Combinators for Bidirectional Tree Transformations: A Linguistic Approach to the View Update Problem - part 1

Monotonic function

From Wikipedia, the free encyclopedia

"Monotonicity" redirects here. For information on monotonicity as it pertains to voting systems, see monotonicity criterion.

"Monotonic" redirects here. For other uses, see Monotone (disambiguation).

Figure 1. A monotonically increasing function. It is strictly increasing on the left and right while just non-decreasing in the middle.

Figure 2. A monotonically decreasing function

Figure 3. A function that is not monotonic

In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves the given order. This concept first arose incalculus, and was later generalized to the more abstract setting of order theory.

Combinatory logic

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Not to be confused with combinational logic, a topic in digital electronics.

Combinatory logic is a notation to eliminate the need for variables in mathematical logic. It was introduced by Moses Schönfinkel and Haskell Curry and has more recently been used in computer science as a theoretical model of computation and also as a basis for the design of functional programming languages. It is based on combinators. A combinator is a higher-order function that uses only function application and earlier defined combinators to define a result from its arguments.

Domain theory

Domain theory is a branch of mathematics that studies special kinds of partially ordered sets (posets) commonly called domains. Consequently, domain theory can be considered as a branch of order theory. The field has major applications in computer science, where it is used to specify denotational semantics, especially for functional programming languages. Domain theory formalizes the intuitive ideas of approximation and convergence in a very general way and has close relations to topology. An alternative important approach to denotational semantics in computer science is that of metric spaces.

Topology (from the Greek τόπος, “place”, and λόγος, “study”) is a major area of mathematics concerned with the most basic properties of space, such as connectedness. More precisely, topology studies properties that are preserved under continuous deformations, including stretching and bending, but not tearing or gluing. The exact mathematical definition is given below. Topology developed as a field of study out of geometry and set theory, through analysis of such concepts as space, dimension, and transformation.

Partially ordered set

From Wikipedia, the free encyclopedia

"Ordered set" redirects here. For totally ordered sets, see total order.

The Hasse diagram of the set of all subsetsof a three-element set {x, y, z}, ordered by inclusion.

In mathematics, especially order theory, a partially ordered set (or poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary relation that indicates that, for certain pairs of elements in the set, one of the elements precedes the other. Such a relation is called a partial orderto reflect the fact that not every pair of elements need be related: for some pairs, it may be that neither element precedes the other in the poset. Thus, partial orders generalize the more familiar total orders, in which every pair is related. A finite poset can be visualized through itsHasse diagram, which depicts the ordering relation.^[1]

A familiar real-life example of a partially ordered set is a collection of people ordered by genealogicaldescendancy. Some pairs of people bear the descendant-ancestor relationship, but other pairs bear no such relationship.