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Abnormal Sexual Psychology/Definition

Paraphilia
A paraphilia is being aroused by factors considered non-standard by mainstream society.

Clinical Meaning
The DSM-IV classifies paraphilias as an Axis II disorder.. This means they are viewed as clinical syndromes or symptomatic personality functions that are evidenced in long lasting symptoms. Typically, all Axis II conditions are believed to affect all parts of the individual's way of interacting with the world, and are called pervasive. [The DSM-TR-IV identifies paraphilias as Axis I disorders. The only disorders recorded on Axis II are Personality Disorders and Mental Retardation. Please check DSM-IV-TR, pg. 28 for clarification.]

Clinically speaking, paraphilias are put in the same category as paranoia, antisocial disorders, and personality disorders.

Paraphilia as disorder
In short, paraphilias are considered to be a component of the person's entire psychological makeup. They cannot be "fixed", but their effects can be suppressed. They should not be looked at as mere preferences. They are seen as being pervasive, as mentioned above, affecting all components of a person's lifestyle, thinking, and emotional states.

A person can like looking at naked members of the opposite sex, and get aroused by this. This is natural behavior. But such a person could also get aroused by risqué talk, or while the person touches them in an erotic manner, or in anticipation of sexual contact in absence of any other arousing factors.

A person with a paraphilia, on the other hand, has varying levels of difficulty in becoming aroused when their specific trigger is not present. A "standard" pedophile might be able to become aroused by adults of the gender(s) they are attracted to. An "exclusive" pedophile is unable to become aroused at all except by a pre-adolescent child.

The reason this is a disorder is that paraphilias tend to skew a person's entire way of thinking. A person with exclusively masochistic tendencies will be unable to function in a relationship without such overtones. They will be incapable of keeping such desires hidden from their sexual partners, and in some cases, may need to engage in autoerotic satisfaction of such desires merely to get through the day. This takes it beyond a mere preference or something that is enjoyed.

The other reason this is classified as a disorder is that the person with a paraphilia has no ability to control their desires. Pyrophilia is extremely dangerous both for the risk of arson and to the person suffering it, but pyrophiliacs are unable to control themselves.

There is research indicating that paraphiliacs have higher libidos than normal people, and are more likely to experience sexual addictions related to their paraphilia.

Levels of Exclusivity
Each type of paraphilia is subclassified (by individual) on the level of exclusivity the person demonstrates for it. Many psychologists try to find patterns of behavior to distinguish between "levels" of paraphilia, to measure how much the paraphilia affects the individual.

DSM-IV does not adopt such terminology, but most psychologists agree that there are three levels of paraphiliac exclusivity.

 * Level One: Referred to as "standard", "optional", and "low level".
 * Level Two: Referred to as "preferred" and "mid level".
 * Level Three:Referred to as "focused", "exclusive", and "high level".

(For the purposes of this book, we will use the "standard-preferred-focused" line of terminology).

A person with a standard paraphilia is very strongly aroused by the subject of the paraphilia, but can still find at least some level of arousal through more conventional methods. A masochist may achieve a higher level of arousal through pain, but can still be aroused normally. This is the lowest functional level of paraphilia, and is the most common. However, indulging in the paraphilia results in the activity becoming more and more preferred. Most fantasies involve the paraphilia, but they are rarely acted on, or if acted on are kept hidden and quiet.

A person with a preferred paraphilia is extremely aroused by the subject of the paraphilia, and achieves primary arousal through that means. They can only achieve arousal through conventional methods with great difficulty, and sexual relations may fail if conducted without referral the paraphilia. A masochist with this level of preference experiences most of their arousal through pain, and has difficulty without it. The person has a harder time not indulging in their preferences on a regular basis, even in a non-sexual way. All fantasies involve the paraphilia. The person may change their lifestyle to more closely fit their needs, and may act on their desires.

A person with focused paraphilia is uncontrollably aroused by the subject of the paraphilia. Their entire thought process begins to focus on their desires. They are incapable of arousal except through the subject of their paraphilia. Interpersonal relationships that are not founded on their desires will also fail. The person will usually begin becoming anti-social or uncommunicative. Sexual addictions and obsessive compulsive disorders are common at this high level of paraphilia.

A person with multiple focused paraphilias is usually incapable of functioning normally in society. In particular, persons who have the following combinations withdraw from society or are criminalized:

 * sadistic/hybristophiliac (cause pain, commit crime)
 * masochistic/blood fetishists
 * masochistic with algolagnia

7 Clinical Paraphilias
There are seven clinically recognized paraphilias. These are considered to be the most common paraphilias, and each one is distinct. DSM-IV is close to recatagorizing everything according to a new system, in which paraphilia would be broken down into three classes: action, reaction, and object based paraphilas.

 * Pedophilia: sexual arousal by and strong sexual attraction to children and adolescents.
 * Exhibitionism: sexual arousal by revealing one's body or performing sexual acts in public.
 * Frotteurism: sexual arousal by rubbing one's body against an unwilling person.
 * Masochism: sexual arousal by having pain inflicted on one's self.
 * Sadism: sexual arousal by inflicting pain on another.
 * Transvestic Fetishism: sexual arousal by dressing and acting as the opposite sex.
 * Voyeurism: sexual arousal by watching others engage in sex without their knowledge.

Fetishism
The DSM-IV is not precisely clear on fetishism. According to the DSM-IV, fetishism is the use of inanimate objects or parts of the human body as a stimulus to achieve sexual arousal and satisfaction. This means that some paraphilias are not clinical paraphilias but do not qualify as fetishes either, particularly "action" based paraphilias like biastophilia or lust murder.

For the purposes of this book, fetishism will include the so-called "Other paraphilias not otherwise specified", code 302.81.

Cultural Meaning
The word paraphilia, or fetish, is a loaded term in society. Most people fail to understand the differing criteria used to identify real paraphilias from "things that turn people on". For example, a male who is aroused by a woman's breasts does not have a breast fetish in the clinical sense of the word. The breast fetishist would be aroused by any breasts, and would have extreme difficulty in achieving arousal or orgasm without such stimulus.

Homosexuality and Bisexuality are important since in older version of the DSM they were considered mental illnesses or paraphilias. Some misinformed people today still term them as such. They are not paraphilias since they encompass a far larger perversion of the normal sexual instinct than what arouses. The term perversion is used non-judgmentally, to indicate deviations from the scientific norm of expected behavior. Unfortunately, this sort of language does not serve to segregate these actions from the stigma of being "unnatural" to most heterosexuals.

Transgender individuals do not have a fetish, either. Transvestic Fetishism is arousal by dressing up as or pretending to be a member of the opposite sex. A transgender or transsexual person is someone who undergoes surgical, hormonal, and psychological procedures to shift their sex to the opposite sex. Transgenderation is not a paraphilia. Culturally, of course, it's deemed non-standard and again, older versions of the DSM lumped it in with Transvestic Fetishism.

Pedophilia is perhaps the most problematic large-scale paraphilia and, since it involves children, the most reviled. Some versions of the DSM-IV would lump pedophilia in with the more arcane term of chronoastiphilia, or attraction to younger people. But pedophilia is specific. Throughout history, younger individuals have engaged in sexual congress with older members of the community through marriage, etc. A 25 year old male engaging in sexual contact with a 15 year old female is not pedophilia. Pedophilia specifically refers to children (under 13 years of age) who are typically not sexually or mentally mature enough to have a developed sex drive or physically developed enough to safely engage in sex. Due to the large number of molestation cases from parents, priests, etc., this paraphilia has been shoved to the forefront of the cultural limelight.

Other paraphilias and fetishes, such as zoosadism, necrophilia, and crush fetishes, are so limited in their exposure that the mere existence of such things often shocks people who were unaware of them. Typically speaking, the more extreme the paraphilia, the less known it tends to be. Unfortunately, the most dangerous and criminal paraphilias are almost unknown outside of certain circles.

Sexuality and Paraphilia
Obviously, since paraphilia is a sexual disorder, it should come as no surprise that any discussion of it is tied up heavily with a discourse on sexuality itself. Paraphilia is, in the simplest terms, an attraction to things beyond the norm.

Interactions
One large issue with sexuality and paraphilia is that they are not always mutual in nature or at their core. Thankfully, many paraphilias go extremely well together. A masochistic raptophiliac who has algolagnia (a person who enjoys pain, enjoys being raped, and derives sexual pleasure from physical pain beyond their control) is a natural match for a sadistic biastophiliac (person who enjoys raping and causing pain). With the popularity and anonymity the internet affords, more and more people are finding others who match their sexual fetish.

There are dangers in paraphilia relationships, however. Due to the violent nature of many paraphilias', one partner may end up injured in a way neither intended, or possibly dead. This is why many paraphilia websites and groups stress the importance of education and offer events where newcomers may learn about various fetishes and how to safely execute them. Safe words and other tools are used to make sure consent is always given, though some members of paraphilia groups have reported and written articles where they claim to feel ashamed if they use their safe word. Another danger to a paraphilia relationship is mental. Some paraphilia fetishes involve emotional and/or mental derogation. A person may find that they enjoy this in fantasy, but it is damaging to them in reality. Partners in such a relationship should keep an eye out for indications of this, as the person being derogated may not realize how it is affecting them. There are sex-positive therapists who will deal with those kinds of situations, allowing the person to not feel ashamed for their needs while at the same time exploring their boundaries so that they know what is too much for them.

While many paraphilia's are harmless when explored by someone who has educated themselves on their fetish and its safety, some cannot be achieved without breaking some moral or legal law. This can lead to sexual frustration, a feeling of being abnormal and "wrong", and cause emotional problems that could potentially lead to violent behavior. This is especially true of those who fetishize life threatening injuries or death. There is a major debate on whether it is better to allow porn or computer generated fake porn of various paraphilias in the theory that they would calm sexual frustration and decrease violent acts, or if it is better to disdain or even make certain paraphilia porn illegal on the theory that extreme porn promotes extreme or violent behavior. Neither side has made any headway in the debate, though both sides have scientific studies they claim are unbiased and prove their argument.

