adjacency as an operator - can any lexer handle it?

Question

Say a language defines adjacency of two mathematical unicode alphanumerical symbols as an operator. Say,

Answer 1

Invisible operators cannot be recognized with lexical analysis, for reasons which should be more or less obvious. You can only deduce the presence of an invisible operator by analyzing the syntactic context, which is the role of a parser.

Of course, most lexical analysis tools allow arbitrary code to be executed for each recognized token, so nothing stops you from building a state machine, or even a complete parser, into the lexical scanner. That is rarely good design.

If your language is unambiguous, then there is no problem handling adjacency in your grammar. But some care must be taken. For example, you would rarely want x-4 to be parsed as a multiplication of x and -4, but a naive grammar which included, eg.,

expr -> term | expr '-' term
term -> factor | term factor | term '*' factor
factor -> ID | NUMBER | '(' expr ')' | '-' factor

would include that ambiguity. To resolve it, you need to disallow the adjacency production with a second operand starting with a unary operator:

expr -> term | expr '-' term
term -> factor | term item | term '*' factor
factor -> item | '-' factor
item -> ID | NUMBER | '(' expr ')'

Note the difference between term -> term '*' factor, which allows x * - y, and term -> term base, which does not allow x - y (expr -> expr '-' term recognizes x - y as a subtraction).

For examples of context-free grammars which allow adjacency as an operator, see, for example, Awk, in which adjacency represents string concatenation, and Haskell, in which it represents function application.

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Since this question comes up from time to time, there are a number of relevant answers already on SO. Here are a few:

• Parsing a sequence of expressions using yacc. Invisible function application operator. Uses yacc/bison; includes both explicit and precedence-based solutions

• yacc - Precedence of a rule with no operator? Invisible string concatenation operator. Uses Ply (Python parser generator)

• Concatenation shift-reduce conflict Another invisible concatenation operator. Uses JavaCUP.

• Parsing a sequence of expressions using yacc Invisible function application operator. Uses fsyacc (F# parser generator)

• Using yacc precedence for rules with no terminals, only non-terminals. Adjacency in ordinary mathematical expressions. Uses yacc/bison with precedence rules.

• bison/yacc - limits of precedence settings. Haskell-like function application adjacency. Uses yacc/bison with precedence rules.

Answer 2

Here is one example using pyparsing in Python:

import pyparsing as pp

integer = pp.pyparsing_common.integer()
variable = pp.oneOf(list("abcdefghijklmnopqrstuvwxyz"))

base_operand = integer | variable

implied_multiplication = pp.Empty().addParseAction(lambda: "*")
expr = pp.infixNotation(base_operand,
                [
                    ("**", 2, pp.opAssoc.LEFT),
                    (implied_multiplication, 2, pp.opAssoc.LEFT),
                    (pp.oneOf("+ -"), 1, pp.opAssoc.RIGHT),
                    (pp.oneOf("* /"), 2, pp.opAssoc.LEFT),
                    (pp.oneOf("+ -"), 2, pp.opAssoc.LEFT),
                ])

This assumes that variables are just single characters. There is also some fudging of precedence of operations to make adjacency, exponentiation, and leading signs work. The parse action added to the implied_multiplication expression is there to show the insertion of the multiplication operator.

Here is some test output:

tests = """
    x-4
    ax**2 + bx +c
    ax**2-bx+c
    mx+b
    """
expr.runTests(tests, fullDump=False)

prints:

x-4
[['x', '-', 4]]

ax**2 + bx +c
[[['a', '*', ['x', '**', 2]], '+', ['b', '*', 'x'], '+', 'c']]

ax**2-bx+c
[[['a', '*', ['x', '**', 2]], '-', ['b', '*', 'x'], '+', 'c']]

mx+b
[[['m', '*', 'x'], '+', 'b']]

Answer 3

Unless tokens have a fixed length, you must separate adjacent tokens of the same type with some other token or whitespace. The Gosu programming language incorporates adjacency to implement "binding expressions" which support units:

var length = 10m  // 10 meters

var work = 5kg * 9.8 m/s/s * 10m
print( work )  // prints 490 J

var investment = 5000 EUR + 10000 USD

var date = 1966-May-5 2:35:53:909 PM PST