Talk:Hindley–Milner type system

3062:

monotypes are only used in the definition of the semantics. In many contemporary papers types are stratified into two classes: polytypes (Damas & Milner's type-schemes) and monotypes (Damas & Milner's types). Milner (1978) only uses two classes but leaves all quantifiers implicit and doesn't have a type annotation on his lambda (or even mention exactly over what it can abstract, I believe). It might be useful to note all this in the article, though.

870:"Readers unfamiliar with the lambda calculus might not only be puzzled by the syntax, which can quickly be straightened out translating, that the application e_1 e_2 represents the function application, often written e_1(e_2) and that the abstraction means anonymous function or function literal, common in most contemporary programming languages, there perhaps spelled only more verbosely \texttt{function}\,(x)\ \texttt{return}\ e\ \texttt{end}." 162: 414: 1320:"Type variables are monotypes. Standing alone, a type variable α is meant to be as concrete as int or β, and clearly different from both. Type variables occurring as monotypes behave as if they were type constants whose identity is unknown. Correspondingly, a function typed α→α only maps values of the particular type α on itself. Such a function can only be applied to values having type α and to no others." 4398:. And yes, stylistically it can become less ornamented in parts, but the introduction's content might essentially be rather misplaced leading to an impression of wrong tone. Most particular, no referable lemma or section for "type" and other prerequisites are present in WP. Thus, the issue might better be fixed by distributing the content of the introduction to other lemmata and then drawing from them. 85: 64: 33: 2958:. This links both examples. In HM, the types of variables introduced in the let-clause can be instanciated as explained in the preceeding section, parameters can not. Thus, the abbreviation is still valid with respect to the computation, but not for the typing, which behaves different on both sides. I agree, that the section can be improved. -- 4292:. While I can't remember the specifics of why a tagged it (as it was a good few months ago now), at the moment I feel the predominant use of lists and lack of inline citations in many places, along with other concern would warrant this tag. Hope this helps, and feel free to copy this message over to the appropriate talk page. 2447:

increasingly common - c++ is a good example, it had no type inference before c+11 but now can infer types in some situations with auto or decltype. Start with an example of the problem and show how the algorithm solves the problem, like you would with an article on binary search or btree insertion or something similar.

4596:. If you have a suitable article at hand, please drop a line. IMO, the point here is the static binding of the type variables left implicit in rule system, which obscures both rule system and the generalization process. I think, this might be clarified for sake of accessibility by looking at type system from point of 2517:

While the suggestions how to "fix" the article are not helpful, I hold the complain for rightful and take it serious. As it currently is, the article has an unclear focus of the audience. This is a Knowledge article and a programmer or CS student interested in type inference should be able to read at

1117:

The "universal function" mentioned seems to be just an example of a function returning values of different types which in all static type systems can only be represented be an uppermost type (often called "top" or "object") or some wrapper-type. The type system thus does not "collapse", but correctly

811:

failed. There's quite some frustration about this expressed in the literature. "outstanding" was meant more as relatively "isolated" in respect of extensions or extensibility. This was a very neutral remark, actually, and I believe one worth to mention. I'll reinsert it later in the end in context of

3170:

Are mono-types constants or can be variables? That is what should be explained in plain English, that is the kind of change that can make this article clearer. Aren't polytypes universally quantified type variables? That should be the starting point, the description in plain English, then the formal

3020:

Another mistake is the way lateral conditions are placed. For example, in the "Declarative Rule System" box, the rule (whose name is not "Var" by the way, it should be "Taut" because it's a "tautology") shows "x : σ ∈ Γ" on top of the line, but that is a side condition and it should be on the side.

2593:

While the article has links to most of these topics, they do not improve readability since the prerequisite is unknown but used. Thus a link only sends the reader away. Not knowing the prerequisites is not the interested readers fault, but a challenge for the author. I did not consciously check this

2391:

I do like the version in the article, it is very clean, and as noted, alludes to a comparison with the syntactic rules. For this reason I have seen this presentation being used on several occasions (e.g. in class). This is, however, a matter of taste; why not also provide the pseudo-code and let the

4588:

could contain quite some of the material herein and could then be just summarized and referred to, allowing to focus on let-polymorphism and the generalization and instantiation rule. Additionally, and that is a question of content as well as presentation, the article has to bridge between the rule

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Now brevity and focus is also a matter of the state of other lemmas or material one can draw from. At this point you wish that the article should get out of the way, i.e. moved to a blog or something, as well as it should remain accessible, which is too unspecific for me. Please see the "wp:length"

3654:

Since i did not revised the article for quite a while and left it with its red line burried deeply under too many distracting side notes making it hard to read and improve, i thank all the other authors, proof-readers and those, who wrote on the talk page for their contributions and patience during

2662:

The second question, whether the algorithm makes use only of rules in the system and of no other rules. Otherwise, it could be inconsistent w.r.t. to the rule system or the rule system incomplete w.r.t. the algorithm. This is not really a 'decision' but rather the implementation of the rules or its

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But my point is actually only that all the substitutions can be carried out cheaply by a side effect eliminating all their various compositions, too. Making the side effects explicit does of course add extra variables and computations, making things "harder" (actually only more laborious) to follow

3248:

is named 'types' in the original article. I did not notice it, perhaps because variables bound in a non-empty initial context are proper constants. As mentioned, monotypes in the original article are a purely semantical feature and syntactically ground terms there. Thus the meaning of variables in

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To fix this, I should perhaps turn around the order and emphasize first on the advantage of the form used in the original article, namely being side-effect free and explicit and reformulate its disadvantage positively, to that the substitutions and their composition needed in a fully explicit form

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at time of writing. But so may be the designers/users of the inference algorithm. IMO, the article could be improved, by first explaining the difference between type checking and type inference. Having the Curry-Howard correspondence at hand, one even has a third option: can we find an expression

2523:

Because quite some technologies and notations uncommon in "normal" programming and CS are used here, virtually everything in the article fails to "parse" and the article becomes incomprehensible. Example current prerequisites are: understanding grammar and terms as a data type, i.e. discriminated

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Which "decision problem" is undecidable for "anything as expressive as the lambda calculus"? The obvious one would be the type inference problem, but AFAIK there are type systems with recursive types for which type inference is decidable and which are able to type every term of the untyped lambda

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means to introduce an axiom which is not true even in classical logic. On the other hand, the algorithm was designed to practically infere types of recursive programs. This contradicts with the otherwise very useful property of strong normalization imposed by typing. Thus, IMO, when dealing with

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Actually, i'm not sure, under which conditions, i.e. with which rules precisely the inference becomes "undecidable". Could you please add some citation? I guess, one could drop the let-clause and allow a more liberal version of generalization. Then one could attempt to construct a variant of the

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I agree that the article is technical, long, an has various other readability issues. But I cannot help that the topic needs a few "formulas", definitions, mostly, to properly present it. Being vague would be wrong. Your reference to graph algorithms does not help, as this is not an algorithm on

4563:, too, have to be balanced, to make the lemma useful for different groups of readers. As a consequence, each group might find sections to simply page over. To facilitate this, the article is ordered from introductory to technical. I agree, that the "Overview" section can be removed or rewritten. 3388:

the point. I could rewrite it, starting with a low bar and quickly raise it without adding more basics at some point, but silently assuming them, indicating the end of the "less technical" part for the various groups of readers, consciously throwing them out, while providing the start for those

3061:

The definitions of the original author do not always stick. Damas & Milner (1982) stratify types into three classes: monotypes (without type variables), types (without quantifiers and with variables; the types of things a lambda can abstract over), and type-schemes (with quantifiers). Their

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but I did not made the change because, as I said before, the reader should be addressed to the lambda calculus article first, explaining here that application is left associative, and how substitution is done, etc.. will extend this article with information that should be in the lambda calculus

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It may not be just the introduction, where I do not express clearly how things play to together. Actually, while reviewing the article and comments, I think, while almost everything else is there, its red thread linking all parts might not be made as proper as I hoped. I'll try to readjust the

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prerequisites but interspersed them very irregularly only making the article unbalanced for every reader. Oh my! This means the article need not only be reviewed but fully remolded. I'll try to sketch a new organization of the article including the other issues, but cannot promise anything. --

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I don't even know where to start - this is incomprehensible. What's worse is that if you look up type inference, it says "the solution to type inference has been known since 1958" and points you here to five pages of formulas. Type inference in static languages is new for most people but

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seems to be tautological, as I consider "different" to mean the same as "other". At least, the part is unclear. Maybe it was intended to mean that different monotype terms always denote different types, i.e. the mapping from terms to types is injective? Or maybe it is better to explain the

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recursion, one either has to reduce type inference of recursive functions to type checking (breaking inference but preserving the logic) or add some illocical rules to make the inference algorithm applicable, which would work well, i think, if the function typed is in fact terminating. --

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I don't use chrome, but I suspect that you need to install some plugin to render math formulas. At 2 years from this comment, that should be solved, if someone has the same problem search in stackexchange or similar forum which plugin should be installed or what else can be the problem.

