Abstraction Pattern Syntax

From Lojban
Revision as of 15:02, 2 June 2020 by Tsani (talk | contribs) (mistype anonymous -> named)
Jump to navigation Jump to search

Expressing certain ideas such as "I like apples more than you like oranges" is tricky with zmadu. We can say lo ni mi nelci lo plise cu zmadu lo ni do nelci lo najnimre, but there is redundancy in this sentence: we repeat the selbri nelci. A solution might be to introduce a new selbri based on zmadu dealing in binary relations instead of unary ones. A potential definition might be: "x1 is related to x2 by x5 (ka) more than x3 is related to x4 by x5 (ka) [by amount x6]". This is not only cumbersome because it has more places than zmadu, but it also doesn't generalize to relations of higher arities. Instead, I propose a pattern interpretation of the (optional) ce'ai-prenex in a ka-abstraction. This interpretation allows expressing several complex ideas, with one application being to give a general solution to the above example problem.

ka-abstractions and the ce'ai-prenex

Recall the identity ko'a ckaji lo ka ce'u broda = ko'a broda. This uses the "anonymous" bound variable ce'u: each appearance of ce'u denotes a new slot in the property-abstraction being constructed. The slot numbering is lexical, meaning that the x1 slot of the property is associated to the first ce'u, the x2 slot is associated to the second ce'u, and so on. Given that multiple occurrences of ce'u are distinct bound variables, it could be tricky to express a reflexive action such as "talking to oneself". One approach is to use goi as in mi ckaji lo ka ce'u goi ko'a tavla ko'a. The alternative, due to la latro'a, is to use the ce'ai-prenex: mi ckaji lo ka ko'a ce'ai ko'a ko'a tavla. In this syntax, ce'u is unused. Instead, variables appearing in the ce'ai-prenex are binders, and their order matters in the same way that ce'u appearance order matters. In case both a ce'ai-prenex and anonymous variables are used, then the anonymous variables are understood as positioned after all named variables, e.g.

lo ka ko'a ko'e ce'ai ko'a ce'u ko'e zukte

is equivalent to

lo ka ko'a ko'e ko'i ce'ai ko'a ko'i ko'e zukte

which refers to a selbri equivalent to se te se zukte.

A pattern syntax for lambdas

Given that ce'ai exists to formally bind variables in ka-abstractions, there is a natural extension to a syntax of patterns. The most basic pattern is a variable, e.g. ko'a. More complex patterns are built from descriptions (using gadri) and connectives (especially ce'o).

Connective-based patterns

Let's jump right in with an example using ce'o,

mi ce'o do ckaji lo ka ko'a ce'o ko'e ce'ai ko'a ko'e rinsa

The interpretation is mi do rinsa.

The way pattern matching works is to line up the argument mi ce'o do' against the pattern ko'a ce'o ko'e. Then mi fills ko'a and do fills ko'e. A natural thing to consider now is what happens in a nontrivial case, where the pattern and the argument don't obviously line up this way. Let's slightly rephrase the example.

mi'o ckaji lo ka ko'a ce'o ko'e ce'ai ko'a ko'e rinsa

Here we use mi'o instead of mi ce'o do. So we line up mi'o against the pattern ko'a ce'o ko'e, and this forces us to interpret mi'o as an ordered pair. There are two natural such interpretations: mi ce'o do and do ce'o mi. The effect is a universal reading, so the whole statement claims

mi do rinsa and do mi rinsa

More generally, when an argument can be decomposed in multiple ways to satisfy a pattern, then all decompositions are considered. Typically though, patterns and arguments are to be constructed to have exactly one meaning. To force an existential reading in this case, one can first convert mi'o to a sequence using vu'i (LAhE). This conversion operator gives an unspecified ordering, so vu'i mi'o ckaji lo ka ko'a ce'o ko'e ce'ai ko'a ko'e rinsa leaves it up to context whether the interpretation is mi do rinsa or do mi rinsa.

Now that we understand ce'o patterns, we can say "I like apples more than you like oranges" using zmadu.

mi ce'o lo plise do ce'o lo najnimre cu zmadu lo ni ko'a ce'o ko'e ce'ai ko'a ko'e nelci

Using the jargon of functional programming, lo ni ko'a ce'o ko'e ce'ai ko'a ko'e broda is said to uncurry the first two places of broda. To make this operation more convenient, we introduce the experimental MOI cmavo ka'oi.

Description-based patterns

Although the original problem statement only motivated the development of connective-based patterns, in particular ones built from ce'o, one can further extend the syntax and the semantics to include descriptions formed from gadri.

Consider the pattern lo pendo be ko'a in the example

mi ckaji lo ka lo pendo be ko'a ce'ai ko'a tadni

This says "My friend studies", albeit in a very roundabout way.

Using the "lining up" interpretation strategy, we match mi with lo pendo be ko'a. This makes sense if in context, I am a friend of someone, and the variable ko'a is made to refer to that person. Using lo to form the pattern is what gives the contextual interpretation. Different gadri could give different interpretations, but I leave developing these new interpretations as future work.