Camlp5 syntax extensions

A few syntax extensions are used in this project.

For testing we use the one from logger-p5 opam package. It allows to convert OCaml expression to its string form. For example, it rewrites let _ = REPR(1+2) to

$ camlp5o `ocamlfind query logger`/pa_log.cmo pr_o.cmo
let _ = "1 + 2", 1 + 2

For OCanren itself we use syntax extension to simplify writing relational programs

$  cat
let _ = fresh (x) z
$  camlp5o _build/camlp5/pa_ocanren.cmo pr_o.cmo
let _ = (fun x -> delay (fun () -> z))

OCanren vs. miniKanren

The correspondence between original miniKanren and OCanren constructs is shown below:







((==) a b)

(a === b)

((=/=) a b)

(a =/= b)

(conde (a b ...) (c d ...) ...)

conde [a &&& b &&& ...; c &&& d &&& ...; ...]

(fresh (x y ...) a b ... )

fresh (x y ...) a b ...

In addition, OCanren introduces explicit disjunction (|||) and conjunction (&&&) operators for goals.

OCanren-specific Camlp5 syntax extension

(Also known as regular extension)

This section is a part of Yue Li’s tutorial on OCanren.

We take a look at the implementation of the ocanren{} quotation which we will call the formula parser in the rest of this lesson. Our terminology follows Camlp5 Reference Manual. We take a top-down approach, starting with an overview of the structure of the parser, then explain its individual parts.

The structure of the parser: an overview

We describe the formula parser as the reader (a Camlp5 novice) sees it, and then putting it in perspective, briefly explain how it works.

What we see

This code links with Camlp5 libraries, in particular the one that provide ways to extend syntax: pa_extend. Loading them amounts to extending the OCaml syntactic category expression with several sub-categories one of which is named extend:

expr = ... | extend ;
extend = "EXTEND", extend-body, "END" ;

An expression that belongs to the category “extend” would be called an EXTEND statement.

Our formula parser has only one EXTEND statement, whose extend-body starts with a global indicator followed by a semicolon separated list of entries (whose names are, exhaustively, long_ident, expr, ocanren_embedding, ocanren_expr, ocanren_term, ocanren_term’ and ctyp). An entry is a (vertical bar separated) list of levels (with a pair of enclosing square brackets); a level is a (vertical bar separated) list of rules (with a pair of enclosing square brackets); a (non-empty) rule is a (semicolon separated) list of “psymbols” (collectively acting as a pattern) followed by an optional semantic action that produces an abstract syntax tree (or AST, of any string that matches the pattern specified by the list of psymbols). The details on the syntax and semantics of the “extend” category can be found in the Extensible Grammars section of the Camlp5 Manual.

Besides the EXTEND statement our formula parser has some auxiliary functions such as decapitalize, ctor and fix_term etc.

How it works

The syntax extension kit pa_extend is fundamental for the cascade of extensions described below. The entries expr and ctyp origin from the module Pcaml that is the core of Camlp5 and is opened by the formula parser. Pcaml initializes the (empty) grammar entries expr and ctyp. The standard OCaml parsing kit of Camlp5 then defines them by means of an EXTEND statement and accordng to the standard syntax of OCaml. Our EXTEND statement further extends these global entries with locally defined entries — entries other than expr and ctyp in our EXTEND statement are locally defined, such as ocanren_embedding, ocanren_expr and ocanren_term etc. Below you can find three stages of extension with links to documentation .. The following table summarizes the stages of extension, providing links to copies of relevant files from either OCanren ource or Camlp5 source, together with their documentations.

  1. Initialization is documented in The Pcaml module and happend in file : expr and ctyp.

  2. Parsing Kit for Standard OCaml : expr, ctyp

  3. OCanren Formula Parser expr, ctyp

As a preprocessing tool, Camlp5 defines its own parser for standard OCaml, so that any standard OCaml code can be converted by it into an AST recongnizable by the OCaml compiler. Is a redundant piece of work for we can just use the OCaml compiler to build the AST? Not exactly, because besides, Camlp5 also provides EXTEND statments so that syntactic categories defined in can be extended. The result is that using the combination of and we can convert code that is not wholly in OCaml into a purely OCaml AST.


The OCanren formula parser has the EXTEND statement as its core, which consists of a list of entries, notably the global entries expr and ctyp that extend the corresponding predefined entries that conform to standard OCaml. Such extension is characterized by the locally defined entries such as ocanren_embedding.

We will next focus on the extension of expr and leave ctyp. As far as the semantics is concerned entries are parsers for syntactic categories. From now on we use the words “entry” and “parser” interchangeably.