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A wrong decision can have two effects. Either it fails to find a proof, i.e. claiming a type error where the is non, or second, producing not the principle type (technically by choosing not to generalize or wrongly instantiating), which could later lead to the first, a purported type

2697:"Wells' result implies that type inference for System F is impossible. A restriction of System F known as "Hindley–Milner", or simply "HM", does have an easy type inference algorithm and is used for many statically typed functional programming languages such as Haskell 98 and ML." 3443:

combinator in Haskell, the systems signals an error, saying that it can not infer the type of the expression. However there are a couple of typed fixed point combinators, the simpler relies on the graph reduction semantics. The other is more complicate to explain it here now.

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unbalanced, and, as you put it, in parts absurd. Looking closer, i aparently decided at time of writing to intersperse basics trying to keep the article going longer as possible for a casual or less prepared reader to the effect of only irritating everyone. Yes. I think this

2426:. While writing the article, i was very eager to demonstrate that a both simple and efficient implementation is possible, while at this point appealing mostly to the reader's metalogical intuition. I'll add algorithm W in a calculus form and hint its use in a proof. -- 2796:

has undecidable type inference, which might be what is referred to. On the other hand, for things like the (λ id . ... (id 3) ... (id "text") ... ) (λ x . x) example, it seems that you could apply a purely syntactic substitution to do your below "abbrevation" first.

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The run-time of HM can be exponential in the size of the expression, so this paragraph is somewhat strongly worded. There are some performance results in the literature (exponential in the depth of the nesting of let-bindings, so linear in practice, if I remember

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If that parapgraph isn't revised, I suggest that a note is added, explaining the deviant notation in the original papers. Moreover, an example should then be given for a function of some type α, but not of type ∀α.α. This could also help making more clear what

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In the latest Chrome browser the notation is not rendered properly, making the article unreadable. I tried to upload a screenshot of the rendering, but it was blocked as "abuse". Categorizing any old thing as "abuse" seems to be the latest advance in abuse.

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While is hesitate to add content to the article while it still is so "incomprehensible", as stated rightfully in other postings, I absolutely like your idea. One can easily put a section in the end noting this relation and known limitations of the HM, etc. --

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Yes, that would be a better way to phrase this. And more part of this article could use some improvements in the clarity of writing. This may have to do something with the original author not being a native speaker, although I'm not quite sure if the

1484:" the proper technical name and meaning. An indeterminate cannot be instantiated in HM and thus remains indeterminate. A type variable can only be instantiated (to any monotype) if quantified. But then we a do have a polytype and not a monotype. 2984:

This is stylized form of academic writing which does not best serve the purpose here. Legitimized by whom? Direct comparison with side-tack against what? And where the heck is the main verb in the long string of words following the comma? —

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Reworking the article, i'll give a proper attributation to the algorithm. Since implementing side effects might not be possible or wanted in a functional language, algorithm W has of course its own rights. But as Milner emphasises, algorithm W

4522:

Phases like “The remainder of this article proceeds as follows” are not a great sign, and the bulk of the content completely lacks inline citations. It would be nice to preserve the content in this article, perhaps as a blog post or a book on

3532: 4600:, replacing the current bridging section, which is too syntactical for my taste. If you have a different suggestion, please drop a line. Otherwise, I disagree with your assertion that the text provides insufficient evidence or references. 2956: 4589:

system and the algorithms in some way. Unfortunately, the rule system is both elegant and obscure and i have not seen this bridging be properly resolved in literature, so that it can be just summarized and referred to as you suggest.

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returning an int. Fine, just call it and receive the number. Thus the first case passes. But what would happen if you call f with an int? One doesn't know and HM soundly refuses to place a bet. Likely if you have another pointer to

3353:

For a student or programmer who wishes to implement or understand it, because that is taught in programming languages courses or some course project to implement some interpreter, a detailed and "technical" exposition is needed.

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This subject is hardly be of the interest of a reader who ignores lambda calculus and logic. The inference rules are very common in logic and formal computer science. For example in semantic rules for programming languages.

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Very simple explanations, even in natural language, can be given on what a type system is and what type systems exist with examples in the more popular programming languages to contrast with the Damas-Hindley-Milner system.

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Actually, I now believe, there are only two points in HM which are difficult to understand. One is the relation between the rule systems and the algorithm. Though not perfect, this point should already made clear in the

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Reword the paragraph starting with "Perhaps a bit irritating, type variables are monotypes" to mention higher-ranked types and why they are not supported in HM, leading to the stratification of types into monotypes and

4369:

Some phrases could benefit from (as counterintuitive as it sounds) the passive voice instead of the active voice, but primarily most of the trouble phrases could be "drier" and should try to be as concise as possible.

4083: 3249:'types' depends of the initial context. If bound, they are constants, if unbound, they are implicitly all-quantified both by the generalization rule. Mentioning this would help to improve the paragraph starting with 2620:

Thus clearly, decisions not present in the logic might have been made constructing the algorithm, which demand a closer look and justifications but would perhaps remain non-obvious without the above differentiation.

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I also agree, that the article should be as focused as possible. As a minimum, it should present the declarative rules system as well as the algorithms W and J, but without just dumping the rules, like e.g. in

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The second is monotype variables or rather, whether a type variable is quantified or not. See Example 1 in 'Free type variables' for another related pitfall, where the variable is bound by another type. It is

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That should be enough for a casual reader, for example a programmer that want to know what means that Haskell has Hindley-Milner types. That intuitive introduction should be enough to satisfy those readers.

4673:

You are right, Stephen70edwards. Thank you. Indeed, proofs that produce the principle type often have a generalization step in the end. I'll look for the proof in the Damas-Milner paper, as you suggest. --

3645:

A most obvious omission is IMO substitution, substitution order, composition of substitution, mgu, etc. But much other stuff is missing, too. Whether the articles referred to will be enough, remains to be

3649:

The "low" start in the introduction distracting towards the use of typing in natural language processing and the like may or may not be received well and may come out debatable, with perhaps interesting

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No, not the same. That unbound type variables are unbound in a type expression does not imply they are universally quantified. They also have to be unbound in the initial environment. See the related

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looking for their particular point. As you say, this would also help to make article denser towards the end. Thus each group would have one continuous section, that could be read with some benefit.

4422:. Additionally, i removed some remarks with questionable tone. As a consequence, this article now lacks some introduction, which is a different issue to filled in later. I close the template. -- 1838: 501: 4609:, you added to the head of the article, to draw some attention to this points, but find them too unspecific as they are and will remove them in while, if no more concrete suggestions are made. 4390:

The examples are all from the introduction. I see some conflict of objectives, as the seemingly "wet" style was intended to give a proper introduction to non-technical readers, too. Please see

2588:, lambda calculus, currying, pattern matching, unification, ... at least half a dozen more. Most of them only at hand for a reader with experience in functional programming or lambda calculus. 207: 3025:

There are varying styles writing rules in the literature and the syntax i used is explained in the box right above and carefully discriminates side conditions (predicates) and judgements. --

2264:

That you find my presentation harder to understand is very unfortunate. I have consciously not presented the original form in favor for the readers. I fail to see the critic, when you write

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While the offending wording has been removed, I like to add that the intention was not to praise HM or something. The thought was rather that many attempts to extend HM, e.g. towards

3586:

Because some critic was raised that the article is not accessible enough for an encyclopedia, i've now added a longer introduction. Adding the introduction has a some disadvantages:

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I do state all this in the article at that point, but what was meant merely as a note on advantages and disadvantages of different forms comes over as a negative personal judgment.

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Since there were no further posts to this topic, i remove the above WP-flags. Before, i considered you points again, but they would equally match almost any CS article in WP, say

4527:, which could then be provided in this article as an external link for further reading. However, the article itself really ought to be trimmed down to a more focused overview. — 2130:, which prevents generalization and with it, instantiation. Thus one can stare at the type syntax, the relevant freeness is there only w.r.t. instantiation. It is in the context 2747:

I've added a second reference to System F. It is now referred to both in "Meta Types" and in the introductory section about "let-polymorphism". -- 14:45, 19 December 2018 (UTC)

3342:

The problem is the style of the presentation, is not clear. The intent to explain here what lambda calculus is, is absurd, it only makes the article more difficult to read.

1187: 264: 202: 3983:. There is no intentention in leaving out the index but only that none of the rule systems have been introduced at that point. And yes, the indexing is nowhere introduced. -- 1247:

Yes, this is wrong and should be changed, especially as "qualifier" does have a very specific and different meaning in some type systems (e.g., Mark Jones' qualified types). —

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It definitely reads like an algorithm: if case 1 then do this, if case 2, then do that, etc, as compared to the "logical" presentation of the algorithm which is found here.

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I've seen the side condition x : σ ∈ Γ gone both above, below and besides the bar; the rule is more often called "var" than "taut"; these are just matters of presentation.

598: 125: 2294: 618: 2475:. The other way round, C++11 inference is unable to go beyond a C++ expression, e.g. cannot infer the type of a function parameter from the functions body, which HM can. 1459:

I do indeed tend to awkward expressions, as in the tautology above. I meant the tautology as an emphasis. (And yes, my German is also not any better in this respect. ;^)

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To the variables in monotypes: I think it is actually as I wrote, and there is in fact a subtlety. More precisely a free type variable can only be interpreted as an

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The second proof under "Typing Rules" (i.e., "To demonstrate generalization") is malformed. It concludes that the type of the let expression is forall α . α -: -->

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Readers unfamiliar with what this entire section is trying to say might be puzzled by its syntax which could sorely use some translation and straightening out.