There are different levels of extreme in any paraphilia relationship, and though one couple may have the same fetish as another they may go to completely different extremes or satisfy their urges in completely different ways.

Criminal concerns
Other paraphilias are so outre and unacceptable, such as pedophilia, that there is no legal way for a person suffering from them to pursue them. A person with focused pedophilia will end up having sex with young children until he or she is caught and imprisoned.

Some paraphilias, such as pyrophilia, can cause property damage. Others, like necrophilia, can wreak enormous psychological damage on a grieving person, who now has to deal with the fact that a loved one's corpse has been sexually violated. People who are aroused by the torture and slaying of pets is also a criminally problematic issue.

The law, as a rule, does not accept paraphilia as a debilitating mental illness that may impact the ability of a person to tell right from wrong. The fact that it is increasingly compulsive is also not seen as a legal basis for consideration on the legislation. Particularly, in the case of pedophilia, where premeditation and clear intent are required, the law is unprepared to alter its viewpoint, that paraphilia should be considered a valid mental illness and paraphilic patients should be given proper treatment. It also interesting to note that police forces take in consideration this factors when dealing with the issue in the investigative process, monitoring of detected "deviants". Facts that are also taken in consideration when attempting rehabilitation of individuals convicted of paraphilia-related crimes.

Mental health issues
Aside from mental health issues caused by negligence in supervising someone's mental health in a relationship, or identity crisis and shame caused by a paraphilia being illegal, a person can have mental health issues that stem from the paraphilia itself. This is especially true of specific paraphilias. An example of this would be a masochist finding release easily with a sadist, compared to a masochist who wants bones broken regularly (a specific rather than generalized paraphilia). A person cannot survive having their bones broken too often in too soon a space of time. They would be considered someone who, if unable to control their urges, would be advised to seek a mental health professional.

When these urges cannot be satisfied, undue stress is often placed on the mind of the sufferer. Common effects of this sort of psychic stress can cause further psychological problems, including obsessive-compulsive disorders, sexual addictions, and depression.

title: Abnormal Sexual Psychology/Definition


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Abnormal Sexual Psychology/Types

Types of Paraphilia
The definition and clinical diagnosis demanded by the DSM-IV for a "mainline" paraphilia is rather restrictive. The previous section covered an overview of paraphilia. This section will detail each component.

The Seven Classic types
The DSM-IV recognized seven types of paraphilia as clinical in nature, and threw all the rest into a single code under the heading "Paraphilias not otherwise classified". The reason these seven are clinical is due to several reasons.

The first reason is that it is believed, to a large extent, that all humans have at least limited paraphiliac tendencies in these seven categories. While this may sound somewhat disturbing, the theory goes that these things are seen as deviant due to the fact that normal people can control these minor urges. In everyday life, people have minor urges towards voyeurism, masochism, exhibitionism, and certainly frotteurism. Many people have sadistic or pedophiliac arousal and feel disgusted over such. These things are suppressed in almost all people, as multiple tests have shown.

The second reason these are "clinical" paraphilias is that they occur in a large enough segment of the population to have been tested and examined under clinical conditions. Literally any action or interaction could be the basis for a paraphilia, including such outlandish thing as ghosts, food, or even tinsel. (These, amusingly, have all been reported) The seven clinical types have clear definitions for diagnosis and clear-cut legal, culture, and moral issues of concern to both psychiatrists and the public at large.

Dr. John Wincze, a abnormal psychology researcher and expert on paraphilia, wrote a long paper on why the clinical paraphilia's were so important, and why so little attention was paid to other paraphilia types:

"The primary consideration in determining the clinical significance of a paraphilial sexual disorder is proportional to the effects such disorders have on society as a whole. The more inherently dangerous paraphilia, such as lust murder syndrome and rape fixation, are so intertwined with the crimes necessary that studying them is a matter for criminal rather than sexual psychology. On the other hand, the actions of the unclassified paraphilias are, in light of their actual focus, quite harmless. One rarely hears about someone who enjoys the feel of silk becoming an issue of law. However, the clinical paraphilias are both widely spread and disturbing in nature. Many of them involve or could involve involuntary participation. All of them involve, with the lamentable and archaic exception of tranvestic fetishism, issues of criminality and mental stability."

Voyeurism
Voyeurism is an uncontrollable attraction to, and arousal from, the act of observing unsuspecting individuals, usually strangers, who may be naked or in the process of disrobing, engaging in sexual activity, or otherwise naked. The voyeur is different than a person who simply enjoys nakedness or watching people have sex, since the arousal comes from the fact the other person is unaware they are being watched. Related behaviors often include various antisocial disorders.

The DSM-IV lists the main criteria for the diagnosis as being:

 * Having intense sexual desires, fantasies or behaviors concerning the act of watching an unsuspecting person who is naked, disrobing or having sex for a long period of time.
 * Voyeuristic activity is the primary mode of sexual arousal and expression. The subject is increasingly incapable of sexuality without this activity.
 * Voyeuristic activity causes clinically important distress or impairs work, social or personal functioning.

Onset is typically before the age of fifteen, and in 60% of cases studied, the individual may become so invested in the voyeuristic activity as to have this as the sole sexual behavior. This often is a chronic condition.

Levels:
 * Standard : A standard voyeur will regularly engage in acts of voyeurism, and will attempt to incorporate voyeuristic themes into any sexual acts with others (watching their sexual partner engage in sex with a third person, for example).
 * Preferred : A preferred voyeur will have increasing difficulty in maintaining personal sexual relationships, instead trying to find ways to view unaware persons in the act of disrobing or having sex.
 * Focused : A focused voyeur will be unable to resist their tendencies. They will almost never actually have direct sexual contact with other persons, and will usually purchase equipment to enhance their voyeuristic tendencies. Some may go so far as to commit breaking and entering to observe directly.

Many people have some voyeuristic tendencies, usually below the clinical threshold. The primary clinical factor is the requirement that the subject be unaware. A voyeur would not, for example, be interested in going to a strip club.

Pedophilia
This disorder is characterized by either intense sexually arousing fantasies, urges, or behaviors involving sexual activity with a pre-adolescent child, usually 13 or younger. To be considered for this diagnosis according to DSM-IV, the person must be at least 16 years old and at least 5 years older than the child.

Levels:
 * Standard : A standard pedophiliac will almost certainly not engage in actual intercourse with children. They will almost certainly use pedophilaic imagery, and may commit acts of voyerism against children. Some will use ageplay, or seek to have sex with young-looking adults.
 * Preferred : A preferred pedophliac will try to commit at least peripheral contact with children in a sexual manner. They will certainly use pedophilaic imagery. Some, if they have children, will seek to molest them.
 * Focused : A focused paedophiliac will be unable to resist attempting to prey on children, and may resort to criminal acts to meet his needs. If they have children, they will almost certainly be molested repeatedly.

Pedophilia's is one of the most common forms of paraphilia. .

title: Abnormal Sexual Psychology/Types


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User talk:24.40.131.211

Hi, just so you don't get confused, the chapter you just edited was moved to Saxophone/Flat notes and sharp notes, to keep up with the naming policy. We "properly redlinked" all the unwritten chapternames for you too... hope you stick around!-- SB_Johnny | talk 01:41, 22 December 2006 (UTC)

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User talk:Djs2~enwikibooks

Your recent addition to Psychiatry appears to be a copy-and-paste move from Clinical_depression. This is a copyright violation because you did not attribute it. Likewise, earlier versions seem to have been copied from Psychiatric Disorders, again without attribution. I am deleting the module as it stands now.

If you would like to use materials from wikipedia in order to create a book, please just as for a copy of the article to be imported on WB:RFI. If you want to make a fork from another book, make sure you note that in the talk page, but also justify the reason for the forking (forks are only permitted for certain reasons). That way all our legal bases are covered. Sorry about this! -- SB_Johnny | talk 02:16, 22 December 2006 (UTC)

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The developer team at Wikimedia is making some changes to how accounts work, as part of our on-going efforts to provide new and better tools for our users like cross-wiki notifications. These changes will mean you have the same account name everywhere. This will let us give you new features that will help you edit and discuss better, and allow more flexible user permissions for tools. One of the side-effects of this is that user accounts will now have to be unique across all 900 Wikimedia wikis. See the announcement for more information.

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User talk:137.158.128.105

Your recent addition to Textbook of Psychopharmacology and Textbook of Psychiatry for Medical Students appears to be a copy-and-paste move from Antidepressant, Benzodiazepine, Antipsychotic, Anxiolytic; and Clinical depression; respectively. These are both copyright violations because you did not attribute the contributors at Wikipedia.

If you would like to use materials from Wikipedia as the basis of a book, please ask for a copy of the article to be imported on WB:RFI. That way all our legal bases are covered. --Iamunknown 02:31, 22 December 2006 (UTC)

title: User talk:137.158.128.105


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Abnormal Sexual Psychology/Editors

Major Contributors (Content)

 * Elaragirl - Paraphilias suffered

Supporting Contributors (Layout)

 * Darklama
 * Whiteknight

title: Abnormal Sexual Psychology/Editors


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Biblical Studies/Christianity/The Church of Jesus Christ of Latter-day Saints/Scriptures

Scriptures
As stated in the church's eighth article of faith, "We believe the Bible to be the word of God as far as it is translated correctly; we also believe the Book of Mormon to be the word of God."

