domain theory

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Domain theory is a branch of
mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It has no generally ...
that studies special kinds of
partially ordered set upright=1.15, Fig.1 The Hasse diagram of the Power set, set of all subsets of a three-element set \, ordered by set inclusion, inclusion. Sets connected by an upward path, like \emptyset and \, are comparable, while e.g. \ and \ are not. In mathem ...
s (posets) commonly called domains. Consequently, domain theory can be considered as a branch of
order theory Order theory is a branch of mathematics which investigates the intuitive notion of order using binary relations. It provides a formal framework for describing statements such as "this is less than that" or "this precedes that". This article intro ...
. The field has major applications in
computer science Computer science deals with the theoretical foundations of information, algorithms and the architectures of its computation as well as practical techniques for their application. Computer science is the study of Algorithm, algorithmic proc ...
, where it is used to specify
denotational semantics In computer science Computer science deals with the theoretical foundations of information, algorithms and the architectures of its computation as well as practical techniques for their application. Computer science is the study of Algorit ...
, especially for
functional programming languages In computer science Computer science deals with the theoretical foundations of information, algorithms and the architectures of its computation as well as practical techniques for their application. Computer science is the study of Alg ...
. Domain theory formalizes the intuitive ideas of approximation and convergence in a very general way and is closely related to
topology s, which have only one surface and one edge, are a kind of object studied in topology. In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structu ...

.

Motivation and intuition

The primary motivation for the study of domains, which was initiated by
Dana Scott Dana Stewart Scott (born October 11, 1932) is an American logician who is the emeritus Hillman University Professor of Computer Science Computer science deals with the theoretical foundations of information, algorithms and the architectures o ...
in the late 1960s, was the search for a
denotational semantics In computer science Computer science deals with the theoretical foundations of information, algorithms and the architectures of its computation as well as practical techniques for their application. Computer science is the study of Algorit ...
of the
lambda calculus Lambda calculus (also written as ''λ''-calculus) is a formal system in mathematical logic for expressing computability, computation based on function Abstraction (computer science), abstraction and function application, application using variable N ...
. In this formalism, one considers "functions" specified by certain terms in the language. In a purely
syntactic In linguistics Linguistics is the science, scientific study of language. It encompasses the analysis of every aspect of language, as well as the methods for studying and modeling them. The traditional areas of linguistic analysis includ ...

way, one can go from simple functions to functions that take other functions as their input arguments. Using again just the syntactic transformations available in this formalism, one can obtain so called
fixed-point combinatorIn mathematics and computer science in general, a ''fixed point (mathematics), fixed point'' of a function is a value that is mapped to itself by the function. In combinatory logic for computer science, a fixed-point combinator (or fixpoint combinato ...
s (the best-known of which is the
Y combinator Y Combinator (YC) is an American seed money startup accelerator launched in March 2005. It has been used to launch over 2,000 companies, including Stripe, Airbnb Airbnb, Inc. (pronounced and stylized as airbnb) operates an onli ...
); these, by definition, have the property that ''f''(Y(''f'')) = Y(''f'') for all functions ''f''. To formulate such a denotational semantics, one might first try to construct a ''model'' for the lambda calculus, in which a genuine (total) function is associated with each lambda term. Such a model would formalize a link between the lambda calculus as a purely syntactic system and the lambda calculus as a notational system for manipulating concrete mathematical functions. The
combinator calculus Combinatory logic is a notation to eliminate the need for Quantifier (logic), quantified variables in mathematical logic. It was introduced by Moses Schönfinkel and Haskell Curry, and has more recently been used in computer science as a theoretic ...
is such a model. However, the elements of the combinator calculus are functions from functions to functions; in order for the elements of a model of the lambda calculus to be of arbitrary domain and range, they could not be true functions, only partial functions. Scott got around this difficulty by formalizing a notion of "partial" or "incomplete" information to represent computations that have not yet returned a result. This was modeled by considering, for each domain of computation (e.g. the natural numbers), an additional element that represents an ''undefined'' output, i.e. the "result" of a computation that never ends. In addition, the domain of computation is equipped with an ''ordering relation'', in which the "undefined result" is the
least element In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
. The important step to find a model for the lambda calculus is to consider only those functions (on such a partially ordered set) that are guaranteed to have
least fixed point In order theory, a branch of mathematics, the least fixed point (lfp or LFP, sometimes also smallest fixed point) of a function (mathematics), function from a partially ordered set to itself is the fixed point (mathematics), fixed point which is les ...
s. The set of these functions, together with an appropriate ordering, is again a "domain" in the sense of the theory. But the restriction to a subset of all available functions has another great benefit: it is possible to obtain domains that contain their own
function space In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
s, i.e. one gets functions that can be applied to themselves. Beside these desirable properties, domain theory also allows for an appealing intuitive interpretation. As mentioned above, the domains of computation are always partially ordered. This ordering represents a hierarchy of information or knowledge. The higher an element is within the order, the more specific it is and the more information it contains. Lower elements represent incomplete knowledge or intermediate results. Computation then is modeled by applying monotone
function Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...
s repeatedly on elements of the domain in order to refine a result. Reaching a fixed point is equivalent to finishing a calculation. Domains provide a superior setting for these ideas since fixed points of monotone functions can be guaranteed to exist and, under additional restrictions, can be approximated from below.