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notation to express it. It is just to make this article less wordy. More concise and clearer. A simple example of type inference should clarify both concepts!

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The wording was corrected a day later. It's my bad English. I ment "also" or "likewise", referring to the ground terms of types in the preceding paragraph. --

3625:

A programmer with no experience in parameteric types or functional programming might find the historical and pragmatic part of the introduction worth to read.

4605:

Alexis, you or anyone else could help by being more specific in what should be removed or kept or how the lemma should be structured better. I leave these

4658:

I would suggest instead including the proof from the Damas-Milner paper, which showed the type for let i = λx.x in i i, also demonstrating generalization.

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What is meant by "too expressive to be meaningfully typed at all"? Is there some kind of limit in expressiveness that type systems can meaningfully handle?

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I've revised the english in the first half of the article. The second half is not too bad. It still needs another — perhaps more aggressive — style pass.

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in the article. Your second quote says the same, but needs the sentences before it as precondion (which are intended to be included by the leading word

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Thanks, Dleonard. Please don't hesitate to rewrite the text more aggressvily. I'm not a native speaker and my clumsiness in expression is unintented. --

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The article is too long. Beside rebalancing most of the introduction with other lemmata, the body of the article could be reorganized into two parts:

3141:. Do a Google Scholar search on "monotypes polytypes" to find many more examples. I didn't see any paper other than Milner's that does not allow this. 3129:

This is the definition of mono- and polytypes used in TaPL (p. 359) and PFfPL (1st ed., p. 202). Some papers that allow type variables in monotypes:

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note in the talk page above. What i think is, that instead, the article should better be distributed to some degree into existing or new lemmas.

2755:

Perhaps it would be a good idea to emphasize that if polymorphic types were allowed for lambda parameters, type inference would be undecidable.

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graphs, thus no pictures or anything else appealing much to "common sense", though I'd tried. I do provide simple examples as early as possible.

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to the left of the turnstile (the former often being used in literature, the latter being how the empty context is defined in the syntax box).

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In type judgements, leaving an empty context completely blank makes the notation slightly confusing. In the second example under Typing rules,

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Additionally, i'm not sure, whether the level of introduction came out too low for an article with a highly technical body. But perhaps not...

3092:"You have seen" side conditions going to wild places doesn't have a place in an encyclopedia, as others might have seen very different things. 4036: 3379:

Agree, and thanks for your suggestions. While writing, I had a veering image of the potential readers. As a consequence the article came out

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lemma. In particular, the conceptually static binding of type variables is currently left implicit, making the rules appear too syntactical.

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I do not agree that C++11 is a good example for the algorithm described in this article. In particular, HM cannot be extended to cope with

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So I think it's unfair to call it "hard to read", it is perhaps "hard to directly relate to the inference rules", but not hard to read. --

4559:, you should make precise, which information should be removed for brevity. At this point, please be aware, that many guidelines, e.g. 4228: 2839: 794: 255: 3637:

Last but not least, and perhaps most important for a public encyclopedia, the article is now better prepared for additions and changes.

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It is legitimized here, for allowing a direct comparison with \vdash_S while expressing an efficient implementation at the same time.

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As it stands, W is hardly an efficient algorithm; substitutions are applied too often. It was formulated to aid the proof of soundness.

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I'll add the substitution version of Algorithm W in deductive style, to have the four variants of calculus in a unified form. (Done)

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Perhaps it would be a good idea to make this section comprehensible to somebody who doesn't already know what it is trying to say.

1085: 4655:α, but this is a polytype (sigma σ) not a monotype (tau τ), and the rule may only conclude that a let expression has a monotype. 3105: 631: 3044: 2999:

This page is completely messed up. The biggest error is the definition of monotypes. In the original paper, the definition of a

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There is a lot of wonderful information in this article, and I respect the amount of effort that has clearly been put into it.

3436:. Can be easily defined, but the later is also known as the paradoxical combinator because it is related to Russell's paradox. 328: 4263: 4182: 3980: 2257:

I just read the algorithm in the original paper and to be honest I found it easier to understand than the one presented here.

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It's true that Pascal and C didn't have type polymorphism, but other languages from the same era as ML did, notably CLU.

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There are still quite some breaches and omission, but i hope, they can be closed or corrected more easily than before:

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People with interest in learning about HM by reimplementing will find easy code examples and test cases to start with.

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does its job by preventing these values to be used in any type-specific operation and this way preventing type errors.

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As a second step, these features will be removed from the body of the article, where the now disturb the flow. (Done)

3083:

What contemporary papers are you talking about? Are they more authoritative than the original Milner&Damas paper?

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Doesn't that mean that these two are the same? The only difference between them is the universal quantification. --

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suggests to consider splitting an introduction of into a seperate article, which i feel would be wrong in this case.

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Some special topics now mentioned in the body will perhaps better discussed at the end in a separate section. (Done)

4209:

bilber seems to be a typo but it is not obvious what it should be - can anyone figure out what it should be? (from

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related articles on Knowledge. If you would like to participate, please visit the project page, where you can join

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The ones working through the proofs in literature seeking hints or readily layouted formulas might be served, too.

1204:"operators (such as infixed for functions and postfixed list for lists); a type-scheme is a type with (possibly) 4570: 860: 846: 365: 4663: 4162:

Finally, a type variable may legally occur unbound in a typing, in which case they are implicitly all-quantified

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Agree, that this could be what 37.175.216.246 meant. Then the reference to undecidibitly might be misplaced. --

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from such rules. But when this would succeed is not obvious to me. I've added your claim to the text anyway. --

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this time. I hope to have managed to work in most of the valuable suggestions and critiques in this revision.

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The first question is, how does the algorithm make use of the rules. The decision is when to apply which rule.

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Yes, Alexis, there is room for improvement, but i disagree that the article violates WP guidelines, namely

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Drop the figure "The Syntax of Rules". This should most be self-explanatory from the previous sections.

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Hmm, at this point, i feel the templates related to the technicity of the article has been addressed.

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First, the introduction will now contain the most important features, without too much details. (Done)

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Finally, i found a bit time to rework the article hopefully addressing the critics and suggestions.

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The typed fixed point combinator is tricky, that section should be more detailed. If you define the

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but has the advantage to be far more logical (i.e. side effect free) than my "logical" presentation.

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Finally, the overhaul of the article is done. It should be more accessible now in several aspects:

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I did not erased that part, only made a little fix in the notation for lambda expressions, because

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The type inference method designed by Hindley and Milner does just this for programming languages.

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This subject is technical. It should only be written in a more readable style, not less technical!

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containing no type variables". The whole Knowledge page assumes monotypes contain type variables.

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Looking in the original papers cited by the article, I found no occurrence of the search string "

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I tagged the last 2 paragraphs in the "note on expressiveness" section as 'original research?'.

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Perhaps one or two ideas for a presentation of HM might come up in those who are preparing one.

3527:{\displaystyle {\mathit {fix}}:\forall \alpha .(\alpha \rightarrow \alpha )\rightarrow \alpha } 2407:

Paging through the literature, i found the algorithm as presented in Milner 1978, P 370 ff. as

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And, if I may add Cobalt, I appreciated the low-tech introduction. It is not distracting.

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can all be eliminated as shown when using procedural means, in particular side effects.

2268:, since I even explicitly explained how to read the rule system as a procedure, saying 1438: 1381: 1249: 733: 319: 161: 1881: 1227:

So, if there is no particular reason I'm not aware of, the name should be changed to "

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Requested articles/Applied arts and sciences/Computer science, computing, and Internet

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with no explanation or reference to what the latter means. Could we either stick to

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in rule TyVar). Placing it above the bar is the most common variant in my experience.

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Usually, the let-clause can be expressed in the lambda-calculus as an abbreviation:

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It is not a typo. That section uses a nonsense word for demonstration purposes. --

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Algorithm aside, HMs behavior makes sense. You have a pointer to something having

1372:"Type variables are monotypes" refers to the fact that plain HM doens't allow for 3471:(i.e. a proof) that has a particular type? In this light, introducing a function 3412:

In untyped lambda calculus, the fixed point combinator is easy to define, either

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in the conclusion where the execution of the premises proceeds from left to right

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The sentence: "Perhaps a bit irritating, type variables are monotypes, either."

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Hmm, not sure what to do with your note. You refer to the second paragraph? --

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lemma and later chopped off, leaving more room to present the let-polymorphism.

1586:{\displaystyle \Gamma =x:\alpha ,\,y:int,\,z:\beta ,\,f:\alpha \rightarrow int} 793:

have been to mark it as different, but it certainly reads rather non-neutral —

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A casual non-programmer would hopefully get in idea what the article is about.

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Readers unfamiliar with lambda calculus should read the lambda calculus first!

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on p.359 of Milner, (1978). In particular, I couldn't find a source for the

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Focusing on this point should help to make the article clearer, I hope --

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allowing it. I had failed to properly present this point in the article.

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Just improve the style, please don't try to make it less "technical".

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consequences and short-cuts. I meant this second also at this point.