The Church of Jesus-Christ of Latter Day Saints uses the Book of Mormon, King James Version of the Bible, Pearl of Great Price, and the Doctrine and Covenants as official church scripture. The Church also believes that the inspired word of its leaders, namely Prophets and Apostles are scriptural in nature - "And whatsoever they shall speak when moved upon by the Holy Ghost shall be scripture, shall be the will of the Lord, shall be the mind of the Lord, shall be the word of the Lord, shall be the voice of the Lord, and the power of God unto salvation." D&C68:4


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Abnormal Sexual Psychology/Society

Paraphilia in the Media
-ControversyBox: Pedopriests

 * Mainstream Paraphilia
 * Special Case: Fur fetish
 * Psychological statements in the media

Paraphilia in "common knowledge"

 * Perceptions of "safe" paraphilias
 * Perceptions of pedophilia
 * Discrimination

Dangerous Misconceptions

 * Homosexuality as paraphilia
 * Paraphila as fetishism

Immorality and Paraphilias
-ControversyBox: Christianity

 * Religious views

title: Abnormal Sexual Psychology/Society


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Adventist Youth Honors Answer Book/First aid/Sunburn

Sunburn is from over-exposure to the harmful ultraviolet rays of the sun. While the symptoms are usually temporary (such as red skin that is painful to the touch), the skin damage is often permanent and can have serious long-term health effects, including skin cancer.
 * There is no such thing as a "healthy tan." Unprotected sun exposure causes premature aging of the skin.
 * Sun exposure can cause first and second degree burns.
 * Skin cancer usually appears in adulthood, but is caused by sun exposure and sunburns that began as early as childhood. You can help prevent skin cancer by protecting your skin and your children's skin from the harmful rays of the sun.

Factors that make sunburn more likely: First Aid
 * Infants and children are especially sensitive to the burning effects of the sun.
 * People with fair skin are more likely to get sunburn. But even dark and black skin can burn and should be protected.
 * The sun's rays are strongest during the hours of 10:00 a.m. to 4:00 p.m. The sun's rays are also stronger at higher altitudes and lower latitudes (closer to the tropics). Reflection off water, sand, or snow can intensify the sun's burning rays.
 * Sun lamps can cause severe sunburn.
 * Some medications (such as the antibiotic doxycycline) can make you more susceptible to sunburn.
 * Try taking a cool bath or shower. Or place wet, cold wash cloths on the burn for 10 to 15 minutes, several times a day. You can mix baking soda in the water to help relieve the pain. (Small children may become easily chilled, so keep the water tepid.)
 * Apply a soothing lotion to the skin.
 * Aloe gel is a common household remedy for sunburns. Aloe contains active compounds that help stop pain and inflammation of the skin.
 * An over-the-counter pain medication, such as acetaminophen or ibuprofen may be helpful. DO NOT give aspirin to children.

Do Not
 * DO NOT apply petroleum jelly, benzocaine, lidocaine, or butter to the sunburn. They make the symptoms worse and can prevent healing.
 * DO NOT wash burned skin with harsh soap.

/Transcluded Modules


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Talk:Adventist Youth Honors Answer Book/First aid/Sunburn

This text was taken from the following public domain resource: http://www.nlm.nih.gov/medlineplus/ency/article/000062.htm


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Muggles' Guide to Harry Potter/Characters/Witherwings

Overview
Witherwings is an alias for Buckbeak, Hagrid's pet Hippogriff.

Half-Blood Prince
When Sirius Black dies at the end of Harry Potter and the Order of the Phoenix, he leaves behind several survivors, notably the house-elf Kreacher and his pet Hippogriff, Buckbeak. While Kreacher was taking care of himself for several years while Sirius was incarcerated, and so is not much of a concern, Buckbeak has no such resources and must be cared for by someone... and who better than Rubeus Hagrid, who had been caring for him originally? Of course, Buckbeak was supposed to be executed, so he cannot return under that name; Hagrid renames him to keep the Ministry from trying again.

While he is referred to as Witherwings a few times in the book, the deception is very thin; people don't seem to be able to remember that he is no longer called Buckbeak. Thus, all the events that occur to Witherwings are cataloged under the character entry for Buckbeak.

Relationships with Other Characters
Witherwings is treated as a pet by Hagrid, sharing his hut much the way that Fang, Hagrid's boarhound, does.


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Adventist Youth Honors Answer Book/First aid/Blisters

Blister Prevention

When it comes to blisters on the feet, it is always best to avoid them. Blisters are caused when your foot rubs against something repeatedly. Blisters can be avoided by wearing footwear that fits you correctly. Also, be sure to break your footwear in before embarking on a long hike. It also helps to wear two pairs of socks. When you do this, the inner sock moves with your foot and the outer sock moves with your boot. All the rubbing takes place between the two socks rather than between your foot and the sock. This will eventually wear a hole in your sock, but replacing socks is a lot less painful than hiking with a blister.

Blister Treatment

Try not to break the blister open. It is far better if the skin continues to cover the injury, as this will keep out germs and prevent infection. If you must open the blister in order to be able to walk, do it as follows.
 * Clean the affected area
 * Sterilize a needle
 * Pierce the blister in several places around its perimeter to drain the fluid, but try to leave as much of the skin intact as possible.
 * Wipe the injury down with rubbing alcohol.
 * Cover a small blister with an adhesive bandage
 * Cover a large blister with gauze and adhesive tape.
 * Be sure to allow the blister to breathe

/Transcluded Modules


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Cantonese/Contributor's Guide

Besides the obvious expansion of lessons, there are many ways for you to contribute, even as a beginning student of Cantonese. Some are listed below. Add your thoughts as to what you think should be done to the Planning page so that discussion is not lost, scattered throughout various discussion pages. =Other Ways to Contribute =

Stroke Order Images
There is a project on Wikicommons to upload images and animations of the stroke order for characters. There are directions on how to contribute. It's easy with the use of some free programs and can be done even with only a basic understanding of stroke order. Please contribute so that we'll have a standardized reference for this and other Wikibook users.

Sound Samples
With a microphone and a read through the recording guidelines for the Spoken Wikipedia you can contribute audio examples. Sound samples are particularly important for beginners, especially ones who are studying alone with no teacher or native speaker on hand.

You can see what has been uploaded so far at Category:Chinese pronunciation.

Translate Pages
Currently, the English version is the most developed of the Cantonese Wikibooks. If you know another language, please check if a Chinese book is started in that language and compare it to this one. You can make changes to their content based on ideas gained here, or do a wholesale translation of this textbook into the target language. Since the Cantonese parts can stay the same across languages, a lot of lesson development work can be saved in this way.

If you can only do a partial translation, leave a note to later contributors linking back to your source (a good candidate for using Templates).

Add InterWiki Links
If you can't translate pages, but know enough of a language to locate the corresponding Cantonese Wikibook, you can make Interwiki links. You can see them on the Wikibooks Sidebar listed under "in other languages" when available. They are typically placed at the very end of a page using a simple format. Here is the list used for the Cantonese TOC page:

In addition to the main page, Lesson pages, Stroke Order pages and anything else can be interwikied as long as the content is the same.

title: Cantonese/Contributor's Guide


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Wikibooks:Wikibooks Gazette/1

Text of the First Newsletter
Wikibooks Newsletter, Volume 1

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title: Wikibooks:Wikibooks Gazette/1


text:
Taxation in the United Kingdom/Corporation tax/Accounting periods

Tax is charged by reference to accounting periods. Often an accounting period coincides with a companies period of account. This is the period for which it draws up accounts, except for a life assurance company, where it is the period for which it draws up its periodical return. However, periods of account and accounting periods do not necessarily coincide.

Basic rules
An accounting period begins when :

 * the company comes within the charge to corporation tax;
 * an accounting period of the company ends, and the company is still within the charge to corporation tax; or
 * the company does not currently have an accounting period and has a chargeable gain or allowable loss.

An accounting period ends on the earliest of the following :

 * the expiration of 12 months from the beginning of the accounting period;
 * an accounting date of the company or, if there is a period for which the company does not draw up accounts, the end of that period;
 * the company beginning or ceasing to trade or to be, in respect of the trade or (if more than one) of all trades carried on by it, within the charge to corporation tax;
 * the company beginning or ceasing to be UK resident;
 * the company ceasing to be within the charge to corporation tax.

There are further rules to deal with windings up, life assurance companies and lessor companies.

A UK-resident company is treated as coming within the charge to corporation tax (if it has not already come within the charge to corporation tax) when it commences to carry on business.

Example
Suppose ABC Ltd, a UK resident company is incorporated on 1 August 20X1. It acquires a source of income (an interest-bearing bank account) on 1 September 20X1 and commences trading on 1 October 20X1 and continues trading throughout all other periods under review. It draws up its accounts for the following periods:

 * 1 August 20X1 to 31 December 20X1
 * 1 January 20X2 to 31 December 20X2
 * 1 January 20X3 to 30 June 20X3
 * 1 July 20X3 to 31 December 20X4

Then ABC Ltd would have the following accounting periods:

 * 1 September 20X1 to 30 September 20X1 (from when it came into charge to corporation tax to the commencement of trade)
 * 1 October 20X1 to 31 December 20X1 (to the end of a period of account)
 * 1 January 20X2 to 31 December 20X2 (to the end of a period of account)
 * 1 January 20X3 to 30 June 20X3 (to the end of a period of account)
 * 1 July 20X3 to 30 June 20X4 (to the expiry of 12 months)
 * 1 July 20X4 to 31 December 20X4 (to the end of a period of account)

Company with more than one trade
If a company carrying on more than one trade draws up accounts of any of them to different dates, and does not make up general accounts for the whole of the company's activities, the company may determine that any one of those dates shall be taken into consideration for the purposes of determining whether an accounting period has ended under the rule in the second bullet point above.

If, however, the Board of HMRC is of the opinion, on reasonable grounds, that the date so chosen by the company is inappropriate, they may, if it is reasonable to do so, give notice that the accounting date of another of the company's trades should be used instead.

Mean accounting date
Some companies draw up financial statements to the same day of the week, a Sunday, say, each year. They will therefore have 364 day (52 week) or 371 day (53 week) periods of account. As long as the company draws its accounts up to within four days of a mean accounting date each year, it may, for the purposes of determining its accounting periods, deem its accounts to be drawn up for a twelve month period. For example, when there is a 371 day period of account, instead of having one accounting period of 365 days (in a non-leap year) and another of 6 days, it will prepare its return on the assumption that the period of account was 365 days long.