A guide to the formal definitions

In this section, the central concepts and definitions of domain theory will be introduced. The above intuition of domains being ''information orderings'' will be emphasized to motivate the mathematical formalization of the theory. The precise formal definitions are to be found in the dedicated articles for each concept. A list of general order-theoretic definitions, which include domain theoretic notions as well can be found in the
order theory glossary This is a glossary of some terms used in various branches of mathematics that are related to the fields of order theory, order, lattice (order), lattice, and domain theory. Note that there is a structured list of order topics available as well. Othe ...
. The most important concepts of domain theory will nonetheless be introduced below.

Directed sets as converging specifications

As mentioned before, domain theory deals with
partially ordered set upright=1.15, Fig.1 The Hasse diagram of the Power set, set of all subsets of a three-element set \, ordered by set inclusion, inclusion. Sets connected by an upward path, like \emptyset and \, are comparable, while e.g. \ and \ are not. In mathem ...
s to model a domain of computation. The goal is to interpret the elements of such an order as ''pieces of information'' or ''(partial) results of a computation'', where elements that are higher in the order extend the information of the elements below them in a consistent way. From this simple intuition it is already clear that domains often do not have a
greatest element In mathematics, especially in order theory, the greatest element of a subset S of a partially ordered set (poset) is an element of S that is greater than every other element of S. The term least element is defined duality (order theory), dually, t ...
, since this would mean that there is an element that contains the information of ''all'' other elements—a rather uninteresting situation. A concept that plays an important role in the theory is that of a directed subset of a domain; a directed subset is a non-empty subset of the order in which any two elements have an
upper bound In mathematics, particularly in order theory Order theory is a branch of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), s ...
that is an element of this subset. In view of our intuition about domains, this means that any two pieces of information within the directed subset are ''consistently'' extended by some other element in the subset. Hence we can view directed subsets as ''consistent specifications'', i.e. as sets of partial results in which no two elements are contradictory. This interpretation can be compared with the notion of a
convergent sequence As the positive integer An integer (from the Latin wikt:integer#Latin, ''integer'' meaning "whole") is colloquially defined as a number that can be written without a Fraction (mathematics), fractional component. For example, 21, 4, 0, ...
in
analysis Analysis is the process of breaking a complexity, complex topic or Substance theory, substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathematics and logic since before Ari ...
, where each element is more specific than the preceding one. Indeed, in the theory of
metric space In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
s, sequences play a role that is in many aspects analogous to the role of directed sets in domain theory. Now, as in the case of sequences, we are interested in the ''limit'' of a directed set. According to what was said above, this would be an element that is the most general piece of information that extends the information of all elements of the directed set, i.e. the unique element that contains ''exactly'' the information that was present in the directed set, and nothing more. In the formalization of order theory, this is just the
least upper bound are equal. Image:Supremum illustration.svg, 250px, A set ''A'' of real numbers (blue circles), a set of upper bounds of ''A'' (red diamond and circles), and the smallest such upper bound, that is, the supremum of ''A'' (red diamond). In mathematic ...
of the directed set. As in the case of limits of sequences, least upper bounds of directed sets do not always exist. Naturally, one has a special interest in those domains of computations in which all consistent specifications ''converge'', i.e. in orders in which all directed sets have a least upper bound. This property defines the class of directed-complete partial orders, or dcpo for short. Indeed, most considerations of domain theory do only consider orders that are at least directed complete. From the underlying idea of partially specified results as representing incomplete knowledge, one derives another desirable property: the existence of a
least element In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
. Such an element models that state of no information—the place where most computations start. It also can be regarded as the output of a computation that does not return any result at all.

Computations and domains

Now that we have some basic formal descriptions of what a domain of computation should be, we can turn to the computations themselves. Clearly, these have to be functions, taking inputs from some computational domain and returning outputs in some (possibly different) domain. However, one would also expect that the output of a function will contain more information when the information content of the input is increased. Formally, this means that we want a function to be monotonic. When dealing with complete partial order, dcpos, one might also want computations to be compatible with the formation of limits of a directed set. Formally, this means that, for some function ''f'', the image ''f''(''D'') of a directed set ''D'' (i.e. the set of the images of each element of ''D'') is again directed and has as a least upper bound the image of the least upper bound of ''D''. One could also say that ''f'' ''limit-preserving function (order theory), preserves directed suprema''. Also note that, by considering directed sets of two elements, such a function also has to be monotonic. These properties give rise to the notion of a Scott-continuous function. Since this often is not ambiguous one also may speak of ''continuous functions''.

