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https://github.com/propensive/denominative

An experiment in numeric semantics to eradicate off-by-one errors
https://github.com/propensive/denominative

off-by-one ordinal-numbers scala

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

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# Denominative



__Distinct ordinal numbers to avoid 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

- `Ordinal`s should be used for values which zero-indexed or one-indexed cardinal numbers could be ambiguous

- `Ordinal`s are distinct from cardinal `Int`s

- conversions between `Ordinal`s and `Int`s may be made only by specifying zero- or one-indexing

- introduces a distinct namespace for the first ten ordinals



## Availability



## Getting Started



### The `Ordinal` type



_Denominative_ introduces a new type, `Ordinal`, which represents an ordinal number. Unlike cardinal numbers

(still represented by `Int`s and `Long`s) 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 `Int`s, 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 `Int`s and `Ordinal`s 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 `Int`s, 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`:

```scala

val ordinal: Ordinal = Ter + 3    // Sen

val ordinal2: Ordinal = 3 + Quin  // Oct

```



A cardinal number may be subtracted from an `Ordinal`:

```scala

val ordinal: Ordinal = Den - 7  // Ter

```



One `Ordinal` may be subtracted from another:

```scala

val cardinal: Int = Non - Sept  // 2

```



### `Interval`s



A range of `Ordinal`s 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,

```scala

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 `Interval`s size with the `size`

method.



#### Iterating over `Interval`s



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, Soundness 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 154 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](https://github.com/propensive/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`](https://github.com/propensive/wrath/releases/latest), 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](https://github.com/propensive/denominative/labels/beginner).



We suggest that all contributors read the [Contributing

Guide](/contributing.md) 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](https://propensive.com/).



## 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, Soundness 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](/license.md).