example, what are all the items that make up this part? If we look upward, from the view of any given child, it is called implosion. The quantity needed within each assembly can be identified. We can ask, what parts is this part used in? In other words, it is dependent on the combination of parent and child. The classical BOM provides the capability to navigate from any parent to all of its component parts.
Modeling Hierarchies | index-art.info
It may be necessary to navigate and flip-flop many, many parent-child relationships to get there, but it is eminently possible. It is important to note one final thing: Retrieval of the entire bill of materials for a given part requires navigation through the succession of parent-child relationships, starting with the ultimate parent for explosion, or the ultimate child for implosion.
This treatment also shows those familiar with current computer implementations of BOM that the construct is, in fact, a network and not a hierarchy. An airplane has a hierarchy of components.
If one is simply decomposing one airplane into greater and greater levels of detail, it appears as a hierarchy. However, when looking at the entire range of airplanes for a manufacturer, the structure is not a hierarchy but rather a network of parts, i. In other words, a given part or assembly such as a carburetor type can be used in many different types of engines.
Likewise, an engine might be used in many different types of airplanes, and so on. So any given assembly can have subassemblies under it and other assemblies above it.
This can be depicted in what is called a product structure tree as shown. Any item can be considered either an assembly or a part depending upon its relationship to other items. For example, item C is a part within assembly B, but it is also an assembly consisting of parts 2, D, and 4.