[Fix] Fix JsonSchemaConverter for numbers "-0. ..." (#462)

This PR fixes the bug in the number format of the JsonSchemaConverter.
Currently, the corresponding rule is:
```
"(\"0\" | \"-\"? [1-9] [0-9]*) (\".\" [0-9]+)? ([eE] [+-]? [0-9]+)?"
```
and it cannot accept numbers like `-0.123`. This PR fixes it by
replacing it with:
```
"\"-\"? (\"0\" | [1-9] [0-9]*) (\".\" [0-9]+)? ([eE] [+-]? [0-9]+)?"
```

---------

Signed-off-by: Yuchuan <[email protected]>

Author: Seven-Streams

cpp/json_schema_converter.cc

@@ -1963,7 +1963,7 @@ std::string JSONSchemaConverter::VisitNumber(
     return converted_regex;
   }
 
-  return "(\"0\" | \"-\"? [1-9] [0-9]*) (\".\" [0-9]+)? ([eE] [+-]? [0-9]+)?";
+  return "\"-\"? (\"0\" | [1-9] [0-9]*) (\".\" [0-9]+)? ([eE] [+-]? [0-9]+)?";
 }
 
 std::string JSONSchemaConverter::VisitString(

tests/python/test_function_calling_converter.py

@@ -27,7 +27,7 @@ def test_string_schema(input_str: str, accepted: bool):
 xml_any ::= basic_number | xml_string | basic_boolean | basic_null | basic_array | basic_object
 basic_any ::= basic_number | basic_string | basic_boolean | basic_null | basic_array | basic_object
 basic_integer ::= ("0" | "-"? [1-9] [0-9]*)
-basic_number ::= ("0" | "-"? [1-9] [0-9]*) ("." [0-9]+)? ([eE] [+-]? [0-9]+)?
+basic_number ::= "-"? ("0" | [1-9] [0-9]*) ("." [0-9]+)? ([eE] [+-]? [0-9]+)?
 basic_string ::= ["] basic_string_sub
 basic_boolean ::= "true" | "false"
 basic_null ::= "null"
@@ -72,7 +72,7 @@ def test_additional_properties_schema(input_str: str, accepted: bool):
 xml_any ::= basic_number | xml_string | basic_boolean | basic_null | basic_array | basic_object
 basic_any ::= basic_number | basic_string | basic_boolean | basic_null | basic_array | basic_object
 basic_integer ::= ("0" | "-"? [1-9] [0-9]*)
-basic_number ::= ("0" | "-"? [1-9] [0-9]*) ("." [0-9]+)? ([eE] [+-]? [0-9]+)?
+basic_number ::= "-"? ("0" | [1-9] [0-9]*) ("." [0-9]+)? ([eE] [+-]? [0-9]+)?
 basic_string ::= ["] basic_string_sub
 basic_boolean ::= "true" | "false"
 basic_null ::= "null"
@@ -116,7 +116,7 @@ def test_not_required_properties_schema(input_str: str, accepted: bool):
 xml_any ::= basic_number | xml_string | basic_boolean | basic_null | basic_array | basic_object
 basic_any ::= basic_number | basic_string | basic_boolean | basic_null | basic_array | basic_object
 basic_integer ::= ("0" | "-"? [1-9] [0-9]*)
-basic_number ::= ("0" | "-"? [1-9] [0-9]*) ("." [0-9]+)? ([eE] [+-]? [0-9]+)?
+basic_number ::= "-"? ("0" | [1-9] [0-9]*) ("." [0-9]+)? ([eE] [+-]? [0-9]+)?
 basic_string ::= ["] basic_string_sub
 basic_boolean ::= "true" | "false"
 basic_null ::= "null"
@@ -201,7 +201,7 @@ def test_inner_object_schema(input_str: str, accepted: bool):
 xml_any ::= basic_number | xml_string | basic_boolean | basic_null | basic_array | basic_object
 basic_any ::= basic_number | basic_string | basic_boolean | basic_null | basic_array | basic_object
 basic_integer ::= ("0" | "-"? [1-9] [0-9]*)
-basic_number ::= ("0" | "-"? [1-9] [0-9]*) ("." [0-9]+)? ([eE] [+-]? [0-9]+)?
+basic_number ::= "-"? ("0" | [1-9] [0-9]*) ("." [0-9]+)? ([eE] [+-]? [0-9]+)?
 basic_string ::= ["] basic_string_sub
 basic_boolean ::= "true" | "false"
 basic_null ::= "null"
@@ -253,7 +253,7 @@ def test_numbers_schema(input_str: str, accepted: bool):
 xml_any ::= basic_number | xml_string | basic_boolean | basic_null | basic_array | basic_object
 basic_any ::= basic_number | basic_string | basic_boolean | basic_null | basic_array | basic_object
 basic_integer ::= ("0" | "-"? [1-9] [0-9]*)
-basic_number ::= ("0" | "-"? [1-9] [0-9]*) ("." [0-9]+)? ([eE] [+-]? [0-9]+)?
+basic_number ::= "-"? ("0" | [1-9] [0-9]*) ("." [0-9]+)? ([eE] [+-]? [0-9]+)?
 basic_string ::= ["] basic_string_sub
 basic_boolean ::= "true" | "false"
 basic_null ::= "null"
@@ -332,7 +332,7 @@ def test_string_format_length_schema(input_str: str, accepted: bool):
 xml_any ::= basic_number | xml_string | basic_boolean | basic_null | basic_array | basic_object
 basic_any ::= basic_number | basic_string | basic_boolean | basic_null | basic_array | basic_object
 basic_integer ::= ("0" | "-"? [1-9] [0-9]*)
-basic_number ::= ("0" | "-"? [1-9] [0-9]*) ("." [0-9]+)? ([eE] [+-]? [0-9]+)?
+basic_number ::= "-"? ("0" | [1-9] [0-9]*) ("." [0-9]+)? ([eE] [+-]? [0-9]+)?
 basic_string ::= ["] basic_string_sub
 basic_boolean ::= "true" | "false"
 basic_null ::= "null"