Inspector determining the accounting period
Where it appears to the tax inspector that the beginning or end of any accounting period is uncertain, he may make an assessment on the company for such period, not exceeding 12 months, as appears to him appropriate. In that event, that period is treated for all purposes as an accounting period unless the inspector sees fit to revise it as a result of receiving further facts, or on an appeal against the assessment in respect of some other matter the company shows the true accounting periods. If on an appeal against an assessment made under this rule the company shows the true accounting periods, the assessment appealed against has effect as an assessment or assessments for the true accounting periods, and assessments may be made for any such periods as might have been made at the time when the assessment appealed against was made.

Administration
An accounting period ends immediately before the day a company enters into administration. For this purpose a company enters administration when it enters administration under Schedule B1 to the Insolvency Act 1986 or is subject to any corresponding procedure otherwise than under that Act (for example, in an overseas jurisdiction). .

An accounting period ends when a company ceases to be in administration. For these purposes a company ceases to be in administration when it ceases to be in administration under Schedule B1 to the Insolvency Act 1986 or any corresponding event occurs otherwise than under that Act (for example, in an overseas jurisdiction).

Winding up
An accounting period ends and a new one begins with the commencement of a winding up. For this purpose a winding up is taken to commence on the passing of a resolution for the winding up of the company, or on the presentation of a winding up petition if no such resolution has previously been passed and a winding up order is made on the petition, or on the doing of any other act for a like purpose in the case of a winding up otherwise than under the Insolvency Act 1986.

After this, an accounting period does not end other than by the expiration of 12 months from its beginning or by the completion of the winding up. However, if the company later enters administration, this rule is disapplied from that point (and with an accounting period ending because the company has entered into administration).

During the course of a winding up, a company may make an estimate of when it will be wound up so that it can prepare tax returns for its final accounting period before that period ends. If the date assumed for the winding up turns out to be before the actual winding up, then an accounting period ends on the date assumed for the winding up, a new accounting period starts, and the immediately preceding paragraph applies as if the winding up had started with that accounting period. .

Cooperatives preparing quarterly and half-yearly accounts
HMRC will, under certain conditions, allow cooperatives that prepare quarterly and half-yearly accounts to merge a number of periods of account for tax purposes so that they have only accounting date each year.

Life assurance companies
Additionally, there are special rules that apply to life assurance companies from which life assurance business is transferred under and insurance business transfer scheme. Different rules apply depending on whether s444AA ICTA applies. For transfers before 1 January 2007, s444AA applies where the whole of the transferor's long-term insurance business is being transferred and the company. It is proposed that the Finance Bill 2007 will amend this for transfers on or after 1 January, so that s444AA applies where the whole or substantially the whole of the transferor's long-term insurance business is transferred, in which case the transfer time is taken to be the time some business is first transferred.

If s444AA does not apply, an accounting period of the transferor company ends with the day of the transfer.

If s444AA applies, an accounting period of the transferor company ends immediately before the transfer, and the transferor also has an accounting period covering the instant of the transfer (except for the purposes of calculating the equalisation provision for tax purposes).

Example where s444AA applies
Suppose a transferor company, XYZ Life Ltd, has in the past prepared periodical returns for a period encompassing the calendar year. It transfers the whole of its business at 11.59pm on 31 December 20X1, after which it retains some cash on deposit. It agrees with the Financial Services Authority that it does not need to prepare a periodical return for 20X1.

XYZ Life Ltd will have the following accounting periods in 20X1:

 * 1 January 20X1 to immediately before the transfer of business
 * An accounting period covering the instant of the transfer (11.59pm on 31 December 20X1)
 * An accounting period covering the period beginning immediately after the transfer to midnight on 31 December 20X1

Lessor companies
If a company carries on a business of leasing plant or machinery, is within the charge to corporation tax in respect of the business and there is a qualifying change of ownership in relation to the company, then an accounting period of the company ends on the day of that qualifying change in ownership, and a new accounting period begins on the following day. There are exceptions to this in Paragraph 40.

Partial deeming provisions
Other provisions in the Taxes Acts sometimes deem there to be accounting periods for specific purposes only. This is typically on the introduction of new legislation. For example, when section 75 of the Income and Corporation Taxes Act 1988 was replaced as a result of the Finance Act 2004, companies, if necessary, deemed themselves to have an accounting period ending on 31 March 2004 for the purposes of calculating management expenses that they were entitled to under the rules applying pre-1 April 2004 and post-31 March 2004.

 * Preceding page: The schedular system of corporation tax
 * Next page: Tax rates


text: Taxation in the United Kingdom/Legislation/Section 12 of the Income and Corporation Taxes Act 1988


text:
Introduction to Game Theory/Overview

Introduction to Game Theory
Introduction to Game Theory

Game theory, according to Wikipedia, is a branch of applied mathematics that uses models to study interactions with formalised incentive structures. These interactions are called "games". Game theory can be used to explain phenomena in the areas of economics, political theory, evolutionary science and military strategy, to name a few. There are several ingredients which are common to all games.

Players/Agents

The first thing we need are two or more players. These can also be called agents. These, like in a board game or sport, are the people, groups, animals etc. who actually interact. These players have preferences and goals. That is to say, like all normal people, these agents prefer some options over another. Peaches over apples, happiness over unhappiness - it doesn't really matter WHAT they prefer, but they must have preferences. These preferences are used to work out exactly what the outcomes of a game will be, because usually, players will choose options that increase the chances of getting their preferred outcomes.

Teoria del joc/Introducció

title: Introduction to Game Theory/Overview


text:
Learn to Play Drums/Lessons/Single strokes and double strokes

Learn how to perform and practice (properly) single and double strokes.

Single Strokes
A technique mainly used in fills, this comprises of you hitting drums with your alternate hands. Oh, I'll just put tab...

CR|| HH|| T1|--RLRL--| Sn|RLRLRL--| T2|--RL| T3|RL--| BD||

This is a simple fill, using single strokes. If you remember, in our description of tab L means hit it with your left stick and R means hit it with your right stick.

Double strokes
The same, but two hits with each hand! Ok then, more tab...

CR|| HH|| T1|| Sn|LLRRLLRRLLRRLL--| T2|| T3|| BD||

I use this technique for convienience and if I find it impractical in one instance to hit a certain drum. You'll see what we mean...

Practicing strokes
When you first start drumming, your strokes might be fast, but they are nothing compared to your potential maximum. To achieve your potential maximum all you need to do is practice the stroke techniques once a day for a long as you want (longer = faster strokes!). You'll need to practice properly, too.

Step-by-step guide to practicing strokes:
 * 1) Starting with the single stroke, start very slow and build up speed.

 * 1) When you begin to feel tension in your arms, slow down back to your original speed.

 * 1) Repeat with double stroke.

If you continue playing either of the strokes after beginning to feel tension, you could cause some serious damage - repetitive strain injury could force you to abandon your drumming career, besides being a nuisance, sometimes making everyday tasks unbearably painful.

Want to see the coolness of playing with single and double strokes? Then play this...

CR|| HH|| T1|--RLRL--| Sn|LLRLRL--| T2|--RL| T3|RL--| BD||

(Ok, that doesn't really prove my point so anyone who can think up a better fill as this example, feel free to put it in).

Back to Starting Out


text:
User:Tannertsf/Historical Documents/Pre 18th Century/Acilian Law on the Right to Recovery of Property Offically Extorted, 122 B.C.

1. MANIUS ACILIUS ... plebeian tribunes duly proposed to the plebs, and the plebs duly resolved on the day before ... tribe ... was the first to vote, and ... cast the first vote for the tribe ...

2. IF FROM ANYONE of the allies either of the Latin name or of foreign nations, or from anyone of those dependent on the discretion, dictation, power, or friendship of the Roman people ... in any single year property amounting to more than ... sesterces in value has been taken, seized, exacted, procured, or appropriated from such person himself or the king or his people or his father, or from anyone who is or has been in his or his father's family, or in bondage to them, or of whom he or his father or his son is heir, by a person in the exercise of his command or power who was dictator, consul, praetor, master of the horse, censor, aedile, plebeian tribune, quaestor, triumvir capitalis, triumvir for granting and assigning lands, military tribune in any of the first four legions, or who is the son of one of these, or who is himself a senator or his father is one: with regard to this matter the wrong person shall have the right to bring suit and report the name of the offender. The praetor shall have jurisdiction over the proceedings, and the trial, judegement, and assessment of damages shall be by the terms of this law in the competence of those persons who constitute the trial court according to this law. ...If anyone asserts that the praetor has not accepted the name of a defendent so reported in accordance with this law, and that a trial has not been granted in accordance of this law so that he might sue: With regard to this matter the plaintiff shall have the right to bring suit and to report the name of the offender and the praetor shall have jurisdiction over the proceedings. The trial, judgement, and assessment of damages shall be by the terms of this law in the competence of those persons who constitute the trial court according to this law. In regard to a person who is judged in accordance with this law, and is accused of having acted contrary to this law, after his case is adjudged, or who is accused collusivelyor whose name is removed from the list of defendants in accordance with this law: if anyone reports his name before the praetor a second time, ... the praetor shall have jurisdiction over the proceedings; and the trial, judgment, and assessment of damages shall be by the terms of this law in the competence of those persons who constitute a trial court according to this law. If


text:
Cookbook talk:Hot and Sour Soup

Just tried this - very good. Didn't have the tofu, bamboo and mushrooms. And I shredded in 3 green finger chilis and 1 clove of garlic, instead of the garlic chili sauce. The egg effect is great.

title: Cookbook talk:Hot and Sour Soup


text:
User talk:Jagged 85

Welcome
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Good luck! -- Herby talk thyme 14:55, 22 December 2006 (UTC)

Copyright issue
Could you please comment on Metal Gear Series/In the Darkness of Shadow Moses's apparent copyright violation - thanks -- Herby talk thyme 14:55, 22 December 2006 (UTC)

The other two pages as well please. If I do not hear from you soon the pages will be deleted -- Herby talk thyme 14:56, 22 December 2006 (UTC)

title: User talk:Jagged 85


text:
Cookbook:Salami

Salami is a popular type of fermented sausage made usually of pork and beef. It is most popular in cold cut sandwiches, but may also be used as a snack or a pizza topping.

title: Cookbook:Salami


text:
Talk:Handbook of Genetic Counseling/Hemoglobin C

''will my child get sickle c trait if i have it? will my child get sc disease? and if my child did get sc disease will it effect the way my child grows and how he\she reacts while he\she is getting older?''


text: Category:Book:Perl Programming

title: Category:Book:Perl Programming


text: Template:ASPCite

title: Template:ASPCite


text:
File:IMovie Interface.png

Summary
The iMovie HD 6 Interface

title: File:IMovie Interface.png


text:
Horticulture/Stellaria media

The common chickweed (Stellaria media) is a cool-season annual plant native to Europe. The plant germinates in fall or late winter, then forms large mats of foliage. Flowers are small and white, followed quickly by the seed pods. This plant flowers and sets seed at the same time, and the seed pods explode when mature to disperse the seeds.