The global entry: expr

This is the major entry of the OCanren formula parser, which starts like:

expr: LEVEL "expr1" [ ...

where we see the entry name expr and the position LEVEL "expr1". We now use OCanren-expr to refer to the expr entry in the OCanren formula parser, and OCaml-expr to refer to the predefined entry expr in the Camlp5 parsing kit for standard OCaml. OCanren-expr extends OCaml-expr in the position LEVEL "expr1" : the first level of the OCanren-expr is merged with the level named “expr1” of the OCaml-expr, i.e., their rules are put together and grouped as a single level named “expr1”; other levels from OCanren-expr are inserted into OCaml-expr as new levels, right below the extended “expr1” level. There are three levels in the OCanren-expr, the third of which is:

[ e=ocanren_embedding -> e ]

This third level of OCanren-expr is inserted as a new level in OCaml-expr, and the entry ocanren_embedding directly corresponds to the ocanren{} quotations we see in the library implementation, so that we can mix ocanren{} quotations with standard OCaml expressions, and Camlp5 will take care to convert such mixture into standard OCaml AST. We now explain the local entry ocanren_embedding.

Local entries I: ocanren_embedding and ocanren_expr

The entry ocanren_embedding directly corresponds to the ocanren{} quotations we see in the library implementation, and it further calls the entry ocanren_expr to parse the content between the braces:

ocanren_embedding: [[ "ocanren"; "{"; e=ocanren_expr; "}" -> e ]];

The ocanren_expr entry has four levels which strongly reminds us of the recursive definition of a formula, i.e, a formula is either atomic, or a conjunction/ disjunction of two formulae, or an existential quantification over a formula, or an explicitly delimited formula (with braces).

1. The first level parses a disjunction:

"top" RIGHTA [ l=SELF; "|"; r=SELF -> <:expr< OCanren.disj $l$ $r$ >> ]
  1. The second level parses a conjunction:

RIGHTA [ l=SELF; "&"; r=SELF -> <:expr< OCanren.conj $l$ $r$ >> ]

3. The third level parses a fresh variable introduction (i.e., existential quantification):

[ "fresh"; vars=LIST1 LIDENT SEP ","; "in"; b=ocanren_expr LEVEL "top" ->
     (fun x b ->
        let p = <:patt< $lid:x$ >> in
        <:expr< OCanren.call_fresh ( fun $p$ -> $b$ ) >>

4. The fourth level parses atomic, named and grouped formulae (and else):

"primary" [
    p=prefix; t=ocanren_term                      -> let p = <:expr< $lid:p$ >> in <:expr< $p$ $t$ >>
  | l=ocanren_term; "==" ; r=ocanren_term         -> <:expr< OCanren.unify $l$ $r$ >>
  | l=ocanren_term; "=/="; r=ocanren_term         -> <:expr< OCanren.diseq $l$ $r$ >>
  | l=ocanren_term; op=operator; r=ocanren_term   -> let p = <:expr< $lid:op$ >> in
                                                     let a = <:expr< $p$ $l$ >> in
                                                     <:expr< $a$ $r$ >>
  | x=ocanren_term                                -> x
  | "{"; e=ocanren_expr; "}"                      -> e
  (* other rules omitted *)

The order of the levels determines the precedence of the logic connectives: the parser first sees if the formula is a disjunction at the top level, if not, sees if it is conjunction, and so on, implying that disjunction has the least precedence, above which is conjunction, then existential quantification, and finally syntactic equality, disequality and braced groups (among others) enjoy the highest precedence. We can justly call a level: a “precedence level”.

The first and second level also have the associativity indicator RIGHTA, requiring that the conjunction and disjunction connectives associate to the right.

The third level refers back to the first level (named “top”) when parsing the <body> part of a formula of the form fresh <vars> in <body>, implying that the scope of fresh extends to the right as far as possible.

Quotations and antiquotations

In every rule above we could see at least one quotation:

quotation = "<:", quotation name, "<", quotation body, ">>"

Within a quotation body we may see an antiquotation:

antiquotation = "$", antiquotation body, "$"

If antiquotations are not allowed, then a quotation body is any expression in the revised syntax of OCaml. At parse time a quotation is expanded by the (loaded and predefined) quotation expander q_MLast.cmo into an AST of the quotation body. An antiquotaion body is usually a pattern variable bound to some other AST which is inserted into the the quotation body’s AST.

Local entries II: ocanren_term and ocanren_term'

The values that we write in an ocanren{} quotation, such as "this is a string", 'c' (a single character), true (a boolean value), S (S O) (a constructor application), (O, S O) (a tuple), 15 (an integer), [1;2;3] (a list) and false :: [] (amending a list) etc., are converted into the injected level from the ground level where they seem to be. For example, the occurrence of S (S O) in the expression below is transformed into s (s (o ())) :

ocanren { fresh x in S (S O) == x }

Such conversion bridges the gap between the programmer’s intuition of writing OCaml values and OCanren’s internal representation of the same values, Inspecting the entries ocanren_term, ocanren_term' and their auxiliary functions help us know precisely how the conversion is performed.