tests/python/test_grammar_matcher_json_schema.py

@@ -11,6 +11,7 @@
 from xgrammar.testing import (
     _get_masked_tokens_from_bitmask,
     _get_matcher_from_grammar_and_tokenizer_info,
+    _is_grammar_accept_string,
 )
 
 
@@ -554,5 +555,20 @@ def test_regression_empty_property_key_regex():
     assert _ is not None
 
 
+def test_json_schema_number_without_constraint():
+    schema = {"type": "object", "properties": {"value": {"type": "number"}}, "required": ["value"]}
+    grammar = xgr.Grammar.from_json_schema(schema)
+    assert _is_grammar_accept_string(grammar, '{"value": -0.5}')
+    assert _is_grammar_accept_string(grammar, '{"value": -1.5}')
+    assert _is_grammar_accept_string(grammar, '{"value": 0}')
+    assert _is_grammar_accept_string(grammar, '{"value": 1234567890}')
+    assert _is_grammar_accept_string(grammar, '{"value": 3.14159}')
+    assert _is_grammar_accept_string(grammar, '{"value": 1e10}')
+    assert _is_grammar_accept_string(grammar, '{"value": -2.5E-3}')
+    assert _is_grammar_accept_string(grammar, '{"value": 0.0}')
+    assert _is_grammar_accept_string(grammar, '{"value": -0.0}')
+    assert not _is_grammar_accept_string(grammar, '{"value": "abc"}')
+
+
 if __name__ == "__main__":
     pytest.main(sys.argv)

