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11.1 BNF Grammars

Syntactic conventions

In the following BNF grammars, literal strings are indicated in single quotes ('). Grammar non-terminals are unquoted strings. A prefix question mark (?A) indicates optionality of A. A postfixed Kleene star (A*) indicates zero or more occurrences of A. The notation A % B, following the Boost Spirit parser library’s notation, is shorthand for ?(A (B A)*), i.e., any number of A (including zero), separated by B. A postfixed, curly-braced number indicates a fixed number of repetitions; e.g., A{6} is equivalent to a sequence of six copies of A.

Programs

program ::= ?functions ?data ?tdata ?params ?tparams ?model ?generated

functions ::= 'functions' function_decls
data ::= 'data' var_decls
tdata ::= 'transformed data' var_decls_statements
params ::= 'parameters' var_decls
tparams ::= 'transformed parameters' var_decls_statements
model ::= 'model' var_decls_statements
generated ::= 'generated quantities' var_decls_statements
function_decls ::= '{' function_decl* '}'
var_decls ::= '{' var_decl* '}'
var_decls_statements ::= '{' var_decl* statement* '}'

Function declarations and definitions

function_decl ::= return_type identifier '(' parameter_decl % ',' ')'
                  statement

return_type ::= 'void' | unsized_type
parameter_decl ::= ?'data' unsized_type identifier
unsized_type ::= basic_type ?unsized_dims
basic_type ::= 'int' | 'real' | 'vector' | 'row_vector' | 'matrix'
unsized_dims ::= '['  ','*  ']'

### Variable declarations and compound definitions {-}

var_decl ::= var_type variable ?dims ?('=' expression) ';'

var_type ::= 'int' range_constraint
           | 'real' constraint
           | 'vector' constraint '[' expression ']'
           | 'ordered' '[' expression ']'
           | 'positive_ordered' '[' expression ']'
           | 'simplex' '[' expression ']'
           | 'unit_vector' '[' expression ']'
           | 'row_vector' constraint '[' expression ']'
           | 'matrix' constraint '[' expression ',' expression ']'
           | 'cholesky_factor_corr' '[' expression ']'
           | 'cholesky_factor_cov' '[' expression ?(',' expression) ']'
           | 'corr_matrix' '[' expression ']'
           | 'cov_matrix' '[' expression ']'

constraint ::= ?('<' range '>')

range ::= 'lower' '=' constr_expression ',' 'upper' = constr_expression
        | 'lower' '=' constr_expression
        | 'upper' '=' constr_expression

dims ::= '['  expressions ']'

variable ::= identifier

identifier ::= [a-zA-Z] [a-zA-Z0-9_]*

Expressions

expressions ::= expression % ','

expression ::= expression `?` expression `:` expression
             | expression infixOp expression
             | prefixOp expression
             | expression postfixOp
             | common_expression

constr_expression ::= constr_expression arithmeticInfixOp constr_expression
                    | prefixOp constr_expression
                    | constr_expression postfixOp
                    | constr_expression '[' indexes ']'
                    | common_expression

common_expression
  ::= real_literal
    | variable
    | '{' expressions '}'
    | '[' expressions ']'
    | function_literal '(' ?expressions ')'
    | function_literal '(' expression ?('|' expression % ',') ')'
    | 'integrate_1d' '(' function_literal (',' expression){5|6} ')'
    | 'integrate_ode' '(' function_literal (',' expression){6} ')'
    | 'integrate_ode_rk45' '(' function_literal (',' expression){6|9} ')'
    | 'integrate_ode_bdf' '(' function_literal (',' expression){6|9} ')'
    | 'algebra_solver' '(' function_literal (',' expression){4|7} ')'
    | 'map_rect' '(' function_literal (',' expression){4} ')'
    | '(' expression ')'

prefixOp ::= ('!' | '-' | '+' | '^')

postfixOp ::= '\''

infixOp ::= arithmeticInfixOp | logicalInfixOp

arithmeticInfixOp ::= ('+' | '-' | '*' | '/' | '%' | '\' | '.*' | './')

logicalInfixOp :: ('||' | '&&' | '==' | '!=' | '<' | '<=' | '>' | '>=')

index ::= ?(expression | expression ':' | ':' expression
        | expression ':' expression)

indexes ::= index % ','

integer_literal ::= [0-9]+

real_literal ::= integer_literal '.' [0-9]* ?exp_literal
               | '.' [0-9]+ ?exp_literal
               | integer_literal exp_literal

exp_literal ::= ('e' | 'E') ?('+' | '-') integer_literal

function_literal ::= identifier

Statements

statement ::= atomic_statement | nested_statement

atomic_statement ::=  lhs assignment_op expression ';'
   | expression '~' identifier '(' expressions ')' ?truncation ';'
   | function_literal '(' expressions ')' ';'
   | 'increment_log_prob' '(' expression ')' ';'
   | 'target' '+=' expression ';'
   | 'break' ';'
   | 'continue' ';'
   | 'print' '(' (expression | string_literal) % ',' ')' ';'
   | 'reject' '(' (expression | string_literal) % ',' ')' ';'
   | 'return' expression ';'
   | ';'

assignment_op ::= '<-' | '=' | '+=' | '-=' | '*=' | '/=' | '.*=' | '/*='

string_literal ::= '"' char* '"'

truncation ::= 'T' '[' ?expression ',' ?expression ']'

lhs ::= identifier ('[' indexes ']')*

nested_statement
::=
  | 'if' '(' expression ')' statement
    ('else' 'if' '(' expression ')' statement)*
    ?('else' statement)
  | 'while' '(' expression ')' statement
  | 'for' '(' identifier 'in' expression ':' expression ')' statement
  | 'for' '(' identifier 'in' expression ')' statement
  | '{' var_decl* statement+ '}'