Below is the definition of the entry ocanren_term :

ocanren_term: [[ t=ocanren_term' -> fix_term t ]];

where the ocanren_term' parser is called immediately to process expressions like S (S O) and the intermediate result (an AST) is bound to the pattern variable t and then passed to the auxiliary function fix_term. The AST returned by fix_term is returned by the parser ocanren_term.

The ocanren_term' parser has four levels, namely:

1. “app”, for applications.

"app"  LEFTA  [ l=SELF; r=SELF -> <:expr< $l$ $r$ >> ]``

Applications are treated as being left associative as indicated by LEFTA. This level not yet converts constructor applications into injection function applications. Instead it simply builds the AST of the application in a straightforward manner, not distinguishing a constructor application from a function application.

  1. “list” , for non-empty lists with :: as the top level constructor.

"list" RIGHTA [ l=SELF; "::"; r=SELF -> <:expr< OCanren.Std.List.cons $l$ $r$ >> ]

The constructor :: is replaced by the OCanren standard library function cons which is the injection function for the constructor OCanren.Std.List.Cons.

3. “primary”, which has rules for:

"!"; "("; e=expr; ")" -> e

So that the ocanren{} quotation would take any <value> from !(<value>) as is without further processing. In other words, the <value> will be parsed using the entry expr.

c=INT -> let n = <:expr< $int:c$ >> in <:expr< OCanren.Std.nat $n$ >>

Thus, occurrences of integers like 15 within the ocanren{} quotation would be converted to values of the Peano number type that is provided by the OCanren standard library OCanren.Std.Nat.

c=CHAR   -> let s = <:expr< $chr:c$ >> in <:expr< OCanren.inj (OCanren.lift $s$) >>      | s=STRING -> let s = <:expr< $str:s$ >> in <:expr< OCanren.inj (OCanren.lift $s$) >>

Characters and strings are injected using the primary injection function !! (see its signature and implementation).

| true"   -> <:expr< OCanren.Std.Bool.truo >>
| "false"  -> <:expr< OCanren.Std.Bool.falso >>

Boolean values are converted into the corresponding injected values from the OCanren standard library LBool.

"["; ts=LIST0 ocanren_term' SEP ";"; "]" ->
  ( match ts with
  | [] -> <:expr< OCanren.Std.nil () >>
  | _  ->
      List.fold_right (fun x l -> <:expr< OCanren.Std.List.cons $x$ $l$ >> )
      <:expr< OCanren.Std.nil () >>

The entry ocanren_term’ is recursively called to process the list members and the injection functions for list constructors are applied.

"("; op=operator_rparen -> <:expr< $lid:op$ >>

Operators are specified by the auxiliary function `is_operator <.>`__ and extracted by another auxiliary function operator_rparen (the name of which reads “operator right parenthesis”).

"("; ts=LIST0 ocanren_term' SEP ","; ")" ->
(match ts with
  | []  -> <:expr< OCanren.inj (OCanren.lift ()) >>
  | [t] -> t
  | _   -> <:expr< ( $list:ts$ ) >> )

There is a recursive call of the entry itself to process members of the tuple, and then the AST of the tuple is built.

  1. The level for long identifiers.

    [ long_ident ]

This level calls the entry long_ident to build AST’s of (possibly qualified) upper / lower case identifiers and operators which are taken as is.

Therefore, given S (S O) the ocanren_term' parser would return a straightforward translation into an AST. The interesting thing is done by fix_term and its helper ctor (read “C-tour”). The latter tests the input: if it is a (possibly qualified) uppercase identifier then sets the initial letter to lowercase and wraps the whole thing by Some, e.g., Mod1.Mod2.ABC becomes (roughly) Some Mod1.Mod2.aBC ; if the input is not a (qualified) uppercase identifier then returns None :

let rec ctor e =
  let loc = MLast.loc_of_expr e in
  match e with
  | <:expr< $uid:u$ >>   -> Some (<:expr< $lid:decapitalize u$ >>)
  | <:expr< $m$ . $e$ >> -> (match ctor e with Some e -> Some (<:expr< $m$ . $e$ >>) | _ -> None)
  | _                    -> None

The fix_term function then recurses down the structure of applications to systematically replace uppercase identifiers with lowercase identifiers produced by ctor. After a constant constructor is changed to lowercase, it is provided with the unit value () as the argument, e.g., O becomes o (). A non-constant constructor is not only set to lowercase, but also has its argument list transformed, e.g., Cons(a,b) becomes (roughly) cons a b. Tuples are also replaced by their OCanren standard library counterpart — logic tuples.

These lowercase identifiers converted from constructors are supposed to be injection functions, which must be defined by the programmer somewhere in the program, otherwise there would be some compile-time error like “unbound identifier”. This explains why the injection function names are always differ from the corresponding constructor names by one letter: the initial letter.