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union, syntax-directed rule system, familiarity with notations like

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to a monotype preventing generalization and released to polytype in

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During browsing the original papers, I also noted that they define "

2150:, which prevents generalization. And this context is restricted in 922:

Lambda calculus is a prerequisite to understand the HM type system.

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Hmm, there are totally four different set of rules in the article

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There's no explanation of the `unify` function in the algorithm.

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Should I explicitly bring it to procedural form as an extra step?

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I removed the whole section, since it does not add to the topic.

573:{\displaystyle \vdash _{D}\lambda x.x:\alpha \rightarrow \alpha } 1355:

A monotype always designates a particular type, while a polytype

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for a random instances. I do agree with your last sentence. --

4340:, I'll point out some examples of what sounds unencyclopedic: 404: 302:

Find pictures for the biographies of computer scientists (see

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This sentence from the second paragraph in the introduction:

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First of all, it makes an already long article even longer.

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Please replace hyperbole like "so outstanding" with facts.

1340:"it is equal only to itself and different from all others" 743:

Ruud, thanks for your comments. I'll work through them. --

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This second part could then draw more material from the

4148:{\displaystyle \lambda \ x.x:\alpha \rightarrow \alpha } 3408:

Type inference of fixed point combinator is not trivial!

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Please discuss or fix the reference in question at its

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Converning the overall style, I guess the questions in

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Type functions are annotated with their arity. Perhaps

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and the Specialization Rule (and the text left to it).

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But, yes, overall this section is awkwardly phrased. —

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I've rewritten the offending section and moved it to

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One and the same thing can be used for many purposes.

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Rereading the portion, I could perhaps better stated

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I actually really confused monotypes and the syntax

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Dubious claim that the algorithm W is "hard to read"

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Type variables may be instantiated only by monotypes