Description
Common chickweed is a low, mat-forming plant, present throughout the cool seasons and winter, but dying during hot weather. The roots are very fine, and the crown easily breaks away if pulled from very dry or frozen soils, allowing the plant to regrow.

Ecology
In both Europe and North America this plant is an invasive weed of gardens, fields, and disturbed grounds. Control is difficult due to the heavy seed sets.

Uses
The plant has uses in folk medicine; 17th century herbalist John Gerard recommended it as a remedy for mange. It can also be tolerated as a winter soil cover.

title: Horticulture/Stellaria media


text:
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text:
Talk:Haskell/Denotational semantics

Comment by Apfelmus

 * Look of ⊥ in a mathematical context, $$\perp$$ is preferred, whereas in a code context,  should be the way to go.
 * Draw real pictures showing the various poset structures. Especially the ones with the recursive data type definitions!
 * It's very cool that semantics can partly be explored in Haskell itself. Are there paragraphs where this opportunity was forgotten?

Apfelmus 19:39, 22 December 2006 (UTC)
 * Improve examples and exercises. Most have been invented very quickly and are not very good. Those that are, are most likely ones that I remembered from different textbooks. Exercises where one must repeat or slightly modify the explanation from the text are helpful for a first understanding but should be held in low quantities. True insight can be only be achieved by tasks where one has to twist one's mind. Train your brain, don't train a pattern.
 * Add more exercises and examples? A spot where readers frequently get stuck is clearly a good place for additional examples and exercises. Hopefully, these will emerge at this Talk-page.

Denotational semantics
Imperative languages can certainly be given denotational semantics. It's just that a direct semantics doesn't work anymore. You need to model the store, and usually continuations as well.

 * Furthermore imperative languages can be given other non-operational semantics such as Floyd-Hoare-Dijkstra style axiomatic semantics. You (Apfelmus) may be interested to search the EWD archive for what Dijkstra had to say about operational reasoning, most of it in regard to imperative programming. It was I who removed the erroneous and misleading paragraph mentioning imperative languages. I apologise for not adding an edit summary this was due to my stupidity not rudeness.

 * Of course, you are right. Indeed, just by using a MonadState, one already gives denotational semantics to imperative stuff. I somehow wanted to pin down that reasoning is much easier in purely functional languages because the semantics are very direct. So I just denied denotational semantics for imperative stuff altogether and hoped nobody would complain :) Compared to pure functional programs, it doesn't matter much whether I have operational ("machine, change your state!") or denotational ("plan of how machine state changes") semantics, the reasoning doesn't get much easier. Denotational semantics tries to recapture as much pure stuff as possible from the operational semantics, similar in power to the finding of a suitable loop invariant. Ah, and thanks for the EWD link. Apfelmus 22:29, 27 December 2006 (UTC)

forall a . a
The type 'forall a. a' doesn't have to give an error. It can also loop forever. (Which is an error, but not observable.)

 * Fixed, "any function of the polymorphic type forall a . a must denote ⊥". Why didn't I chose that compelling formulation in first place? Apfelmus 21:04, 27 December 2006 (UTC)

Mistake in the definition of $$\sqsubset$$ ?
Isn't there a mistake in the definition, which states that "$$a \sqsubset b$$ denotes that a is more defined than b" ? It seems inconsistent with "every concrete number is more defined than $$\perp$$: $$\perp\ \sqsubset 1\ ,\ \perp\ \sqsubset 2\, \dots$$". Shouldn't it be "less defined" ?

 * I was going to point this out too. How about "$$a \sqsubset b$$ denotes that b is more defined than a"? 71.202.203.117 00:34, 27 December 2006 (UTC)

Oops, it's a mistake of course. Fixed. At the same time, I reverted "completely equal programs" to "equal", because of the subtetly that programs can be syntactically equal as well. Of course, they can be syntactically different and still denote the same "mathematical object". Apfelmus 20:47, 27 December 2006 (UTC)

Love it!
Has really helped my understanding. Thankyou.

I did some images, they're perhaps a bit big, anyone who cares can resize them. DavidHouse 17:32, 26 December 2006 (UTC)

Non-strict constructors
"because we can write a function that tells them apart" f (Just _) = 4 f _       = 7 "As  ignores the contents of the Just constructor,   is 4 but   is 7." This argument is truly ingenious but unfortunately wrong :) With the above definition, we have . The stated properties cannot hold because they violate monotonicity. The fact that   is different from ⊥ cannot be "proved", it's a design decision. How can this be formulated nicely? The author above clearly wanted to appeal to a "proof" by example like in the case of  . Apfelmus 13:38, 28 December 2006 (UTC)

What's currently in the article is clearly wrong, as you point out, but the fact that f _|_ = _|_ and f (Just _|_) = 4 should be enough to achieve the author's point, that _|_ and (Just _|_) are distinct. -Chinju

Approximations of the Factorial Function
The sentence "That is, it transforms less defined functions to more defined ones." seems to mean that $$x\sqsubseteq g(x)$$ for all x which isn't true.

 * You're right. I just put in a clarification that hopefully makes more sense.  Stereotype441 15:28, 8 February 2007 (UTC)

Denotional semantics & Equational reasoning
I just found an interesting article Fast and loose reasoning is morally correct. Basically it says that even though having Bottom results in inconsistent logic, with a few simple constraints equational proofs still work. I wonder whether some of it can be useful here. I'll be away the rest of the weekend but then will try to read the paper more carefully. I also wonder if the subject of paraconsistent logic has any applicability here. 64.160.39.153 05:28, 4 March 2007 (UTC)

 * Hello, sorry for this late response. I planned a chapter Haskell/Equation reasoning that covers reasoning formally about Haskell programs. As you point out, it would be great to have a subsection based on the article you mentioned that shows how equational reasoning keeps working with ⊥ although it usually assumes that all programs terminate. Thanks for your suggestion :) I don't know enough about paraconsistent logic, but equational reasoning that ignores ⊥ is like calculating with infinitesimal numbers: done right, it works, done wrong, it doesn't :). Related to that, I'd like also to outsource the infinite lists from this chapter to a place where all forms of program proofs (equational, denotational semantics, coinduction) live together, maybe even as a separate chapter. -- apfeλmus 12:10, 24 April 2007 (UTC)

Continuity
The text says that for any type, any monotone function   must be continuous. Why? I can only prove $$\sup\{f(x_0), f(x_1), \ldots \} \sqsubseteq f(\sup\{x_0, x_1, \ldots \})$$. Why there is not a type  and a sequence of values  $$x_0 \sqsubseteq x_1 \sqsubseteq \ldots$$ of type   such that, say, $$f(x_i) = \perp$$ for any $$i$$ and $$f(\sup \{x_0, x_1, \ldots\}) = 0$$?

 * In the light of your comment, the argument in the text indeed doesn't show anything.
 * The fundamental question I wanted to answer is that although the choice allowing or disallowing continuity is an arbitrary choice, every programming language can only formulate continous function. What magic makes this possible? I think that one can of course prove for every given language (given by its programming constructs like,  , recursion and so on) that only continous functions can be written down. However, I feel that this is not the right way to prove it because this property is universal for all programming languages, hence there ought to be a proof working for all languages at the same time.
 * If the type  has only chains with finitely many elements, continuity is granted for free. But for example,   has infinite chains, one being

$$x_i = n\mapsto\mbox{ if } n \le i \mbox{ then } 0 \mbox{ else } \perp$$.
 * Of course, $$x_i\sqsubset x_{i+1}$$. Now, imagine a non-continuous function  with the requirements

$$f(x_i)=\perp$$ $$f(\sup \{x_0, x_1, \ldots\})=0$$
 * Can this funtion be written down in Haskell? No, for it would have to test it's function argument at all possible value and check whether the outcome is . Clearly, this would take an infinite amount of time. What I had in mind when writing the text is that the transition of $$f$$ from   to   must happen at some finite point in the chain, the decision whether it's   or   may not be postponed to infinity. This is due to the finite nature of computation. With this in mind, the current text argues that the image of any function has in essence only finite chains so that the decision about the result value can indeed be done after a finite number of steps.
 * Of course, this whole argument isn't formally correct, but I think it hints on what's going on. I currently don't know a correct argument.
 * Clearly, the text needs a rewrite, though I'm unsure on how to do that. -- apfeλmus 11:31, 24 April 2007 (UTC)

I have an informal counterexample (maybe), but I haven't the knowledge to formalize it. It is inspired by Fran (indeed, I found this book when trying to understand Fran semantics). Let us define

A value  means "time is  " and a value   with $$t_1 \le t_2$$ means "time is in between   and  ". Thus, according to this meaning, we define  $$\sqsubset$$   for any $$t \in [t_1,t_2]$$ and   $$\sqsubset$$   iff $$[s_1,s_2] \subset [t_1,t_2]$$. Now consider the sequence, which converges to  , and consider the function   where

and  is any function such that   and   for any $$x \ne 0$$. This is only an informal example, I don't know if it is valid.

