Prefix and suffix vectors: Difference between revisions

Latest revision as of 01:59, 8 March 2024

In Iverson notation, the prefix vector $\alpha ^{j}(n)$ and suffix vector $\omega ^{j}(n)$ of weight $j$ and length $n$ are Boolean vectors that can be used with Compress to select a prefix or suffix of a vector, a task now performed with Take. Primitives ⍺ and ⍵ based on this notation were defined in IVSYS/7090 and APL\360, but removed from APL\360 before its public release in 1968.

The prefix vector $\alpha ^{j}(n)$ is defined to be the length- $n$ vector where the first $j$ elements (or $n$ , if smaller) are 1 and the rest are 0. Likewise, the suffix vector has that many 1s at the end. In 1-indexed APL, N⍺J was defined to be J≥⍳N and N⍵J to be (1+N-J)≤⍳N. Because Iverson notation allows the length of a vector to be inferred from context, the first $j$ elements of vector $v$ can be taken with $\alpha ^{j}/v$ , and similarly $\alpha ^{j}/M$ and $\alpha ^{j}/\!\!/M$ take the first $j$ columns or rows of a matrix. In APL, what is now J↑V (for J≥0 and J≤N) was written ((⍴V)⍺J)/V, and (-J)↑V was ((⍴V)⍵J)/V (although most likely the shape ⍴V would have been saved elsewhere in the program rather than being written out). An instance of Drop such as J↓V was written (~(⍴V)⍺J)/V.

The prefix and suffix vectors were considered two out of a class of "special vectors", with the others being the interval vector $\iota (n)$ , now ⍳, and the "full vector" $\epsilon (n)$ of all 1s and "unit vector" $\epsilon ^{j}(n)$ , where the value at index $j$ is set to 1. However, $\epsilon$ was also used for Membership (the "characteristic vector"), which is closely related in that $\epsilon ^{j}(n)$ is (⍳N)∊J; only this meaning was defined in APL. The glyphs ⍺ and ⍵ came to be used as argument names in direct definition and later dfns.

Very little has been rendered obsolete in the course of this. One thing that has been, and gave us some pain, (I don’t know if it gives you any pain) and that was the removal of the alpha and omega from the list of operators; but I don’t think the released systems had those, so you never missed them. We had them and it was a bit of a jar to lose them. But they are gone now.
—Adin Falkoff^[1]

References

↑ Adin Falkoff. APL\360 History at APL69.

APL built-ins [edit]
Primitives (Timeline)	Functions
Scalar
		Monadic	Conjugate ∙ Negate ∙ Signum ∙ Reciprocal ∙ Magnitude ∙ Exponential ∙ Natural Logarithm ∙ Floor ∙ Ceiling ∙ Factorial ∙ Not ∙ Pi Times ∙ Roll ∙ Type ∙ Imaginary ∙ Square Root ∙ Round
		Dyadic	Add ∙ Subtract ∙ Times ∙ Divide ∙ Residue ∙ Power ∙ Logarithm ∙ Minimum ∙ Maximum ∙ Binomial ∙ Comparison functions ∙ Boolean functions (And, Or, Nand, Nor) ∙ GCD ∙ LCM ∙ Circular ∙ Complex ∙ Root
Non-Scalar
		Structural	Shape ∙ Reshape ∙ Tally ∙ Depth ∙ Ravel ∙ Enlist ∙ Table ∙ Catenate ∙ Reverse ∙ Rotate ∙ Transpose ∙ Raze ∙ Mix ∙ Split ∙ Enclose ∙ Nest ∙ Cut (K) ∙ Pair ∙ Link ∙ Partitioned Enclose ∙ Partition
		Selection	First ∙ Pick ∙ Take ∙ Drop ∙ Unique ∙ Identity ∙ Stop ∙ Select ∙ Replicate ∙ Expand ∙ Set functions (Intersection ∙ Union ∙ Without) ∙ Bracket indexing ∙ Index ∙ Cartesian Product ∙ Sort
		Selector	Index generator ∙ Grade ∙ Index Of ∙ Interval Index ∙ Indices ∙ Deal ∙ Prefix and suffix vectors
		Computational	Match ∙ Not Match ∙ Membership ∙ Find ∙ Nub Sieve ∙ Encode ∙ Decode ∙ Matrix Inverse ∙ Matrix Divide ∙ Format ∙ Execute ∙ Materialise ∙ Range
Operators	Monadic	Each ∙ Commute ∙ Constant ∙ Replicate ∙ Expand ∙ Reduce ∙ Windowed Reduce ∙ Scan ∙ Outer Product ∙ Key ∙ I-Beam ∙ Spawn ∙ Function axis ∙ Identity (Null, Ident)
Operators	Dyadic	Bind ∙ Compositions (Compose, Reverse Compose, Beside, Withe, Atop, Over) ∙ Inner Product ∙ Determinant ∙ Power ∙ At ∙ Under ∙ Rank ∙ Depth ∙ Variant ∙ Stencil ∙ Cut ∙ Direct definition (operator) ∙ Identity (Lev, Dex)
Quad names	Index origin ∙ Comparison tolerance ∙ Migration level ∙ Atomic vector

[1] Adin Falkoff. APL\360 History at APL69.