96:, a collaborative effort to improve the coverage of 1349:between mono- and polytypes, rather than trying to 4147: 4104: 4077: 4025: 3966: 3937: 3908: 3879: 3835: 3808: 3788: 3761: 3738: 3526: 3424: 3240: 3220: 2950: 2580: 2548: 2356:Right, this sound judging and should be re-worded. 2308: 2288: 2270:The rules now specify a procedure with parameters 2206: 2174: 2142: 2122: 2090: 1923: 1870: 1833:{\displaystyle \Gamma \vdash f:int\rightarrow int} 1832: 1782: 1762: 1736: 1716: 1679: 1642: 1605: 1585: 1181: 1106:Maybe someone could clarify the following points: 1065: 1013: 981: 946: 648:Consider moving this under Polymorphic type order. 612: 592: 572: 208:Computer science articles needing expert attention 2732:Apropos proper context: HM predates System F. -- 1405:are also caused by poor understandability of the 1353:what a monotype is and what a polytype is? E.g. " 4160: 4005: 1952:in register x and a function f that deals with 348:WikiProject Computer science/Unreferenced BLPs 4586:Type inference (simply typed lambda calculus) 4447:Type inference (simply typed lambda calculus) 4189:). This point is really somewhat subtile. -- 3279:Notation is not rendered correctly in Chrome: 1744:(may) differ from any other type and so from 8: 4445:The first part could be shaped towards some 1021:eliminate the ambiguity of the scope of the 494:A list of implementations would be helpful 4650:Second proof under "Typing Rules" is broken 2694:How about a back link to System F. I read: 1296:µ is a type containing no type variables." 697:Ok. I thought this would be present in the 265:Computer science articles without infoboxes 203:Computer science articles needing attention 4214: 3445: 3358: 3312: 3285: 3172: 3109: 3048: 2500:You may be more interested in the article 1075: 621: 495: 169:Here are some tasks awaiting attention: 143: 58: 4704:High-importance Computer science articles 4117: 4096: 4090: 4038: 4017: 4011: 3958: 3952: 3929: 3923: 3900: 3894: 3871: 3865: 3827: 3821: 3801: 3780: 3774: 3754: 3730: 3724: 3479: 3478: 3476: 3417: 3233: 3213: 2942: 2927: 2898: 2886: 2885: 2879: 2861: 2860: 2858: 2700:This could put HM into a proper context. 2561: 2529: 2301: 2275: 2187: 2155: 2135: 2103: 2059: 1883: 1845: 1795: 1775: 1749: 1729: 1692: 1655: 1618: 1598: 1506: 1402: 1308:is a type containing no type variables." 1171: 1054: 1042: 1033: 994: 959: 930: 789:Yes, please do. I appreciate the meaning 605: 585: 540: 534: 4475:Discussion of polymorphism is misleading 4326:Since this is (probably) already in the 3021:Same for the rule and possibly others. 2518:least parts of the article with benefit. 1717:{\displaystyle \Gamma \vdash f\,z:\tau } 1680:{\displaystyle \Gamma \vdash f\,y:\tau } 1643:{\displaystyle \Gamma \vdash f\,x:\tau } 2091:{\displaystyle x\not \in free(\Gamma )} 1703: 1666: 1629: 1560: 1546: 1526: 1482:type constant whose identity is unknown 1327:type constant whose identity is unknown 1048: 901:Sure, but any concrete suggestions? -- 60: 30: 4362: 4357: 4348: 4343: 4270:, what in particular caught your eye. 837:It's the 'either' that's throwing me. 701:article referred to, but it isn't. -- 502:2601:645:8103:3CC8:F12C:66F9:7787:ADE6 110:Knowledge:WikiProject Computer science 4709:WikiProject Computer science articles 4520:But it is decidedly not encyclopedic. 2581:{\displaystyle \Gamma \vdash E:\tau } 1300:on p.3 of Damas, Milner (1982), and 1208:of type variables at the outermost." 113:Template:WikiProject Computer science 7: 4211:Knowledge:Correct typos in one click 1288:" different from this article, viz. 90:This article is within the scope of 3452:2806:107E:4:ADEE:218:DEFF:FE2B:1215 3365:2806:107E:4:ADEE:218:DEFF:FE2B:1215 3319:2806:107E:4:ADEE:218:DEFF:FE2B:1215 3179:2806:107E:4:ADEE:218:DEFF:FE2B:1215 2887: 2862: 1871:{\displaystyle \Gamma \vdash x:int} 1790:. In particular, one cannot derive 1189:" prefix, which is commonly named " 1082:2806:107E:4:ADEE:218:DEFF:FE2B:1215 49:It is of interest to the following 4057: 4033:is incomplete, as one cannot show 3494: 3419: 3102:2620:0:1040:1D:7824:DAED:F681:D77F 2563: 2277: 2137: 2082: 1847: 1797: 1694: 1657: 1620: 1508: 1173: 830:Something that didn't parse for me 628:2001:6B0:2:2801:8C67:A355:7F2:B217 587: 284:Timeline of computing 2020–present 25: 4699:B-Class Computer science articles 1436:is much better in this respect. — 1336:section's 2nd sentence, the part 580:, could maybe be written with an 526:should be discussed here as well? 310:Computing articles needing images 4714:Knowledge pages with to-do lists 4358:and much like letters in a book 3041:2620:0:1040:1D:EC47:412:B787:E3F 1220:at the outermost; for example," 1212:on p.2 of Damas, Milner (1982), 1129:) 11:43, 3 January 2013 (UTC) • 866:Syntax discussion fails to parse 724:Mention (HM limitations w.r.t.) 412: 160: 83: 62: 31: 3999:Question about "all-quantified" 3144:Side-condition placed besides ( 2442:Incomprehensible esp in context 2424:was formulated to aid the proof 1182:{\displaystyle \forall \alpha } 130:This article has been rated as 4684:14:16, 30 September 2023 (UTC) 4537:20:25, 30 September 2022 (UTC) 4139: 4069: 3518: 3515: 3509: 3503: 3466:Right, i was not aware of the 2975:Sometimes too academic in tone 2933: 2911: 2201: 2189: 2169: 2157: 2117: 2105: 2085: 2079: 1918: 1903: 1897: 1885: 1818: 1571: 1162:I'm irritated by the name of " 1066:{\displaystyle (e_{1}\,e_{2})} 1060: 1035: 1008: 1002: 976: 961: 753:15:47, 14 September 2011 (UTC) 738:12:06, 12 September 2011 (UTC) 699:Unification (computer science) 564: 1: 4622:10:44, 21 December 2022 (UTC) 4514:Article reads like a textbook 3251:Type variables are monotypes. 3228:at time of writing. In fact, 2968:13:39, 24 November 2018 (UTC) 2829:17:12, 25 November 2018 (UTC) 2807:11:50, 25 November 2018 (UTC) 2785:09:37, 25 November 2018 (UTC) 982:{\displaystyle (\lambda x.e)} 894:06:42, 27 December 2012 (UTC) 688:16:28, 22 November 2013 (UTC) 364:Tag all relevant articles in 104:and see a list of open tasks. 4644:08:35, 8 February 2023 (UTC) 4592:This may be related to your 4432:11:26, 12 October 2020 (UTC) 4249:10:54, 21 October 2019 (UTC) 4233:23:34, 20 October 2019 (UTC) 3304:06:16, 21 October 2016 (UTC) 2848:21:52, 8 November 2018 (UTC) 2765:11:22, 22 January 2016 (UTC) 2742:08:35, 3 December 2018 (UTC) 2549:{\displaystyle \lambda ,x.E} 2351:16:30, 10 October 2013 (UTC) 2266:reads like an algorithm: ... 1398:" would be more accurate - ? 1394:If I understood you right, " 1224:on p.360 of Milner, (1978). 