 * I don't think it's a counterexample because the meaning of $$\sqsubset$$ is fixed as "less defined than". I mean, neither  $$\sqsubset$$   nor   $$\sqsupset$$   hold. In other words, $$\sqsubset$$ is quite different. Your example basically tries to introduce the real numbers into denotational semantics by redefining $$\sqsubset$$. Of course, not all functions on real numbers are continuous anymore. (In fact, they all are continuous again in intuitionistic logic but that leads too far here)
 * Also note that  is a bad choice, because there is a smallest nonzero floating point number, i.e. the sequence   doesn't converge to  . But you can use an exact representation of rational numbers instead to make at least this convergence work.
 * -- apfeλmus 15:50, 26 April 2007 (UTC)

Not all Functions in Strict Languages are Strict
I think this section is wrong. The reason that the lifting trick works is not because functions are not strict in function types, but because lambda terms like  or   are in normal form and so cannot take a step of evalution. Other terms of type, like   will be reduced.

In fact (ignoring type classes), evaluation in a Haskell-like language ought to be completely independent of type information, which can be erased and forgotten once type checking is completed. In a strict language, if, then.

The following Standard ML code supports this:

In the above code,  has type   and , which has equivalent behaviour to Haskell's  , has type.

The line  gives an exception (bottom), not 1.

A strict language shown in chapter 14 of TAPL has an 'error' term, very similar to Haskell's 'undefined'. It also has this behaviour.

Rebooted 15:32, 6 September 2007 (UTC)

 * Good point. The problem is that ML has "two" undefined functions, namely
 * and
 * or better
 * In other words, we can distinguish between a function that is an exception and a function that raises an exception on every value. Put differently, the function type is lifted by an additional  and every function is indeed strict with respect to this ⊥. Unfortunately, this ⊥ is useless for fixpoint iteration which depends on   (as defined in the beginning of the chapter).
 * So, yes, the section is not correct but the denotational semantics of call-by-value are even trickier than the section is wrong :-) I'm unsure what to do. I wanted the section to somehow show the  trick which can simulate call-by-name in a call-by-value language. Hm, maybe it doesn't belong to denotational semantics at all?
 * -- apfeλmus 16:59, 6 September 2007 (UTC)
 * In other words, we can distinguish between a function that is an exception and a function that raises an exception on every value. Put differently, the function type is lifted by an additional  and every function is indeed strict with respect to this ⊥. Unfortunately, this ⊥ is useless for fixpoint iteration which depends on   (as defined in the beginning of the chapter).
 * So, yes, the section is not correct but the denotational semantics of call-by-value are even trickier than the section is wrong :-) I'm unsure what to do. I wanted the section to somehow show the  trick which can simulate call-by-name in a call-by-value language. Hm, maybe it doesn't belong to denotational semantics at all?
 * -- apfeλmus 16:59, 6 September 2007 (UTC)

 * I'd recommend removing the section and replacing it with one entitled "Emulating non-strict behavior in strict languages" which just demonstrates the trick.
 * Rebooted 15:31, 9 September 2007 (UTC)

 * On second thought, I'm not sure whether this solves the problem because the issue is even larger now: the discussion of "strict language" is wrong, too. I mean, the whole chapter implicitly assumes that  which doesn't work for call-by-value. Also, the trick works still thanks to some kind of non-strictness, namely the fact that   in contrast to.
 * It's probably best to look up the exact denotational semantics of call-by-value in a paper/book to figure out how (or whether) to present strict semantics.
 * --apfeλmus 08:33, 10 September 2007 (UTC)

 * Self-note: fixpoints in call-by-value are fine, it's just that the iterated functional is not strict in the value (and thus not internal to the language). In other words, a function  that starts the iteration with ⊥ would be rather useless in a call-by-value language.
 * --apfeλmus 20:05, 6 October 2007 (UTC)

Partiality of $$\sqsubseteq$$ (in "Strict functions")
FIXME: $$\sqsubseteq$$ defines a totally ordered set (toset) on [ ], so we have either $$2 \sqsubseteq 5$$ or $$5 \sqsubseteq 2$$ or both ; if both were true, then 2 = 5, which would be effectively a contradiction, but we can still have either $$2 \sqsubseteq 5$$ or $$5 \sqsubseteq 2$$ where one of these two predicates is true and the other is false. Such thing is easy to define by defining a semantic to $$x \sqsubseteq y$$ where x and y are both non-undefined integers, using a "friend" toset on [ ]-{ } (such as $$\le$$). So 2 and 5 can be compared with $$\sqsubseteq$$.

Isn't there a confusion here between the toset $$\sqsubseteq$$ and the poset $$\sqsubset$$ ?

-- 90.50.99.233

 * Eh? The main point about $$\sqsubseteq$$ (i.e. $$\sqsubset$$) is that it's not a total ordering, exemplified in Haskell/Denotational semantics. The choice that neither $$2 \sqsubset 5$$ nor $$2 \sqsupset 5$$ hold is intentional, the semantic approximation order only says whether one value (like ⊥) is an approximation to another one (like ). The usual ordering $$\leq$$ of integers is irrelevant here.
 * --apfeλmus 20:02, 6 October 2007 (UTC)

Definition of $$\sqsubseteq$$
The definition of $$\sqsubseteq$$ seems pretty counter-intuitive for me: Are not $$2\sqsubseteq 5$$ at equally defined?

 * $$2=5$$ still means "equal". In other words, $$\sqsubseteq$$ is "the same value, less defined" but not "some values ordered by some degree of definedness". apfeλmus 08:54, 18 June 2008 (UTC)

There seem to be several people confused about the details of the $$\sqsubseteq$$ relation (there's a chat about it under the 'Partial Functions and Approximation Order' heading, and I found it non-obvious upon first reading as well). Perhaps it would be clearer if the $$\sqsubseteq$$ relation were defined in one go instead of being spread over several examples - e.g. $$a \sqsubseteq b$$ iff "you can replace parts of b with $$\bot$$ to get a" Yet Another Haskell User (discuss • contribs) 12:40, 21 August 2020 (UTC)

Functions with several Arguments
Should not the code associated with "We see that it is strict in  depending on whether ..." be

cond True x ⊥ = x; cond False x ⊥ = ⊥

instead the current one? --AndresSR (talk) 21:46, 28 January 2008 (UTC)

 * That's right, thanks :-) Fixed now. -- apfeλmus 08:23, 29 January 2008 (UTC)

take 2 (map (+1) nats) = take 3 nats

Is this really true? Doesn't the left expression evaluate to [2, 3] while the right expression evaluates to [1, 2, 3]?

--Md2perpe (talk) 00:03, 17 February 2008 (UTC)

 * Oops, that's wrong of course. I'm going to fix that. -- apfeλmus 22:01, 20 February 2008 (UTC)

Recursive data types and infinite lists diagram
In the illustration of the data type

data List = [] | : List

The value :[] is shown below :_|_. Is this a mistake? Thanks.

 * Oops, it indeed is. -- apfeλmus 21:11, 8 May 2008 (UTC)

Naïve Sets are unsuited for Recursive Data Types
Nevermind, didn't read carefully enough 76.105.193.33 (talk) 13:45, 22 September 2008 (UTC)

Is g in section 3.1 a monotone functional?
Take f such that f(2)=1 and f = _|_ everywhere else. Now g(f)(2) = 2*f(1) = 2 * _|_ = _|_, which is less defined than f(2) ?

 * Oh. Right, thank you for pointing this out. Looks like there's a fundamental flaw in the argument presented. I'll fix that.
 * -- apfeλmus 08:06, 12 October 2008 (UTC)
 * Ah wait, no that's no problem. Monotone doesn't mean that the result is more defined than the orginal ($$x \sqsubseteq g(x)$$), it means that a larger orgininal will be turned into a larger result ($$x \sqsubseteq y \Rightarrow g(x) \sqsubseteq g(y)$$). So, the fixed point iteration only works if we start with $$x_0=\perp$$.
 * -- apfeλmus 12:24, 12 October 2008 (UTC)

Functions is a (d)cpo
I don't understand why that monotone sequences may be of infinite length implies functions  is a (d)cpo. $$\forall n \in \mbox{ Integer }. \sup_{\sqsubseteq} \{\dots f_{j-1}(n) \sqsubseteq f_j(n) \sqsubseteq f_{j+1}(n) \sqsubseteq \dots\} = i_n$$ $$f(n) = i_n$$ $$\forall j \mbox{ is index of chain }. \forall m \in \mbox{ Integer }. f_j(m) \sqsubseteq i_m \Rightarrow \forall j \mbox{ is index of chain }. f_j \sqsubseteq f$$ $$\therefore f \mbox{ is upper bound of } \{\dots f_{j-1}(n) \sqsubseteq f_j(n) \sqsubseteq f_{j+1}(n) \sqsubseteq \dots\}$$ Define $$k \sqsubset g \mbox{ if } \forall x. k(x) \sqsubseteq g(x) \mbox{ and } \exists y. k(y) \sqsubset g(y)$$ $$\forall h. h \sqsubset f \Rightarrow \exists n. h(n) \sqsubset f(n) = i_n$$ $$\Rightarrow h(n) \mbox{ is not upper bound of chain } f_j(n) \Rightarrow h \mbox{ is not upper bound of chain } f_j$$ $$\therefore \sup_{\sqsubseteq} \{\dots f_{j-1} \sqsubseteq f_j \sqsubseteq f_{j+1} \sqsubseteq \dots\} = f$$ $$\therefore$$every chain of functions  has a least upper bound. Therefore, functions  is a (d)cpo. --LungZeno (talk) 03:55, 7 November 2008 (UTC)