[1]

@@ Line 6: / Line 6: @@
 The prefix vector <math>\alpha^j(n)</math> is defined to be the length-<math>n</math> [[vector]] where the first <math>j</math> elements (or <math>n</math>, if smaller) are 1 and the rest are 0. Likewise, the suffix vector has that many 1s at the end. In 1-indexed APL, <code>N⍺J</code> was defined to be <syntaxhighlight lang=apl inline>J≥⍳N</syntaxhighlight> and <code>N⍵J</code> to be <syntaxhighlight lang=apl inline>(1+N-J)≤⍳N</syntaxhighlight>. Because Iverson notation allows the length of a vector to be inferred from context, the first <math>j</math> elements of vector <math>v</math> can be taken with <math>\alpha^j/v</math>, and similarly <math>\alpha^j/M</math> and <math>\alpha^j/\!\!/M</math> take the first <math>j</math> columns or rows of a matrix. In APL, what is now <syntaxhighlight lang=apl inline>J↑V</syntaxhighlight> (for <syntaxhighlight lang=apl inline>J≥0</syntaxhighlight> and <syntaxhighlight lang=apl inline>J≤N</syntaxhighlight>) was written <syntaxhighlight lang=apl inline>((⍴V)⍺J)/V</syntaxhighlight>, and <syntaxhighlight lang=apl inline>(-J)↑V</syntaxhighlight> was <syntaxhighlight lang=apl inline>((⍴V)⍵J)/V</syntaxhighlight> (although most likely the shape <syntaxhighlight lang=apl inline>⍴V</syntaxhighlight> would have been saved elsewhere in the program rather than being written out). An instance of [[Drop]] such as <syntaxhighlight lang=apl inline>J↓V</syntaxhighlight> was written <syntaxhighlight lang=apl inline>(~(⍴V)⍺J)/V</syntaxhighlight>.
-The prefix and suffix vectors were considered two out of a class of "special vectors", with the others being the [[interval vector]] <math>\iota(n)</math>, now <syntaxhighlight lang=apl inline>⍳</syntaxhighlight>, and the "full vector" <math>\epsilon(n)</math> of all 1s and "unit vector" <math>\epsilon^j(n)</math>, where the value at index <math>j</math> is set to 1. However, <math>\epsilon</math> was also used for [[Membership]] (the "characteristic vector"), which is closely related in that <math>\epsilon^j(n)</math> is <syntaxhighlight lang=apl inline>(⍳N)∊J</syntaxhighlight>; only this meaning was defined in APL. The [[glyph]]s <code>⍺</code> and <code>⍵</code> came to be used as argument names in [[direct definiton]] and later [[dfn]]s.
+The prefix and suffix vectors were considered two out of a class of "special vectors", with the others being the [[interval vector]] <math>\iota(n)</math>, now <syntaxhighlight lang=apl inline>⍳</syntaxhighlight>, and the "full vector" <math>\epsilon(n)</math> of all 1s and "unit vector" <math>\epsilon^j(n)</math>, where the value at index <math>j</math> is set to 1. However, <math>\epsilon</math> was also used for [[Membership]] (the "characteristic vector"), which is closely related in that <math>\epsilon^j(n)</math> is <syntaxhighlight lang=apl inline>(⍳N)∊J</syntaxhighlight>; only this meaning was defined in APL. The [[glyph]]s <code>⍺</code> and <code>⍵</code> came to be used as argument names in [[Direct definition (notation)|direct definition]] and later [[dfn]]s.
 {{quote|Very little has been rendered obsolete in the course of this. One thing that has been, and gave us some pain, (I don’t know if it gives you any pain) and that was the removal of the alpha and omega from the list of operators; but I don’t think the released systems had those, so you never missed them. We had them and it was a bit of a jar to lose them. But they are gone now.|—[[Adin Falkoff]]<ref>[[Adin Falkoff]]. [https://www.jsoftware.com/papers/apl360history.htm APL\360 History] at [[APL69]].</ref>}}

Prefix and suffix vectors: Difference between revisions

Latest revision as of 01:59, 8 March 2024

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