1014:{\displaystyle \lambda x(e)} 829: 803:10:30, 2 December 2013 (UTC) 780:21:43, 6 November 2012 (UTC) 373:WikiProject Computer science 149:WikiProject Computer science 93:WikiProject Computer science 4470:01:57, 9 October 2020 (UTC) 4411:01:57, 9 October 2020 (UTC) 4386:15:55, 6 October 2020 (UTC) 4320:14:24, 6 October 2020 (UTC) 4306:09:19, 2 October 2020 (UTC) 4199:13:18, 12 August 2019 (UTC) 4158:But earlier, we said this: 4105:{\displaystyle \vdash _{S}} 4026:{\displaystyle \vdash _{S}} 3993:12:51, 12 August 2019 (UTC) 3967:{\displaystyle \vdash _{W}} 3938:{\displaystyle \vdash _{J}} 3909:{\displaystyle \vdash _{S}} 3880:{\displaystyle \vdash _{D}} 3836:{\displaystyle \vdash _{D}} 3789:{\displaystyle \vdash _{D}} 3739:{\displaystyle \vdash _{D}} 3711:20:55, 19 August 2018 (UTC) 3697:13:44, 19 August 2018 (UTC) 3682:14:19, 18 August 2018 (UTC) 3468:Curry-Howard correspondence 2773:Post correspondence problem 2710:01:06, 5 January 2015 (UTC) 1135:11:38, 3 January 2013 (UTC) 1028:Application should also be 947:{\displaystyle \lambda x.e} 477:Suggestions for improvement 304:List of computer scientists 4730: 4668:02:54, 30 March 2023 (UTC) 4584:For instance, a new lemma 4489:16:14, 27 April 2022 (UTC) 4264:Hindley–Milner type system 3749:The article switches from 3545:02:53, 15 April 2018 (UTC) 3402:12:15, 15 April 2018 (UTC) 3265:06:10, 15 April 2018 (UTC) 3035:06:10, 15 April 2018 (UTC) 3009:while the definition of a 2726:21:11, 23 March 2015 (UTC) 2684:21:00, 23 March 2015 (UTC) 2642:18:33, 6 August 2014 (UTC) 2604:01:12, 24 March 2015 (UTC) 2511:14:49, 23 March 2015 (UTC) 2494:13:24, 23 March 2015 (UTC) 2386:22:39, 22 March 2015 (UTC) 2336:22:39, 22 March 2015 (UTC) 2241:19:45, 23 March 2015 (UTC) 2017:12:06, 23 March 2015 (UTC) 1374:higher-ranked polymorphism 1152:18:58, 23 March 2015 (UTC) 911:13:13, 23 March 2015 (UTC) 822:21:28, 23 March 2015 (UTC) 711:21:09, 26 March 2015 (UTC) 636:12:51, 24 April 2023 (UTC) 593:{\displaystyle \emptyset } 423:Hindley–Milner type system 136:project's importance scale 4571:calculus of constructions 4262:you have added a flag to 4176:02:26, 23 June 2019 (UTC) 4112:, for instance, but only 3853:02:08, 23 June 2019 (UTC) 3460:11:13, 2 April 2018 (UTC) 3373:10:37, 2 April 2018 (UTC) 3327:10:55, 2 April 2018 (UTC) 3187:10:51, 2 April 2018 (UTC) 2990:05:10, 6 March 2016 (UTC) 2457:19:25, 29 June 2014 (UTC) 2402:17:06, 12 July 2016 (UTC) 2289:{\displaystyle \Gamma ,e} 1443:10:15, 15 June 2013 (UTC) 1419:22:44, 14 June 2013 (UTC) 1386:22:32, 14 June 2013 (UTC) 1367:18:37, 14 June 2013 (UTC) 1316:section's last paragraph 1272:22:52, 14 June 2013 (UTC) 1262:Ok, I changed the names. 1254:22:17, 14 June 2013 (UTC) 1242:17:56, 14 June 2013 (UTC) 1090:10:10, 2 April 2018 (UTC) 861:06:28, 28 June 2012 (UTC) 847:23:09, 18 June 2012 (UTC) 613:{\displaystyle \epsilon } 510:19:25, 24 June 2017 (UTC) 366:Category:Computer science 142: 129: 116:Computer science articles 78: 57: 4506:10:30, 23 May 2022 (UTC) 4273:Thanks and kind regards 3979:The index is omitted in 3668:15:15, 30 May 2018 (UTC) 3614:10:03, 29 May 2018 (UTC) 3582:14:10, 26 May 2018 (UTC) 3166:09:33, 18 May 2016 (UTC) 3116:13:36, 17 May 2016 (UTC) 2436:05:04, 28 May 2018 (UTC) 2409:a simplified algorithm J 1334:Hindley–Milner#Monotypes 1314:Hindley–Milner#Monotypes 368:and sub-categories with 3809:{\displaystyle \vdash } 3762:{\displaystyle \vdash } 3425:{\displaystyle \Theta } 3075:10:43, 4 May 2016 (UTC) 3055:18:34, 3 May 2016 (UTC) 2143:{\displaystyle \Gamma } 1737:{\displaystyle \alpha } 4164: 4156: 4149: 4106: 4079: 4027: 3968: 3939: 3910: 3881: 3837: 3810: 3790: 3763: 3740: 3528: 3426: 3242: 3222: 2982: 2952: 2582: 2550: 2310: 2290: 2208: 2176: 2144: 2124: 2092: 1925: 1872: 1834: 1784: 1783:{\displaystyle \beta } 1764: 1738: 1718: 1681: 1644: 1607: 1587: 1357:...(whatsoever)..." ? 1342: 1322: 1310: 1298: 1222: 1210: 1193:", at least in logic. 1183: 1067: 1015: 983: 948: 812:limitations again. -- 614: 594: 574: 329:Computer science stubs 39:This article is rated 4394:and, more explictly, 4150: 4107: 4080: 4028: 3969: 3940: 3911: 3882: 3838: 3811: 3791: 3764: 3741: 3719:Sudden appearance of 3551:Reworking the article 3529: 3427: 3243: 3241:{\displaystyle \tau } 3223: 3221:{\displaystyle \tau } 2978: 2953: 2583: 2551: 2311: 2309:{\displaystyle \tau } 2291: 2209: 2177: 2145: 2125: 2093: 1980:occuring unquantified 1926: 1873: 1835: 1785: 1765: 1739: 1719: 1682: 1645: 1608: 1606:{\displaystyle \tau } 1593:one can infer a type 1588: 1501:The example is: with 1403:#Original_research.3F 1338: 1318: 1302: 1290: 1214: 1202: 1184: 1068: 1016: 984: 949: 726:polymorphic recursion 615: 595: 575: 4116: 4089: 4037: 4010: 3951: 3922: 3893: 3864: 3820: 3800: 3773: 3753: 3723: 3475: 3416: 3232: 3212: 2857: 2560: 2528: 2300: 2274: 2186: 2154: 2134: 2102: 2058: 1882: 1844: 1794: 1774: 1748: 1728: 1691: 1654: 1617: 1597: 1505: 1170: 1032: 993: 958: 954:is confusing either 929: 604: 584: 533: 147:Things you can help 1763:{\displaystyle int} 524:kind (type theory)s 18:Talk:Hindley–Milner 4302: 4145: 4102: 4075: 4023: 3964: 3935: 3916:syntactical system 3906: 3887:declarative system 3877: 3833: 3806: 3786: 3759: 3736: 3524: 3422: 3238: 3218: 3013:is (literally) "a 3006:τ = α | ι | τ → τ 2948: 2578: 2546: 2411:. He notes there: 2392:reader decide? -- 2306: 2286: 2204: 2172: 2140: 2120: 2088: 1921: 1868: 1830: 1780: 1760: 1734: 1714: 1704: 1677: 1667: 1640: 1630: 1603: 1583: 1561: 1547: 1527: 1407:respective section 1241:_Quantifier?": --> 1179: 1099:Original research? 1063: 1049: 1011: 979: 944: 610: 590: 570: 452:Updated 2018-10-06 45:content assessment 4349:When no chair is 4300: 4235: 4219:comment added by 4123: 4044: 3462: 3450:comment added by 3375: 3363:comment added by 3329: 3317:comment added by 3306: 3292:Stuart.clayton.22 3290:comment added by 3189: 3177:comment added by 3118: 3057: 2937: 2910: 2905: 2893: 2889: 2868: 2864: 2632:comment added by 1924:{\displaystyle ,} 1332:Moreover, in the 1157:Qualifier --: --> 1092: 1080:comment added by 897: 880:comment added by 770:comment added by 678:comment added by 638: 626:comment added by 512: 500:comment added by 474: 473: 403: 402: 399: 398: 395: 394: 391: 390: 387: 386: 16:(Redirected from 4721: 4660:Stephen70edwards 4561:WP:ACCESSIBILITY 4374: 4339: 4333: 4330:'s backlog from 4303: 4296: 4205:bilber - a typo? 4154: 4152: 4151: 4146: 4122: 4111: 4109: 4108: 4103: 4101: 4100: 4084: 4082: 4081: 4076: 4043: 4032: 4030: 4029: 4024: 4022: 4021: 3973: 3971: 3970: 3965: 3963: 3962: 3944: 3942: 3941: 3936: 3934: 3933: 3915: 3913: 3912: 3907: 3905: 3904: 3886: 3884: 3883: 3878: 3876: 3875: 3842: 3840: 3839: 3834: 3832: 3831: 3815: 3813: 3812: 3807: 3795: 3793: 3792: 3787: 3785: 3784: 3768: 3766: 3765: 3760: 3745: 3743: 3742: 3737: 3735: 3734: 3533: 3531: 3530: 3525: 3490: 3489: 3431: 3429: 3428: 3423: 3247: 3245: 3244: 3239: 3227: 3225: 3224: 3219: 2957: 2955: 2954: 2949: 2947: 2946: 2936: 2932: 2931: 2909: 2904: 2903: 2902: 2892: 2891: 2890: 2884: 2883: 2867: 2866: 2865: 2751:Let-polymorphism 2644: 2587: 2585: 2584: 2579: 2555: 2553: 2552: 2547: 2315: 2313: 2312: 2307: 2295: 2293: 2292: 2287: 2213: 2211: 2210: 2207:{\displaystyle } 2205: 2181: 2179: 2178: 2175:{\displaystyle } 2173: 2149: 2147: 2146: 2141: 2129: 2127: 2126: 2123:{\displaystyle } 2121: 2097: 2095: 2094: 2089: 1930: 1928: 1927: 1922: 1877: 1875: 1874: 1869: 1839: 1837: 1836: 1831: 1789: 1787: 1786: 1781: 1769: 1767: 1766: 1761: 1743: 1741: 1740: 1735: 1723: 1721: 1720: 1715: 1686: 1684: 1683: 1678: 1649: 1647: 1646: 1641: 1612: 1610: 1609: 1604: 1592: 1590: 1589: 1584: 1411:Jochen Burghardt 1359:Jochen Burghardt 1280:Def. of monotype 1264:Jochen Burghardt 1233:Jochen Burghardt 1188: 1186: 1185: 1180: 1072: 1070: 1069: 1064: 1059: 1058: 1047: 1046: 1020: 1018: 1017: 1012: 988: 986: 985: 980: 953: 951: 950: 945: 896: 874: 782: 690: 653:Deductive system 619: 617: 616: 611: 599: 597: 596: 591: 579: 577: 576: 571: 545: 544: 453: 416: 415: 405: 377: 371: 246:Computer science 175:Article requests 164: 157: 156: 144: 118: 117: 114: 111: 108: 107:Computer science 98:Computer science 87: 80: 79: 74: 70:Computer science 66: 59: 42: 36: 35: 27: 21: 4729: 4728: 4724: 4723: 4722: 4720: 4719: 4718: 4689: 4688: 4652: 4516: 4477: 4439: 4372: 4354:Emphasis added. 