 * What the text wants to say is the following: if monotone sequences are always finite, like in the case of, then it's clear that the ordering is complete. Of course, this doesn't work for  , but there is a (slightly) different argument that does show that the partial functions are a cpo, too.
 * apfeλmus 09:51, 7 November 2008 (UTC)

 * Does "finite" mean that number of element in a monotone sequence is finite?
 * Or mean that number of monotone sequences in  is finite?
 * --LungZeno (talk) 11:10, 7 November 2008 (UTC)
 * I saw the explaination.
 * You mean that number of elements in every monotone sequence in  is finite.
 * --LungZeno (talk) 11:42, 7 November 2008 (UTC)

 * My proof is wrong. Because
 * $$\forall h . h \sqsubset f \Rightarrow \exists n . h(n) \sqsubset f(n) = i_n$$
 * $$\Rightarrow h(n) \mbox{ is not upper bound of chain } f_j(n) \Rightarrow h \mbox{ is not upper bound of chain } f_j$$
 * $$\therefore \sup_{\sqsubseteq} \{\dots f_{j-1} \sqsubseteq f_j \sqsubseteq f_{j+1} \sqsubseteq \dots\} = f$$
 * Only prove that every chain of functions  has a minimum upper bound.
 * Changing my proof to
 * $$\forall n \in \mbox{ Integer } . \mbox{ cardinal of } \{f_j(n) | f_j(n) \sqsubseteq f_{j+1}(n)\} \mbox{ is finite }$$
 * $$\Rightarrow \mbox{ supremum } i_n \in \{f_j(n) | f_j(n) \sqsubseteq f_{j+1}(n)\}$$
 * $$\therefore \forall j, h , n. f_j \sqsubseteq h \Rightarrow f_j(n) \sqsubseteq h(n) \Rightarrow i_n \sqsubseteq h(n) \Rightarrow f(n) \sqsubseteq h(n) \Rightarrow f \sqsubseteq h$$
 * $$\therefore$$every chain of functions  has a least upper bound.
 * --LungZeno (talk) 12:24, 7 November 2008 (UTC)

Why is Nothing less defined than Just True?
And why is it at the same level as Just _|_ ? --Colin-adams (talk) 21:03, 22 November 2008 (UTC)

 * It's not.  and   are incomparable, and   is less defined than.
 * In the diagrams, "every link between two nodes specifies that the one above is more defined than the one below" (Section 2.2). Should there be a reminder in the section about data types?
 * -- apfeλmus 10:09, 23 November 2008 (UTC)

Haskell Recursion Example
I think it might be useful to point out that the "let x = f x in x" construction for the "fix" function actually uses the same least fixpoint semantics under the hoods. Thus "x = f x" semantically means that "x = fix(g)", where g = \x -> f x, leading to x = lfp(f); Thus you are using the same semantics for recursive functions to denote recursive elements.

I think some extra explanation such as above is really useful as I can see many people becoming puzzled by reading this "let x = f x in x" construction because there could be many fixpoint of f and it is not clear that the haskell semantics for functions are the same for these values. jjanssen (talk) 18:51, 2 April 2009 (UTC)

 * Sounds good. A new subsection on how the discussion applies to Haskell (touching on the examples,   and  ) in the recursion chapter is probably the best way to incorporate that.
 * -- apfeλmus 20:51, 2 April 2009 (UTC)

Request additional Algebraic Data Types examples
More examples of Data types(such as bool, lifted bool,unit, nat) and operations (lift, cartesian conjunction, square, disjunctive join) in the form of trees (one picture has the weight of a thousand words..) would help understand the topic. Can more examples be added? --194.1.130.37 (talk) 09:48, 16 November 2010 (UTC)

"Recursive Data Types and Infinite Lists", the diagram
There is a mistake in the diagram in the chapter "Recursive Data Types and Infinite Lists": the nodes ":⊥" and ":[]" are swapped. Does anyone have a source code for that diagram? --Beroal (discuss • contribs) 15:07, 23 January 2011 (UTC)

 * I think it was an SVG file; maybe it's still available? In any case, I very much like the diagrams by E.Z. Yang. Maybe we can ask him to release them with a CC license?
 * --apfeλmus 08:16, 27 January 2011 (UTC)

Syntax and semantics
The word semantics is usually singular, much like physics: “Denotational semantics gives meanings to programs,” “this language has this semantics,” etc. The CS literature is consistent on this point AFAIK. (The word can also be plural: “We consider several possible semantics, including call-by-value and call-by-name.” But that's not the usage anywhere in this text.)

Luke Maurer (discuss • contribs) 08:18, 2 February 2011 (UTC)

A "wrong" section?
I just noticed that someone added a brisk "This section is wrong." note to the top of the "Not all Functions in Strict Languages are Strict". I don't know enough theory to evaluate that statement; so could anyone more knowledgeable than me tell us whether that section is wrong (and, if so, why)? Thanks, Duplode (discuss • contribs) 04:51, 25 April 2012 (UTC)

Section missing
In Section 6.3 "Naïve Sets are unsuited for Recursive Data Types" there's a link to Naïve Sets are unsuited for Recursive Definitions, but it does not exist.
 * The section was renamed to What to choose as Semantic Domain?. The link is now fixed. Duplode (discuss • contribs) 03:14, 11 April 2014 (UTC)

Stumbling block: Partial Functions and Approximation Order
Under "Partial Functions and Approximation Order", I found the section discussing $$\sqsubseteq$$, rather indigestible. We have the para:

 * This is contrasted to ordinary order predicate $$\le$$, which can compare any two numbers. A quick way to remember this is the sentence: "1 and 2 are different in terms of information content but are equal in terms of information quantity". That's another reason why we use a different symbol: $$\sqsubseteq$$.

"A quick way to remember this" -- What is "this" in this sentence? Does it pertain to the immediately preceding $$\le$$? Or is it trying to describe $$\sqsubseteq$$ ? And what is content vs quantity?

Further, the sentence "1 and 2 are [...in some sense equal, in another sense not equal...]" is followed by this:

 * neither $$1 \sqsubseteq 2$$ nor $$2 \sqsubseteq 1$$ hold, but $$1 \sqsubseteq 1$$ holds.

Huh? How does this statement using $$\sqsubseteq$$, where both expressions involving 1 and 2 do not hold, relate to the previous sentence which says that something about 1 and 2 does hold?

Hopefully someone can clarify this. Gwideman (discuss • contribs) 06:17, 5 February 2013 (UTC)

-- I am similarly confused by this section. First it defines $$\sqsubseteq$$ as: ... $$a\sqsubseteq b$$ will denote that either $$b$$ is more defined than $$a$$ or both are equal (and so have the same definedness).

A little later we read: neither $$1 \sqsubseteq 2$$ nor $$2 \sqsubseteq 1$$ hold,

I would think that given the definition of $$\sqsubseteq$$ that both $$1 \sqsubseteq 2$$ and $$2 \sqsubseteq 1$$ would hold. Aren't $$1$$ and $$2$$ equally defined, so both the relations hold?

Thanks in advance. --Dougc (discuss • contribs) 11:57, 13 April 2013 (UTC)

-- disclamer: I'm no expert

If both $$1 \sqsubseteq 2$$ and $$2 \sqsubseteq 1$$ would hold, then the antisymmetry law of posets would be broken, since $$1 \neq 2$$

Antisymmetry: if both $$x \sqsubseteq y$$ and $$y \sqsubseteq x$$ hold, then $$x$$ and $$y$$ must be equal: $$x=y$$.

So you either don't define $$1 \sqsubseteq 2$$ and $$2 \sqsubseteq 1$$ or you say both are false. In both cases you break a law of the total ordered sets, which seems to be the reason Integer is a poset instead of a total ordered set.

Maybe you could just deny that Integer is a poset(or total order) and make $$1 \sqsubseteq 2 \land 2 \sqsubseteq 1$$ true, but I guess that would be unneccecarily complicated.

--89.246.167.57 (discuss) 19:48, 14 July 2013 (UTC)

Why is f0 = _|_ and why do we have to start at f0?
Section: Approximations of the Factorial Function.

And why $$\bot=f_0 \sqsubseteq f_1 \sqsubseteq f_2 \sqsubseteq \dots $$?

$$f \sqsubseteq g \mbox{ if } \forall x. f(x) \sqsubseteq g(x)$$ is not true here.

For example $$ f_2(0) \not \sqsubseteq f_3(2) $$ or  $$ f_1(0) \not \sqsubseteq f_2(\bot) $$

--89.246.167.57 (discuss) 20:30, 14 July 2013 (UTC)

Interpreting factorial as a set
The article currently says:

Now that we have a means to describe partial functions, we can give an interpretation to recursive definitions. Lets take the prominent example of the factorial function $$f(n)=n!$$ whose recursive definition is

$$f(n) = \mbox{ if } n == 0 \mbox{ then } 1 \mbox{ else } n \cdot f(n-1)$$

Although we saw that interpreting this directly as a set description may lead to problems, …

When did we saw that? What is that referring to? Is it another article? (In that case, I think a link should be there.) Or is it something that was removed since? (Then this should be rewritten.) Or did I just miss something? Svick (discuss • contribs) 22:04, 19 March 2014 (UTC)
 * The reference is to the discussion about recursive functions in Haskell/Denotational semantics. I reworded the passage slightly to make it more obvious. Duplode (discuss • contribs) 03:18, 11 April 2014 (UTC)

Bottom type vs. Bottom value
Someone should note the difference between the bottom value (without type) of this article and the bottom type (requiring subtyping) from Wikipedia: Bottom type. At first I assumed they were the same and couldn't follow the point of creating a bottom value requiring definedness to avoid solving the halting problem, when you could just create a bottom type without value.