tests/python/test_grammar_matcher_structural_tag.py

@@ -53,19 +53,19 @@ class Schema(BaseModel):
 basic_integer_1_1 ::= ("" | ("-")) (=([1-9] [0-9]*))
 basic_escape_2 ::= (([\"\\/bfnrt]) | ("u" [A-Fa-f0-9] [A-Fa-f0-9] [A-Fa-f0-9] [A-Fa-f0-9])) (=(basic_string_sub_2))
 basic_string_sub_2 ::= (("\"") | ([^\0-\x1f\"\\\r\n] basic_string_sub_2) | ("\\" basic_escape_2 basic_string_sub_2)) (=([ \n\t]* [,}\]:]))
-basic_number_9 ::= ((basic_number_7_2 basic_number_3_2 basic_number_6_2)) (=(root_part_0_2 [ \n\t]* "}"))
+basic_number_9 ::= ((basic_number_1_2 basic_number_7_2 basic_number_3_2 basic_number_6_2)) (=(root_part_0_2 [ \n\t]* "}"))
 basic_string_2 ::= (("\"" basic_string_sub_2))
 root_prop_1 ::= (("[" [ \n\t]* basic_string_2 root_prop_1_1 [ \n\t]* "]") | ("[" [ \n\t]* "]"))
 root_part_0_2 ::= (([ \n\t]* "," [ \n\t]* "\"arg4\"" [ \n\t]* ":" [ \n\t]* root_prop_1)) (=([ \n\t]* "}"))
 root_2 ::= (("{" [ \n\t]* "\"arg3\"" [ \n\t]* ":" [ \n\t]* basic_number_9 root_part_0_2 [ \n\t]* "}")) (=("</function>"))
-basic_number_1_2 ::= ("" | ("-")) (=([1-9] [0-9]*))
+basic_number_1_2 ::= ("" | ("-")) (=(basic_number_7_2 basic_number_3_2 basic_number_6_2))
 basic_number_2_2 ::= (([0-9] basic_number_2_2) | ([0-9]))
 basic_number_3_2 ::= ("" | ("." basic_number_2_2)) (=(basic_number_6_2))
 basic_number_4_2 ::= ("" | ([+\-])) (=(basic_number_5_2))
 basic_number_5_2 ::= (([0-9] basic_number_5_2) | ([0-9]))
 basic_number_6_2 ::= ("" | ([eE] basic_number_4_2 basic_number_5_2))
 root_prop_1_1 ::= ("" | ([ \n\t]* "," [ \n\t]* basic_string_2 root_prop_1_1)) (=([ \n\t]* "]"))
-basic_number_7_2 ::= (("0") | (basic_number_1_2 [1-9] [0-9]*)) (=(basic_number_3_2 basic_number_6_2))
+basic_number_7_2 ::= (("0") | ([1-9] [0-9]*)) (=(basic_number_3_2 basic_number_6_2))
 triggered_tags_group ::= (("1>" root "</function>") | ("2>" root_1 "</function>"))
 triggered_tags_group_1 ::= ((">" root_2 "</function>"))
 triggered_tags ::= TagDispatch(
@@ -82,7 +82,7 @@ class Schema(BaseModel):
 basic_string_sub ::= (("\"") | ([^\0-\x1f\"\\\r\n] basic_string_sub) | ("\\" basic_escape basic_string_sub)) (=([ \n\t]* [,}\]:]))
 basic_any ::= ((basic_number) | (basic_string) | (basic_boolean) | (basic_null) | (basic_array) | (basic_object))
 basic_integer ::= (("0") | (basic_integer_1 [1-9] [0-9]*))
-basic_number ::= ((basic_number_7 basic_number_3 basic_number_6))
+basic_number ::= ((basic_number_1 basic_number_7 basic_number_3 basic_number_6))
 basic_string ::= (("\"" basic_string_sub))
 basic_boolean ::= (("true") | ("false"))
 basic_null ::= (("null"))
@@ -99,12 +99,12 @@ class Schema(BaseModel):
 basic_number_6 ::= ("" | ([eE] basic_number_4 basic_number_5))
 basic_array_1 ::= ("" | ([ \n\t]* "," [ \n\t]* basic_any basic_array_1))