Approximation and finiteness

Domain theory is a purely ''qualitative'' approach to modeling the structure of information states. One can say that something contains more information, but the amount of additional information is not specified. Yet, there are some situations in which one wants to speak about elements that are in a sense much simpler (or much more incomplete) than a given state of information. For example, in the natural subset-inclusion ordering on some powerset, any infinite element (i.e. set) is much more "informative" than any of its ''finite'' subsets. If one wants to model such a relationship, one may first want to consider the induced strict order < of a domain with order ≤. However, while this is a useful notion in the case of total orders, it does not tell us much in the case of partially ordered sets. Considering again inclusion-orders of sets, a set is already strictly smaller than another, possibly infinite, set if it contains just one less element. One would, however, hardly agree that this captures the notion of being "much simpler".

Way-below relation

A more elaborate approach leads to the definition of the so-called order of approximation, which is more suggestively also called the way-below relation. An element ''x'' is ''way below'' an element ''y'', if, for every directed set ''D'' with supremum such that :$y \sqsubseteq \sup D$, there is some element ''d'' in ''D'' such that :$x \sqsubseteq d$. Then one also says that ''x'' ''approximates'' ''y'' and writes :$x \ll y$. This does imply that :$x \sqsubseteq y$, since the singleton set is directed. For an example, in an ordering of sets, an infinite set is way above any of its finite subsets. On the other hand, consider the directed set (in fact, the Total order#Chains, chain) of finite sets :$\, \, \, \ldots$ Since the supremum of this chain is the set of all natural numbers N, this shows that no infinite set is way below N. However, being way below some element is a ''relative'' notion and does not reveal much about an element alone. For example, one would like to characterize finite sets in an order-theoretic way, but even infinite sets can be way below some other set. The special property of these finite elements ''x'' is that they are way below themselves, i.e. :$x \ll x$. An element with this property is also called compact element, compact. Yet, such elements do not have to be "finite" nor "compact" in any other mathematical usage of the terms. The notation is nonetheless motivated by certain parallels to the respective notions in set theory and
topology s, which have only one surface and one edge, are a kind of object studied in topology. In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structu ...

. The compact elements of a domain have the important special property that they cannot be obtained as a limit of a directed set in which they did not already occur. Many other important results about the way-below relation support the claim that this definition is appropriate to capture many important aspects of a domain.

Bases of domains

The previous thoughts raise another question: is it possible to guarantee that all elements of a domain can be obtained as a limit of much simpler elements? This is quite relevant in practice, since we cannot compute infinite objects but we may still hope to approximate them arbitrarily closely. More generally, we would like to restrict to a certain subset of elements as being sufficient for getting all other elements as least upper bounds. Hence, one defines a base of a poset ''P'' as being a subset ''B'' of ''P'', such that, for each ''x'' in ''P'', the set of elements in ''B'' that are way below ''x'' contains a directed set with supremum ''x''. The poset ''P'' is a continuous poset if it has some base. Especially, ''P'' itself is a base in this situation. In many applications, one restricts to continuous (d)cpos as a main object of study. Finally, an even stronger restriction on a partially ordered set is given by requiring the existence of a base of ''finite'' elements. Such a poset is called algebraic poset, algebraic. From the viewpoint of denotational semantics, algebraic posets are particularly well-behaved, since they allow for the approximation of all elements even when restricting to finite ones. As remarked before, not every finite element is "finite" in a classical sense and it may well be that the finite elements constitute an uncountable set. In some cases, however, the base for a poset is countable. In this case, one speaks of an ω-continuous poset. Accordingly, if the countable base consists entirely of finite elements, we obtain an order that is ω-algebraic.

Special types of domains

A simple special case of a domain is known as an elementary or flat domain. This consists of a set of incomparable elements, such as the integers, along with a single "bottom" element considered smaller than all other elements. One can obtain a number of other interesting special classes of ordered structures that could be suitable as "domains". We already mentioned continuous posets and algebraic posets. More special versions of both are continuous and algebraic complete partial order, cpos. Adding even further completeness (order theory), completeness properties one obtains Lattice (order)#Continuity and algebraicity, continuous lattices and algebraic lattices, which are just complete lattices with the respective properties. For the algebraic case, one finds broader classes of posets that are still worth studying: historically, the Scott domains were the first structures to be studied in domain theory. Still wider classes of domains are constituted by SFP-domains, L-domains, and bifinite domains. All of these classes of orders can be cast into various category (mathematics), categories of dcpos, using functions that are monotone, Scott-continuous, or even more specialized as morphisms. Finally, note that the term ''domain'' itself is not exact and thus is only used as an abbreviation when a formal definition has been given before or when the details are irrelevant.

Important results

A poset ''D'' is a dcpo if and only if each chain in ''D'' has a supremum. (The 'if' direction relies on the axiom of choice.) If ''f'' is a continuous function on a domain ''D'' then it has a least fixed point, given as the least upper bound of all finite iterations of ''f'' on the least element ⊥: :$\operatorname\left(f\right) = \bigsqcup_ f^n\left(\bot\right)$. This is the Kleene fixed-point theorem. The $\sqcup$ symbol is the Join and meet, directed join.

Generalizations

*
Topological domain theory
*A continuity space is a generalization of metric spaces and posets, that can be used to unify the notions of metric spaces and domains.