Parsing a Miller DSL (domain-specific language) expression goes through three representations:

Source code which is a string of characters.
Abstract syntax tree (AST)
Concrete syntax tree (AST)

The job of the PGPG parser is to turn the DSL string into an AST.

The job of the CST builder is to turn the AST into a CST.

The job of the put and filter transformers is to execute the CST statements on each input record.

Source-code representation

For example, the part between the single quotes in

mlr put '$v = $i + $x * 4 + 100.7 * $y' myfile.dat

AST representation

Use put -v to display the AST:

mlr -n put -v '$v = $i + $x * 4 + 100.7 * $y'
RAW AST:
* StatementBlock
    * SrecDirectAssignment "=" "="
        * DirectFieldName "md_token_field_name" "v"
        * Operator "+" "+"
            * Operator "+" "+"
                * DirectFieldName "md_token_field_name" "i"
                * Operator "*" "*"
                    * DirectFieldName "md_token_field_name" "x"
                    * IntLiteral "md_token_int_literal" "4"
            * Operator "*" "*"
                * FloatLiteral "md_token_float_literal" "100.7"
                * DirectFieldName "md_token_field_name" "y"

Note the following about the AST:

Parentheses, commas, semicolons, line endings, whitespace are all stripped away
Variable names and literal values remain as leaf nodes of the AST
Operators like = + - * / **, function names, and so on remain as non-leaf nodes of the AST
Operator precedence is clear from the tree structure

Operator-precedence examples:

$ mlr -n put -v '$x = 1 + 2 * 3'
RAW AST:
* StatementBlock
    * SrecDirectAssignment "=" "="
        * DirectFieldName "md_token_field_name" "x"
        * Operator "+" "+"
            * IntLiteral "md_token_int_literal" "1"
            * Operator "*" "*"
                * IntLiteral "md_token_int_literal" "2"
                * IntLiteral "md_token_int_literal" "3"

$ mlr -n put -v '$x = 1 * 2 + 3'
RAW AST:
* StatementBlock
    * SrecDirectAssignment "=" "="
        * DirectFieldName "md_token_field_name" "x"
        * Operator "+" "+"
            * Operator "*" "*"
                * IntLiteral "md_token_int_literal" "1"
                * IntLiteral "md_token_int_literal" "2"
            * IntLiteral "md_token_int_literal" "3"

$ mlr -n put -v '$x = 1 * (2 + 3)'
RAW AST:
* StatementBlock
    * SrecDirectAssignment "=" "="
        * DirectFieldName "md_token_field_name" "x"
        * Operator "*" "*"
            * IntLiteral "md_token_int_literal" "1"
            * Operator "+" "+"
                * IntLiteral "md_token_int_literal" "2"
                * IntLiteral "md_token_int_literal" "3"

CST representation

There's no -v display for the CST, but it's simply a reshaping of the AST with pre-processed setup of function pointers to handle each type of statement on a per-record basis.

The if/else and/or switch statements to decide what to do with each AST node are done at CST-build time, so they don't need to be re-done when the syntax tree is executed once on every data record.

Source directories/files

The AST logic is in ./ast*.go. I didn't use a pkg/dsl/ast naming convention, although that would have been nice, in order to avoid a Go package-dependency cycle.
The CST logic is in ./cst. Please see cst/README.md for more information.