 * On a tangential note, the type with no values (sans bottom) is usually known in Haskell as . --Duplode (discuss • contribs) 00:29, 11 May 2019 (UTC)

Approximations of the Factorial Function: Use of next/previous
There are two ways to iterate the factorial function:

$$f(n) = \mbox{ if } n == 0 \mbox{ then } 1 \mbox{ else } n \cdot f(n-1)$$

The first is the operational approach:

$$f(n) = n \cdot T, n \cdot \left [ (n-1) \cdot T \right ], \dots, n \cdot \ldots \cdot (3 \cdot (2 \cdot 1))$$

The second is the denotational approach:

$$f(n) = \bot, g(\bot), g(g(\bot)), \dots$$

The operational approach iterates from $$n$$ towards $$\bot$$ while the denotational approach iterates from $$\bot$$ towards $$n$$. In both cases the function call order remains unchanged: call the next function with diminishing $$n$$, i.e. $$n-1$$. What changes is the definition of 'next' or 'previous'. In the operational case, next is towards $$\bot$$. In the denotational case, next is towards $$n$$. Try as an exercise recreating the steps using:

$$f(n) = f(n+1)$$

In this sentence:

"we calculate a sequence of functions $f_k$ with the property that each one consists of the right hand side applied to the previous one"

the use of 'previous' corresponds to the denotational approach. Intepreted operationally it seems that the function call order has to be reversed, which means:

$$f_{k+1}(n) = f_k(n-1)$$

This is wrong! The correct solution is simply:

$$f_{k+1}(n) = f_k(n+1)$$

This mistake isn't possible with the factorial function because its call order can't be inverted (possibly related to the fact that it converges).

 * I believe that sentence of the text can be made clearer by replacing "one" with "function", so I'm doing that. --Duplode (discuss • contribs) 05:50, 6 May 2019 (UTC)

Monotonicity Inverse
It is also crucial that the inverse does not hold:

$$ x \not\sqsubseteq y \not\Rightarrow f(x) \not\sqsubseteq f(y) $$

For example, $$1 \not\sqsubseteq 2$$ does not imply $$f(1) \not\sqsubseteq f(2)$$. If that were the case, then $$f(1) \neq f(2) \land f(1) \not\sqsubset f(2)$$, and these examples would not be valid:

$$ f(1) = 2, f(2) = 2 $$

$$ f(1) = \bot, f(2) = 2 $$

Convergence Terminology
A directed-complete partial order refers to a partial order in which every directed subset ("every monotone sequence") is complete ("has a least upper bound"). A dcpo with a least element is termed a pointed directed-complete partial order, which this book abbreviates simply as cpo ("we will only meet dcpos which have a least element $$\bot$$ which are called complete partial orders (cpo)"). See Wikipedia: Complete Partial Order.

Continuity refers to Scott continuity. See Wikipedia: Scott Continuity.

Interpretation as Least Fixed Point: Multiple fixed points
The example with multiple fixed points is not an assignment of $$n \mapsto \dots$$ to $$f$$ (that would use $$:=$$ rather than $$=$$); rather it defines a a fixed-point equality (i.e., an equation):

$$f = n\mapsto\mbox{ if } n == 0 \mbox{ then } 1 \mbox{ else } f(n+1)$$,

The corresponding solution would be a fixed point to:

$$g(x) = n\mapsto\mbox{ if } n == 0 \mbox{ then } 1 \mbox{ else } x(n+1)$$,

Here are two fixed-point solutions (thanks to monochrom). The first has a correspondence from  and is also the least fixed point:

$$ f(n) = \begin{cases} 1 & \mbox{ if } n \leq 0 \\ \bot & \mbox{ else } \end{cases} $$

This second solution is clearly more defined than the first:

$$ f(n) = \begin{cases} 1 & \mbox{ if } n \leq 0 \\ 3 & \mbox{ else } \end{cases} $$

Strict Functions: power2
The question regarding  was somewhat vague. Yes a proof via $$2^n$$ is possible, but what does a proof involving fixed point iteration mean? If you iterate this functional:

$$ g(x) = n \mapsto \mbox{ if } n == 0 \mbox{ then } 1 \mbox{ else } 2 \cdot x(n-1) $$

It converges to. If it wasn't for the if-then-else stop condition, it would simply converge to $$\bot$$. Unfortunately these iterations don't answer the question; rather it was intended to simply to iterate  starting with the input $$\bot$$.

Recursive Data Types and Infinite Lists: iterate in Haskell
I think it's interesting to see a fixed-point iteration of  in Haskell:

ones :: [Integer] ones = 1 : ones g :: [Integer] -> [Integer] g x = 1 : x x0 :: [Integer] x0 = undefined (f0:f1:f2:f3:f4:fs) = iterate g x0

Recursive Data Types and Infinite Lists: Exercise #4 possible solution
$$\sup_{\sqsubseteq} \{\bot \sqsubseteq (1:\bot = \bot) \sqsubseteq (1:1:\bot = \bot) \sqsubseteq (1:\bot = \bot) \sqsubseteq \dots\} = \bot $$

Hidden assumption
Expanding the definition of factorial: $$f_{k+1}(n) = \mbox{ if } n == 0 \mbox{ then } 1 \mbox{ else } n \cdot f_k(n-1)$$ at some point we have to assume that $$n \cdot \bot = \bot$$, e.g., in: $$f_1 (2) = 2 \cdot f_0 (1) = 2 \cdot \bot$$ Is this something obvious?

(Bartosz (discuss • contribs) 19:34, 7 September 2019 (UTC))

 * Fortunately, almost exactly this gets explained a few paragraphs later, in the 'Strict Functions Section', which shows why.

 * Unfortunately, it does come a few paragraphs too late. Yet Another Haskell User (discuss • contribs) 12:22, 24 August 2020 (UTC)

Stumbling Block: sup function reads like argmax
The `sup` function for finding the supremum of a set is used in the "Convergence" section without explaining what it does.

On first reading, it looks like it just selects a maximum element from a (potentially infinite) set (i.e. `sup`=`argmax`), whereas in fact one of its critical properties is that it gets to draw its results from some broader set than that which it is passed.

The term 'least upper bound' is mentioned a few times, but doesn't help all that much for a first-time reader since if you don't know it has a precise technical meaning, it's very easy to just consider it a synonym for 'maximum'.

I suggest adding a few words to emphasize that the supremum is allowed to be from outside the set in question. Yet Another Haskell User (discuss • contribs) 12:48, 21 August 2020 (UTC)

'Interpretation as Least Fixed Point' subsection seems out of place
The whole of the 'Recursive Definitions as Fixed Point Iterations' section seems like its purpose is to assemble all the mathematical machinery needed to give recursive functions a solid denotation (it even says as much at the start - "Now that we have the means to describe partial functions, we can give an interpretation to recursive definitions. ").

Once we get to the end of reading the 'Convergence' subsection, it seems like the problem is solved; i.e. we can define the denotation of a recursive function to be the supremum of a particular infinite set of partial functions.

As such, it's quite confusing to read on and find that there is a whole other (quite long) subsection about fixed point iterations and continuity.

It's not at all clear to me why it's necessary to have this subsection at all - the preceding text seems to offer a satisfying enough explanation.

Is the subsection there just because it happens to be interesting that recursive functions satisfy a fixed-point equation? Or is the subsection necessary to prove some details (e.g. existence, uniqueness) which the previous explanation doesn't?

It would be helpful to either add a short intro explaining why we can't just stop after the 'Convergence' (+'Bottom includes non-termination') subsections, or to move the subsection into the 'Other Selected Topics' section where it wouldn't break the article's flow as much). Yet Another Haskell User (discuss • contribs) 13:26, 21 August 2020 (UTC)

 * After doing some further reading, I get the impression that:
 * - The supremum-based definition is actually OK for defining the dentotation
 * - Interpretation as a fixed point is mainly of use for manually simplifying the denotation of particular expressions to something usable in proofs about (e.g.) program correctness

 * Is this the case? Yet Another Haskell User (discuss • contribs) 12:28, 26 August 2020 (UTC)

Is 'converges' the best term to use?
This may be nit-picking, but using the word 'converges' (e.g. "... the sequence converges to the factorial function..") seems to suggest that there is a Metric Space involved.

I know next to nothing about metric spaces, but the general idea seems to be that 'points' (here, partial functions) need to have 'distances' between them (difference in definedness?). And that it's necessary to have all of this (plus some axioms) in order to be using the word 'converges' precisely.

While this may well be the case, it seems a bit beyond the scope of this article.

How about using a more informal word like 'approaches'? Yet Another Haskell User (discuss • contribs) 14:10, 21 August 2020 (UTC)

title: Talk:Haskell/Denotational semantics


text:
User:Smmurphy

Graduate economics student using wikibooks to study for comprehensive exams. Going through the old exams and adding a few problems, entering the theory as the introduction to each chapter/page, and then looking at and solving the problem or the problem in general form. Primarily a wikipedian.

User:Smmurphy/sandbox

title: User:Smmurphy


text:
Talk:Digital Signal Processing/Z Transform

mention of the link to computing analog-digital filter equivalence is needed, especially as relates to the bilinear tranform...--Billymac00 20:30, 22 December 2006 (UTC)


text:
User:Herraotic/Sandbox Two

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New Title Policy
As there are but a few people working on this project, the community has decided to let contributors focus their energy on a single book each quarter. For this reason, there is a New Title Policy. You can also help decide as to which title the Wikijunior community will develop next by voting.

To suggest a title for development, please see New Title Suggestions -

Stubs
Stubs are short and incomplete articles. A list of Wikijunior article stubs can be found at Wikijunior Stubs. -

Discussion
You can discuss the project at Wikijunior meta talk page and Wikibooks Staff Lounge as well as the Wikijunior Talk page.

You can also read and comment on
 * A Case for an Online Wiki for Children.
 * A Wikijunior Style Guide.

To chat about Wikijunior on IRC, use Freenode or join #wikijunior on Freenode.

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Wikijunior Wikichicos Wikijunior Wikijunior ja:Wikijunior:メインページ Wikijunior Wikijunior

title: User:Herraotic/Sandbox Two