4337: 4331: 4310:copied over -- 4299: 4257: 4207: 4114: 4113: 4092: 4087: 4086: 4035: 4034: 4013: 4008: 4007: 4001: 3954: 3949: 3948: 3925: 3920: 3919: 3896: 3891: 3890: 3867: 3862: 3861: 3823: 3818: 3817: 3816:or define what 3798: 3797: 3776: 3771: 3770: 3751: 3750: 3747: 3726: 3721: 3720: 3553: 3473: 3472: 3414: 3413: 3410: 3336: 3281: 3230: 3229: 3210: 3209: 2997: 2977: 2938: 2923: 2894: 2875: 2855: 2854: 2753: 2692: 2634:188.126.200.132 2627: 2612: 2558: 2557: 2526: 2525: 2444: 2298: 2297: 2272: 2271: 2255: 2184: 2183: 2152: 2151: 2132: 2131: 2100: 2099: 2056: 2055: 1880: 1879: 1842: 1841: 1792: 1791: 1772: 1771: 1746: 1745: 1726: 1725: 1689: 1688: 1652: 1651: 1615: 1614: 1595: 1594: 1503: 1502: 1480:, to give the " 1282: 1168: 1167: 1160: 1123:Björn Engelmann 1101: 1050: 1038: 1030: 1029: 991: 990: 956: 955: 927: 926: 875: 868: 832: 765: 673: 602: 601: 582: 581: 536: 531: 530: 479: 470: 469: 427: 413: 383: 380: 375: 369: 357:Project-related 352: 333: 314: 288: 269: 250: 231: 212: 188: 132:High-importance 115: 112: 109: 106: 105: 73:High‑importance 72: 43:on Knowledge's 40: 23: 22: 15: 12: 11: 5: 4727: 4725: 4717: 4716: 4711: 4706: 4701: 4691: 4690: 4687: 4686: 4651: 4648: 4647: 4646: 4632:Turing machine 4624: 4610: 4603: 4601: 4590: 4582: 4580: 4576: 4574: 4566: 4564: 4553: 4551: 4544:WP:NOTTEXTBOOK 4515: 4512: 4511: 4510: 4509: 4508: 4481:128.84.125.220 4476: 4473: 4458: 4457: 4450: 4438: 4435: 4420:type inference 4416: 4415: 4414: 4413: 4399: 4392:WP:EXPLAINLEAD 4367: 4366: 4365: 4360: 4355: 4346: 4324: 4323: 4322: 4256: 4253: 4252: 4251: 4206: 4203: 4202: 4201: 4144: 4141: 4138: 4135: 4132: 4129: 4126: 4121: 4099: 4095: 4074: 4071: 4068: 4065: 4062: 4059: 4056: 4053: 4050: 4047: 4042: 4020: 4016: 4000: 3997: 3996: 3995: 3977: 3976: 3975: 3961: 3957: 3946: 3932: 3928: 3917: 3903: 3899: 3888: 3874: 3870: 3830: 3826: 3805: 3783: 3779: 3758: 3746: 3733: 3729: 3717: 3716: 3715: 3714: 3713: 3652: 3651: 3647: 3639: 3638: 3635: 3632: 3629: 3626: 3623: 3599: 3598: 3595: 3570: 3569: 3566: 3563: 3560: 3552: 3549: 3548: 3547: 3523: 3520: 3517: 3514: 3511: 3508: 3505: 3502: 3499: 3496: 3493: 3488: 3485: 3482: 3421: 3409: 3406: 3405: 3404: 3390: 3335: 3332: 3331: 3330: 3280: 3277: 3276: 3275: 3274: 3273: 3272: 3271: 3270: 3269: 3268: 3267: 3253: 3237: 3217: 3197: 3196: 3195: 3194: 3193: 3192: 3191: 3190: 3157: 3153:), and below ( 3142: 3122: 3121: 3120: 3119: 3114:comment added 3096: 3095: 3094: 3093: 3087: 3086: 3085: 3084: 3078: 3077: 3066: 3063: 3053:comment added 3038: 3037: 2996: 2993: 2976: 2973: 2972: 2971: 2945: 2941: 2935: 2930: 2926: 2922: 2919: 2916: 2913: 2908: 2901: 2897: 2882: 2878: 2874: 2871: 2836: 2835: 2834: 2833: 2832: 2831: 2812: 2811: 2810: 2809: 2788: 2787: 2757:37.175.216.246 2752: 2749: 2745: 2744: 2729: 2728: 2691: 2688: 2687: 2686: 2671: 2670: 2665: 2664: 2659: 2658: 2654: 2650: 2649: 2624: 2623: 2622: 2621: 2611: 2608: 2607: 2606: 2590: 2589: 2577: 2574: 2571: 2568: 2565: 2545: 2542: 2539: 2536: 2533: 2520: 2519: 2514: 2513: 2502:Type inference 2497: 2496: 2482: 2477: 2476: 2468: 2467: 2443: 2440: 2439: 2438: 2419: 2418: 2417: 2389: 2388: 2374: 2369: 2368: 2364: 2363: 2358: 2357: 2339: 2338: 2324: 2320: 2319: 2305: 2285: 2282: 2279: 2254: 2251: 2250: 2249: 2248: 2247: 2246: 2245: 2244: 2243: 2222: 2221: 2220: 2219: 2218: 2217: 2216: 2215: 2203: 2200: 2197: 2194: 2191: 2171: 2168: 2165: 2162: 2159: 2139: 2119: 2116: 2113: 2110: 2107: 2087: 2084: 2081: 2078: 2075: 2072: 2069: 2066: 2063: 2044: 2043: 2042: 2041: 2040: 2039: 2038: 2037: 2026: 2025: 2024: 2023: 2022: 2021: 2020: 2019: 1998: 1997: 1996: 1995: 1994: 1993: 1992: 1991: 1988:standing alone 1969: 1968: 1967: 1966: 1965: 1964: 1963: 1962: 1939: 1938: 1937: 1936: 1935: 1934: 1933: 1932: 1920: 1917: 1914: 1911: 1908: 1905: 1902: 1899: 1896: 1893: 1890: 1887: 1867: 1864: 1861: 1858: 1855: 1852: 1849: 1829: 1826: 1823: 1820: 1817: 1814: 1811: 1808: 1805: 1802: 1799: 1779: 1759: 1756: 1753: 1733: 1713: 1710: 1707: 1702: 1699: 1696: 1676: 1673: 1670: 1665: 1662: 1659: 1639: 1636: 1633: 1628: 1625: 1622: 1602: 1582: 1579: 1576: 1573: 1570: 1567: 1564: 1559: 1556: 1553: 1550: 1545: 1542: 1539: 1536: 1533: 1530: 1525: 1522: 1519: 1516: 1513: 1510: 1492: 1491: 1490: 1489: 1488: 1487: 1486: 1485: 1467: 1466: 1465: 1464: 1463: 1462: 1461: 1460: 1450: 1449: 1448: 1447: 1446: 1445: 1434:German version 1424: 1423: 1422: 1421: 1399: 1389: 1388: 1377: 1281: 1278: 1277: 1276: 1275: 1274: 1257: 1256: 1206:quantification 1178: 1175: 1159: 1155: 1120: 1119: 1115: 1111: 1100: 1097: 1096: 1095: 1094: 1093: 1062: 1057: 1053: 1045: 1041: 1037: 1026: 1010: 1007: 1004: 1001: 998: 978: 975: 972: 969: 966: 963: 943: 940: 937: 934: 923: 920: 914: 913: 867: 864: 831: 828: 827: 826: 825: 824: 805: 784: 783: 761: 760: 756: 755: 729: 728: 721: 720: 719:Further topics 716: 715: 714: 713: 692: 691: 670: 665: 664: 660: 659: 655: 654: 650: 649: 645: 644: 643:Principal type 640: 639: 609: 589: 569: 566: 563: 560: 557: 554: 551: 548: 543: 539: 527: 519: 518: 514: 513: 492: 489: 484: 483: 478: 475: 472: 471: 468: 467: 455: 410: 408: 401: 400: 397: 396: 393: 392: 389: 388: 385: 384: 382: 381: 379: 378: 361: 353: 351: 350: 344: 334: 332: 331: 325: 315: 313: 312: 307: 299: 289: 287: 286: 280: 270: 268: 267: 261: 251: 249: 248: 242: 232: 230: 229: 223: 213: 211: 210: 205: 199: 189: 187: 186: 180: 168: 166: 165: 153: 152: 140: 139: 128: 122: 121: 119: 102:the discussion 88: 76: 75: 67: 55: 54: 48: 37: 24: 14: 13: 10: 9: 6: 4: 3: 2: 4726: 4715: 4712: 4710: 4707: 4705: 4702: 4700: 4697: 4696: 4694: 4685: 4681: 4677: 4672: 4671: 4670: 4669: 4665: 4661: 4656: 4649: 4645: 4641: 4637: 4633: 4629: 4625: 4623: 4619: 4615: 4611: 4608: 4604: 4602: 4599: 4595: 4591: 4587: 4583: 4581: 4577: 4575: 4572: 4567: 4565: 4562: 4558: 4554: 4552: 4549: 4545: 4541: 4540: 4539: 4538: 4534: 4530: 4526: 4521: 4513: 4507: 4503: 4499: 4495: 4494: 4493: 4492: 4491: 4490: 4486: 4482: 4474: 4472: 4471: 4467: 4463: 4455: 4451: 4448: 4444: 4443: 4442: 4436: 4434: 4433: 4429: 4425: 4421: 4412: 4408: 4404: 4400: 4397: 4393: 4389: 4388: 4387: 4383: 4379: 4375: 4368: 4364: 4361: 4359: 4356: 4353: 4352: 4347: 4345: 4342: 4341: 4336: 4329: 4325: 4321: 4317: 4313: 4309: 4308: 4307: 4304: 4297: 4291: 4287: 4286: 4285: 4283: 4279: 4274: 4271: 4269: 4265: 4260: 4254: 4250: 4246: 4242: 4238: 4237: 4236: 4234: 4230: 4226: 4222: 4221:Bellowhead678 4218: 4212: 4204: 4200: 4196: 4192: 4188: 4184: 4180: 4179: 4178: 4177: 4173: 4169: 4163: 4159: 4155: 4142: 4136: 4133: 4130: 4127: 4124: 4119: 4097: 4093: 4072: 4066: 4063: 4060: 4054: 4051: 4048: 4045: 4040: 4018: 4014: 4004: 4003:We say this: 3998: 3994: 3990: 3986: 3982: 3978: 3959: 3955: 3947: 3930: 3926: 3918: 3901: 3897: 3889: 3872: 3868: 3860: 3859: 3857: 3856: 3855: 3854: 3850: 3846: 3828: 3824: 3803: 3781: 3777: 3756: 3731: 3727: 3712: 3708: 3704: 3700: 3699: 3698: 3694: 3690: 3686: 3685: 3684: 3683: 3679: 3675: 3670: 3669: 3665: 3661: 3656: 3648: 3644: 3643: 3642: 3636: 3633: 3630: 3627: 3624: 3621: 3620: 3619: 3616: 3615: 3611: 3607: 3602: 3596: 3593: 3589: 3588: 3587: 3584: 3583: 3579: 3575: 3567: 3564: 3561: 3558: 3557: 3556: 3550: 3546: 3542: 3538: 3521: 3512: 3506: 3500: 3497: 3491: 3469: 3465: 3464: 3463: 3461: 3457: 3453: 3449: 3442: 3437: 3435: 3407: 3403: 3399: 3395: 3391: 3387: 3382: 3378: 3377: 3376: 3374: 3370: 3366: 3362: 3355: 3351: 3347: 3343: 3340: 3333: 3328: 3324: 3320: 3316: 3309: 3308: 3307: 3305: 3301: 3297: 3293: 3289: 3278: 3266: 3262: 3258: 3254: 3252: 3235: 3215: 3207: 3206: 3205: 3204: 3203: 3202: 3201: 3200: 3199: 3198: 3188: 3184: 3180: 3176: 3169: 3168: 3167: 3164: 3163: 3158: 3155: 3152: 3149: 3146: 3143: 3140: 3137: 3134: 3131: 3128: 3127: 3126: 3125: 3124: 3123: 3117: 3113: 3107: 3103: 3100: 3099: 3098: 3097: 3091: 3090: 3089: 3088: 3082: 3081: 3080: 3079: 3076: 3073: 3072: 3067: 3064: 3060: 3059: 3058: 3056: 3052: 3046: 3042: 3036: 3032: 3028: 3024: 3023: 3022: 3018: 3016: 3012: 3007: 3004: 3002: 2994: 2992: 2991: 2988: 2981: 2974: 2969: 2965: 2961: 2943: 2939: 2928: 2924: 2920: 2917: 2914: 2906: 2899: 