From 248c15ba7c69a37c186819c11cb7d98bea36b3c7 Mon Sep 17 00:00:00 2001 From: Marshall Lochbaum Date: Tue, 29 Dec 2020 20:38:22 -0500 Subject: Specify identity values for reductions --- docs/spec/inferred.html | 58 +++++++++++++++++++++++++++++++++++++++++++++++++ 1 file changed, 58 insertions(+) (limited to 'docs') diff --git a/docs/spec/inferred.html b/docs/spec/inferred.html index f18197f3..724014c9 100644 --- a/docs/spec/inferred.html +++ b/docs/spec/inferred.html @@ -7,6 +7,64 @@

Specification: BQN inferred properties

BQN includes some simple deductive capabilities: detecting the type of empty array elements, and the Undo (⁼) and Under (⌾) modifiers. These tasks are a kind of proof-based or constraint programming, and can never be solved completely (some instances will be undecidable) but can be solved in more instances by ever-more sophisticated algorithms. To allow implementers to develop more advanced implementations while offering some stability and portability to programmers, two kinds of specification are given here. First, constraints are given on the behavior of inferred properties. These are not exact and require some judgment on the part of the implementer. Second, behavior for common or useful cases is specified more precisely. Non-normative suggestions are also given as a reference for implementers.

For the specified cases, the given functions and modifiers refer to those particular representations. It is not necessary to detect equivalent representations, for example to reduce (+-×)⁼ to ∨⁼. However, it is necessary to identify computed functions and modifiers: for example F⁼ when the value of F in the expression is ∨, or (1⊑∧‿∨)⁼.

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Identities

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When monadic Fold (´) or Insert (˝) is called on an array of length 0, BQN attempts to infer a right identity value for the function in order to determine the result. A right identity value for a dyadic function 𝔽 is a value r such that e≡e𝔽r for any element e in the domain. For such a value r, the reduction r 𝔽´ l is equivalent to 𝔽´ l for a non-empty list l, because the first application (¯1⊑l) 𝔽 r gives ¯1⊑l, which is the starting point when no initial value is given. It's thus reasonable to define 𝔽´ l to be r 𝔽´ l for an empty list l as well, giving a result r.

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More specifically, the identity of a dyadic function 𝔽 is defined to be a right identity value for the range of 𝔽, if exactly one such value exists. Otherwise, there is no identity and 𝔽´ or 𝔽˝ on an argument with length 0 results in an error.

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Identity values for the arithmetic primitives below must be recognized.

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
IdFnFnId
0+-0
1×÷1
1⋆¬1
∞⌊⌈¯∞
0∨∧1
0≠=1
0>≥1

Undo

The Undo 1-modifier ⁼, given an operand 𝔽 and argument 𝕩, and possibly a left argument 𝕨, finds a value y such that 𝕩≡𝕨𝔽y, that is, an element of the pre-image of 𝕩 under 𝔽 or 𝕨𝔽⊢. Thus it satisfies the constraint 𝕩 ≡ 𝕨𝔽𝕨𝔽⁼𝕩 (𝕨𝔽⁼⊢ is a right inverse of 𝕨𝔽⊢) provided 𝔽⁼ and 𝔽 both complete without error. 𝔽⁼ should of course give an error if no inverse element exists, and can also fail if no inverse can be found. It is also preferred for 𝔽⁼ to give an error if there are many choices of inverse with no clear way to choose one of them: for example, 0‿0⍉m returns the diagonal of matrix m; 0‿0⍉⁼2‿3 requires values to be chosen for the off-diagonal elements in its result. It is better to give an error, encouraging the programmer to use a fully-specified approach like 2‿3⌾(0‿0⊸⍉) applied to a matrix of initial elements, than to return a result that could be very different from other implementations.

When working with limited-precision numbers, it may be difficult or impossible to exactly invert the operand function. Instead, it is generally acceptable to perform a computation that, if done with unlimited precision, would exactly invert 𝔽 computed with unlimited precision. This principle is the basis for the numeric inverses specified below. It is also acceptable to find an inverse by numeric methods, provided that the error in the inverse value found relative to an unlimited-precision inverse can be kept close to the inherent error in the implementation's number format.

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