propensive / denominative

An experiment in numeric semantics to eradicate off-by-one errors

Denominative

An experiment in numeric semantics to eradicate off-by-one errors.

Denominative introduces a distinction between the types of cardinal and ordinal numbers with the clear and ambitious goal of avoiding off-by-one errors, without compromising performance.

Features

• provides an Ordinal type representing ordinal numbers
• Ordinals should be used for values which zero-indexed or one-indexed cardinal numbers could be ambiguous
• Ordinals are distinct from cardinal Ints
• conversions between Ordinals and Ints may be made only by specifying zero- or one-indexing
• introduces a distinct namespace for the first ten ordinals

Availability Plan

Denominative has not yet been published. The medium-term plan is to build Denominative with Fury and to publish it as a source build on Vent. This will enable ordinary users to write and build software which depends on Denominative.

Subsequently, Denominative will also be made available as a binary in the Maven Central repository. This will enable users of other build tools to use it.

For the overeager, curious and impatient, see building.

Getting Started

The Ordinal type

Denominative introduces a new type, Ordinal, which represents an ordinal number. Unlike cardinal numbers (still represented by Ints and Longs) an Ordinal has a first element, called Prim, which unambiguously refers to the first element of any sequence without the need to specify if the sequence is zero- or one-indexed. (There is, absolutely fundamentally, no concept of a "zeroth" Ordinal.)

Ordinal is represented internally by an Int, so shares the performance characteristics of using Ints, but is a distinct type. Thus, an Int such as 1, 138 or -12, cannot be used where an Ordinal is expected, and an Ordinal cannot be used where an Int is expected.

Conversions between Ints and Ordinals may only

The first ten Ordinal numbers have names, which arise from the first part of the sequence that begins, "primary", "secondary", "tertiary", etc.:

• Prim
• Sec
• Ter
• Quat
• Quin
• Sen
• Sept
• Oct
• Non
• Den

However, in practice, only Prim and Sec are likely to find regular use.

Given a sequence of elements, it's often useful to be able to refer to the last or second-to-last elements. This is possible with the ult (short for "ultimate") and pen (short for "penultimate") extension methods that are available on any countable value, and which return the Ordinal referring to these elements. A countable value typically means a Seq or one of its subtypes, but is actually an instance of any type that implements the Countable typeclass.

The ante extension method refers to the ordinal before pen, that is, the third-to-last (or antepenultimate) Ordinal index.

Arithmetic

Certain arithmetic operations are possible between Ordinal values and Ints, but many operations that exist for cardinal numbers (such as multiplication and division) do not make sense for ordinal numbers.

Here are some valid operations.

A cardinal number may be added to an Ordinal:

val ordinal: Ordinal = Ter + 3    // Sen
val ordinal2: Ordinal = 3 + Quin  // Oct

A cardinal number may be subtracted from an Ordinal:

val ordinal: Ordinal = Den - 7  // Ter

One Ordinal may be subtracted from another:

val cardinal: Int = Non - Sept  // 2

Intervals

A range of Ordinals is represented by an Interval. In all cases, this is a closed or "inclusive" interval, and is specified by its first Ordinal and its (included) final Ordinal. For example,

val interval: Interval = Ter ~ Sen

would represent the elements 2, 3, 5, 8 of the Fibonacci sequence, 1, 1, 2, 3, 5, 8, 13, 21, etc.

If, on the other hand, we had a finite sequence, xs, 2, 4, 6, 8, 10, 12, we could refer to all but the first and last elements by writing Sec to xs.pen. This would produce an interval representing the ordinals, Sec, Ter, Quat, Quin; omitting Prim and Sen (which would be xs.ult).

The size of an Interval is a cardinal number, thus an Int. We can get an Intervals size with the size method.

Iterating over Intervals

An Interval is most useful as a way of specifying a range of Ordinal values because we want to perform some operation iteratively using each of the values. Two methods are provided: foreach and foldLeft, which behave exactly as their familiar counterparts in Scala's standard collections library.

Status

Denominative is classified as embryotic. For reference, Scala One projects are categorized into one of the following five stability levels:

• embryonic: for experimental or demonstrative purposes only, without any guarantees of longevity
• fledgling: of proven utility, seeking contributions, but liable to significant redesigns
• maturescent: major design decisions broady settled, seeking probatory adoption and refinement
• dependable: production-ready, subject to controlled ongoing maintenance and enhancement; tagged as version 1.0.0 or later
• adamantine: proven, reliable and production-ready, with no further breaking changes ever anticipated

Projects at any stability level, even embryonic projects, can still be used, as long as caution is taken to avoid a mismatch between the project's stability level and the required stability and maintainability of your own project.

Denominative is designed to be small. Its entire source code currently consists of 89 lines of code.

Building

Denominative will ultimately be built by Fury, when it is published. In the meantime, two possibilities are offered, however they are acknowledged to be fragile, inadequately tested, and unsuitable for anything more than experimentation. They are provided only for the necessity of providing some answer to the question, "how can I try Denominative?".

1. Copy the sources into your own project

   Read the fury file in the repository root to understand Denominative's build structure, dependencies and source location; the file format should be short and quite intuitive. Copy the sources into a source directory in your own project, then repeat (recursively) for each of the dependencies.

   The sources are compiled against the latest nightly release of Scala 3. There should be no problem to compile the project together with all of its dependencies in a single compilation.

2. Build with Wrath

   Wrath is a bootstrapping script for building Denominative and other projects in the absence of a fully-featured build tool. It is designed to read the fury file in the project directory, and produce a collection of JAR files which can be added to a classpath, by compiling the project and all of its dependencies, including the Scala compiler itself.

   Download the latest version of wrath, make it executable, and add it to your path, for example by copying it to /usr/local/bin/.

   Clone this repository inside an empty directory, so that the build can safely make clones of repositories it depends on as peers of denominative. Run wrath -F in the repository root. This will download and compile the latest version of Scala, as well as all of Denominative's dependencies.

   If the build was successful, the compiled JAR files can be found in the .wrath/dist directory.

Contributing

Contributors to Denominative are welcome and encouraged. New contributors may like to look for issues marked beginner.

We suggest that all contributors read the Contributing Guide to make the process of contributing to Denominative easier.

Please do not contact project maintainers privately with questions unless there is a good reason to keep them private. While it can be tempting to repsond to such questions, private answers cannot be shared with a wider audience, and it can result in duplication of effort.

Author

Denominative was designed and developed by Jon Pretty, and commercial support and training on all aspects of Scala 3 is available from Propensive OÜ.

Name

To denominate is to assign a name to, which is denominative. Denominative assigns new names to the ordinal numbers in order to distinguish them from the cardinals.

In general, Scala One project names are always chosen with some rationale, however it is usually frivolous. Each name is chosen for more for its uniqueness and intrigue than its concision or catchiness, and there is no bias towards names with positive or "nice" meanings—since many of the libraries perform some quite unpleasant tasks.

Names should be English words, though many are obscure or archaic, and it should be noted how willingly English adopts foreign words. Names are generally of Greek or Latin origin, and have often arrived in English via a romance language.

Logo

The logo shows an on/off symbol, itself comprised of a 0 (off) and a 1 (on), alluding to the ideo of being "off by one".

License

Denominative is copyright © 2024 Jon Pretty & Propensive OÜ, and is made available under the Apache 2.0 License.