 basic_object_1 ::= ("" | ([ \n\t]* "," [ \n\t]* basic_string [ \n\t]* ":" [ \n\t]* basic_any basic_object_1))
-basic_number_7 ::= (("0") | (basic_number_1 [1-9] [0-9]*))
+basic_number_7 ::= (("0") | ([1-9] [0-9]*))
 basic_escape_1 ::= (([\"\\/bfnrt]) | ("u" [A-Fa-f0-9] [A-Fa-f0-9] [A-Fa-f0-9] [A-Fa-f0-9]))
 basic_string_sub_1 ::= (("\"") | ([^\0-\x1f\"\\\r\n] basic_string_sub_1) | ("\\" basic_escape_1 basic_string_sub_1)) (=([ \n\t]* [,}\]:]))
 basic_any_1 ::= ((basic_number_8) | (basic_string_1) | (basic_boolean_1) | (basic_null_1) | (basic_array_2) | (basic_object_2))
 basic_integer_2 ::= (("0") | (basic_integer_1_1 [1-9] [0-9]*))
-basic_number_8 ::= ((basic_number_7_1 basic_number_3_1 basic_number_6_1))
+basic_number_8 ::= ((basic_number_1_1 basic_number_7_1 basic_number_3_1 basic_number_6_1))
 basic_string_1 ::= (("\"" basic_string_sub_1))
 basic_boolean_1 ::= (("true") | ("false"))
 basic_null_1 ::= (("null"))
@@ -121,12 +121,12 @@ class Schema(BaseModel):
 basic_number_6_1 ::= ("" | ([eE] basic_number_4_1 basic_number_5_1))
 basic_array_1_1 ::= ("" | ([ \n\t]* "," [ \n\t]* basic_any_1 basic_array_1_1))
 basic_object_1_1 ::= ("" | ([ \n\t]* "," [ \n\t]* basic_string_1 [ \n\t]* ":" [ \n\t]* basic_any_1 basic_object_1_1))
-basic_number_7_1 ::= (("0") | (basic_number_1_1 [1-9] [0-9]*))
+basic_number_7_1 ::= (("0") | ([1-9] [0-9]*))
 basic_escape_2 ::= (([\"\\/bfnrt]) | ("u" [A-Fa-f0-9] [A-Fa-f0-9] [A-Fa-f0-9] [A-Fa-f0-9]))
 basic_string_sub_2 ::= (("\"") | ([^\0-\x1f\"\\\r\n] basic_string_sub_2) | ("\\" basic_escape_2 basic_string_sub_2)) (=([ \n\t]* [,}\]:]))
 basic_any_2 ::= ((basic_number_9) | (basic_string_2) | (basic_boolean_2) | (basic_null_2) | (basic_array_3) | (basic_object_3))
 basic_integer_3 ::= (("0") | (basic_integer_1_2 [1-9] [0-9]*))
-basic_number_9 ::= ((basic_number_7_2 basic_number_3_2 basic_number_6_2))
+basic_number_9 ::= ((basic_number_1_2 basic_number_7_2 basic_number_3_2 basic_number_6_2))
 basic_string_2 ::= (("\"" basic_string_sub_2))
 basic_boolean_2 ::= (("true") | ("false"))
 basic_null_2 ::= (("null"))
@@ -145,7 +145,7 @@ class Schema(BaseModel):
 basic_array_1_2 ::= ("" | ([ \n\t]* "," [ \n\t]* basic_any_2 basic_array_1_2))
 basic_object_1_2 ::= ("" | ([ \n\t]* "," [ \n\t]* basic_string_2 [ \n\t]* ":" [ \n\t]* basic_any_2 basic_object_1_2))
 root_prop_1_1 ::= ("" | ([ \n\t]* "," [ \n\t]* basic_string_2 root_prop_1_1))
-basic_number_7_2 ::= (("0") | (basic_number_1_2 [1-9] [0-9]*))
+basic_number_7_2 ::= (("0") | ([1-9] [0-9]*))
 triggered_tags_group ::= (("1>" root "</function>") | ("2>" root_1 "</function>"))
 triggered_tags_group_1 ::= ((">" root_2 "</function>"))
 triggered_tags ::= TagDispatch(
@@ -178,7 +178,6 @@ class Schema2(BaseModel):
     triggers = ["<function=f", "<function=g"]
 