2895: 2880: 2876: 2872: 2869: 2852: 2851: 2850: 2849: 2845: 2841: 2840:97.68.160.106 2830: 2826: 2822: 2818: 2817: 2816: 2815: 2814: 2813: 2808: 2804: 2800: 2795: 2792: 2791: 2790: 2789: 2786: 2782: 2778: 2774: 2769: 2768: 2767: 2766: 2762: 2758: 2750: 2748: 2743: 2739: 2735: 2731: 2730: 2727: 2723: 2719: 2714: 2713: 2712: 2711: 2707: 2703: 2698: 2695: 2689: 2685: 2681: 2677: 2673: 2672: 2667: 2666: 2661: 2660: 2655: 2652: 2651: 2647: 2646: 2645: 2643: 2639: 2635: 2631: 2619: 2618: 2617: 2616: 2615: 2609: 2605: 2601: 2597: 2592: 2591: 2575: 2572: 2569: 2566: 2543: 2540: 2537: 2534: 2531: 2522: 2521: 2516: 2515: 2512: 2509: 2508: 2503: 2499: 2498: 2495: 2491: 2487: 2483: 2479: 2478: 2474: 2470: 2469: 2465: 2461: 2460: 2459: 2458: 2454: 2450: 2441: 2437: 2433: 2429: 2425: 2420: 2416: 2413: 2412: 2410: 2406: 2405: 2404: 2403: 2399: 2395: 2387: 2383: 2379: 2375: 2371: 2370: 2366: 2365: 2360: 2359: 2355: 2354: 2353: 2352: 2348: 2344: 2343:Humble reader 2337: 2333: 2329: 2325: 2322: 2321: 2317: 2303: 2283: 2280: 2267: 2263: 2262: 2261: 2258: 2252: 2242: 2238: 2234: 2230: 2229: 2228: 2227: 2226: 2225: 2224: 2223: 2198: 2195: 2192: 2166: 2163: 2160: 2114: 2111: 2108: 2076: 2073: 2070: 2067: 2064: 2061: 2052: 2051: 2050: 2049: 2048: 2047: 2046: 2045: 2034: 2033: 2032: 2031: 2030: 2029: 2028: 2027: 2018: 2014: 2010: 2006: 2005: 2004: 2003: 2002: 2001: 2000: 1999: 1989: 1985: 1981: 1977: 1976: 1975: 1974: 1973: 1972: 1971: 1970: 1960: 1955: 1951: 1947: 1946: 1945: 1944: 1943: 1942: 1941: 1940: 1915: 1912: 1909: 1906: 1900: 1894: 1891: 1888: 1865: 1862: 1859: 1856: 1853: 1850: 1827: 1824: 1821: 1815: 1812: 1809: 1806: 1803: 1800: 1777: 1757: 1754: 1751: 1731: 1724:in HM, since 1711: 1708: 1705: 1700: 1697: 1674: 1671: 1668: 1663: 1660: 1637: 1634: 1631: 1626: 1623: 1600: 1580: 1577: 1574: 1568: 1565: 1562: 1557: 1554: 1551: 1548: 1543: 1540: 1537: 1534: 1531: 1528: 1523: 1520: 1517: 1514: 1511: 1500: 1499: 1498: 1497: 1496: 1495: 1494: 1493: 1483: 1479: 1478:Indeterminate 1475: 1474: 1473: 1472: 1471: 1470: 1469: 1468: 1458: 1457: 1456: 1455: 1454: 1453: 1452: 1451: 1444: 1441: 1440: 1435: 1430: 1429: 1428: 1427: 1426: 1425: 1420: 1416: 1412: 1408: 1404: 1400: 1397: 1393: 1392: 1391: 1390: 1387: 1384: 1383: 1378: 1375: 1371: 1370: 1369: 1368: 1364: 1360: 1356: 1352: 1348: 1341: 1337: 1335: 1330: 1328: 1321: 1317: 1315: 1309: 1307: 1301: 1297: 1295: 1289: 1287: 1279: 1273: 1269: 1265: 1261: 1260: 1259: 1258: 1255: 1252: 1251: 1246: 1245: 1244: 1243: 1238: 1234: 1230: 1225: 1221: 1219: 1213: 1209: 1207: 1201: 1199: 1194: 1192: 1176: 1165: 1156: 1154: 1153: 1149: 1145: 1140: 1137: 1136: 1132: 1128: 1124: 1116: 1112: 1109: 1108: 1107: 1104: 1098: 1091: 1087: 1083: 1079: 1055: 1051: 1043: 1039: 1027: 1024: 1005: 999: 996: 973: 970: 967: 964: 941: 938: 935: 932: 924: 921: 918: 917: 916: 915: 912: 908: 904: 900: 899: 898: 895: 891: 887: 883: 879: 871: 865: 863: 862: 858: 854: 849: 848: 844: 840: 835: 823: 819: 815: 810: 806: 804: 800: 796: 795:152.78.176.27 792: 788: 787: 786: 785: 781: 777: 773: 769: 763: 762: 758: 757: 754: 750: 746: 742: 741: 740: 739: 736: 735: 727: 723: 722: 718: 717: 712: 708: 704: 700: 696: 695: 694: 693: 689: 685: 681: 677: 671: 667: 666: 662: 661: 657: 656: 652: 651: 647: 646: 642: 641: 637: 633: 629: 625: 607: 567: 561: 558: 555: 552: 549: 546: 541: 537: 528: 525: 521: 520: 516: 515: 511: 507: 503: 499: 493: 491:Fix/recursion 490: 486: 485: 481: 480: 476: 466: 462: 459: 458: 457: 454: 451: 448: 445: 442: 439: 436: 433: 430: 426: 424: 420: 409: 407: 406: 374: 367: 363: 362: 360: 358: 354: 349: 346: 345: 343: 341: 340: 335: 330: 327: 326: 324: 322: 321: 316: 311: 308: 305: 301: 300: 298: 296: 295: 290: 285: 282: 281: 279: 277: 276: 271: 266: 263: 262: 260: 258: 257: 252: 247: 244: 243: 241: 239: 238: 233: 228: 225: 224: 222: 220: 219: 214: 209: 206: 204: 201: 200: 198: 196: 195: 190: 185: 182: 181: 179: 177: 176: 171: 170: 167: 163: 159: 158: 155: 154: 150: 146: 145: 141: 137: 133: 127: 124: 123: 120: 103: 99: 95: 94: 89: 86: 82: 81: 77: 71: 68: 65: 61: 56: 52: 46: 38: 34: 29: 28: 19: 4657: 4653: 4594:second point 4517: 4478: 4459: 4440: 4417: 4396:WP:TECHNICAL 4350: 4275: 4272: 4267: 4261: 4258: 4215:— Preceding 4208: 4186: 4165: 4161: 4157: 4006: 4002: 3748: 3671: 3657: 3653: 3640: 3617: 3603: 3600: 3592:WP:TECHNICAL 3585: 3571: 3554: 3446:— Preceding 3440: 3438: 3433: 3411: 3385: 3380: 3359:— Preceding 3356: 3352: 3348: 3344: 3341: 3337: 3313:— Preceding 3286:— Preceding 3282: 3250: 3173:— Preceding 3160: 3069: 3039: 3019: 3014: 3010: 3008: 3005: 3000: 2998: 2983: 2979: 2837: 2754: 2746: 2699: 2696: 2693: 2628:— Preceding 2625: 2613: 2610:Thus clearly 2505: 2504:, instead. — 2449:98.169.58.20 2445: 2423: 2414: 2408: 2394:Corwin.amber 2390: 2340: 2269: 2265: 2259: 2256: 1987: 1983: 1979: 1958: 1953: 1949: 1650:but not for 1481: 1437: 1395: 1380: 1354: 1350: 1346: 1343: 1339: 1331: 1326: 1323: 1319: 1311: 1305: 1303: 1299: 1293: 1291: 1285: 1283: 1248: 1228: 1226: 1223: 1217: 1215: 1211: 1205: 1203: 1197: 1195: 1190: 1163: 1161: 1141: 1138: 1121: 1105: 1102: 1076:— Preceding 1022: 876:— Preceding 872: 869: 850: 836: 833: 790: 772:63.145.62.78 766:— Preceding 732: 730: 680:38.88.172.69 674:— Preceding 622:— Preceding 496:— Preceding 456: 449: 443: 437: 431: 417: 411: 356: 355: 339:Unreferenced 337: 336: 318: 317: 292: 291: 273: 272: 254: 253: 235: 234: 216: 215: 192: 191: 173: 172: 131: 91: 51:WikiProjects 4548:WP:ORIGINAL 4529:Alexis King 4259:Hi Ed6767, 3974:algorithm W 3945:algorithm J 3110:—Preceding 3049:—Preceding 1986:instead of 1878:in HM. See 1158:Quantifier? 669:correctly). 663:Algorithm W 465:Algorithm G 461:Algorithm M 4693:Categories 4676:Cobalt pen 4636:Cobalt pen 4614:Cobalt pen 4498:Cobalt pen 4462:Cobalt pen 4424:Cobalt pen 4403:Cobalt pen 4373:Tenryuu 🐲 4312:Cobalt pen 4290:Cobalt pen 4278:Cobalt pen 4191:Cobalt pen 3985:Cobalt pen 3703:Cobalt pen 3660:Cobalt pen 3606:Cobalt pen 3574:Cobalt pen 3537:Cobalt pen 3394:Cobalt pen 3257:Cobalt pen 3147:), above ( 3027:Cobalt pen 2960:Cobalt pen 2821:Cobalt pen 2777:Cobalt pen 2734:Cobalt pen 2718:Cobalt pen 2676:Cobalt pen 2596:Cobalt pen 2486:Cobalt pen 2428:Cobalt pen 2378:Cobalt pen 2328:Cobalt pen 2233:Cobalt pen 2009:Cobalt pen 1347:difference 1229:quantifier 1218:quantified 1191:quantifier 1144:Cobalt pen 1074:article. 903:Cobalt pen 853:Cobalt pen 814:Cobalt pen 745:Cobalt pen 703:Cobalt pen 488:polytypes. 419:To-do list 4557:first one 4525:Wikibooks 4437:wp:length 4268:talk page 3981:one place 3843:means? -- 2702:Jan Burse 2473:subtyping 2296:yielding 1959:that type 1954:this type 1950:this type 1770:and (or) 1329:" means. 1166:" for a " 1164:qualifier 1114:calculus. 1025:variable. 809:subtyping 227:Computing 4628:AVL tree 4598:System F 4555:For the 4454:System F 4229:contribs 4217:unsigned 3689:DLeonard 3674:DLeonard 3650:results. 3448:unsigned 3361:unsigned 3315:unsigned 3300:contribs 3288:unsigned 3175:unsigned 3011:monotype 2794:System F 2690:System F 2630:unsigned 2036:article. 1687:nor for 1306:monotype 1294:monotype 1286:monotype 1131:contribs 1078:unsigned 890:contribs 882:Gwideman 878:unsigned 839:Kashikom 768:unsigned 676:unsigned 624:unsigned 498:unsigned 275:Maintain 218:Copyedit 4351:at hand 4328:WP:GOCE 4255:wp:tone 4187:finally 4168:Doradus 3845:Doradus 3112:undated 3051:undated 2098:in the 1200:", but 482:Missing 447:refresh 435:history 256:Infobox 194:Cleanup 134:on the 41:B-class 4607:WP:SCs 2995:Errors 2987:MaxEnt 2669:focus. 2657:error. 2464:origin 1351:define 517:Syntax 237:Expand 47:scale. 4301:talk! 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