     grammar = xgr.Grammar.from_structural_tag(tags, triggers)
-
     assert str(grammar) == expected_grammar_test_structural_tag_before_optimization
 
     accepted_inputs = [
@@ -212,7 +211,6 @@ class Schema2(BaseModel):
 
     compiler = xgr.GrammarCompiler(xgr.TokenizerInfo([]))
     compiled_grammar = compiler.compile_structural_tag(tags, triggers)
-
     assert str(compiled_grammar.grammar) == expected_grammar_test_structural_tag_after_optimization
 
 

tests/python/test_json_schema_converter.py

@@ -19,7 +19,7 @@
 basic_string_sub ::= ("\"" | [^\0-\x1f\"\\\r\n] basic_string_sub | "\\" basic_escape basic_string_sub) (= [ \n\t]* [,}\]:])
 basic_any ::= basic_number | basic_string | basic_boolean | basic_null | basic_array | basic_object
 basic_integer ::= ("0" | "-"? [1-9] [0-9]*)
-basic_number ::= ("0" | "-"? [1-9] [0-9]*) ("." [0-9]+)? ([eE] [+-]? [0-9]+)?
+basic_number ::= "-"? ("0" | [1-9] [0-9]*) ("." [0-9]+)? ([eE] [+-]? [0-9]+)?
 basic_string ::= ["] basic_string_sub
 basic_boolean ::= "true" | "false"
 basic_null ::= "null"
@@ -31,7 +31,7 @@
 basic_string_sub ::= ("\"" | [^\0-\x1f\"\\\r\n] basic_string_sub | "\\" basic_escape basic_string_sub) (= [ \n\t]* [,}\]:])
 basic_any ::= basic_number | basic_string | basic_boolean | basic_null | basic_array | basic_object
 basic_integer ::= ("0" | "-"? [1-9] [0-9]*)
-basic_number ::= ("0" | "-"? [1-9] [0-9]*) ("." [0-9]+)? ([eE] [+-]? [0-9]+)?
+basic_number ::= "-"? ("0" | [1-9] [0-9]*) ("." [0-9]+)? ([eE] [+-]? [0-9]+)?
 basic_string ::= ["] basic_string_sub
 basic_boolean ::= "true" | "false"
 basic_null ::= "null"

tests/python/test_structural_tag_converter.py

@@ -126,7 +126,7 @@ def test_const_string_format(
 basic_string_sub ::= (("\"") | ([^\0-\x1f\"\\\r\n] basic_string_sub) | ("\\" basic_escape basic_string_sub)) (=([ \n\t]* [,}\]:]))
 basic_any ::= ((basic_number) | (basic_string) | (basic_boolean) | (basic_null) | (basic_array) | (basic_object))
 basic_integer ::= (("0") | (basic_integer_1 [1-9] [0-9]*))
-basic_number ::= ((basic_number_7 basic_number_3 basic_number_6))
+basic_number ::= ((basic_number_1 basic_number_7 basic_number_3 basic_number_6))
 basic_string ::= (("\"" basic_string_sub))
 basic_boolean ::= (("true") | ("false"))
 basic_null ::= (("null"))
@@ -142,7 +142,7 @@ def test_const_string_format(
 basic_number_6 ::= ("" | ([eE] basic_number_4 basic_number_5))
 basic_array_1 ::= ("" | ([ \n\t]* "," [ \n\t]* basic_any basic_array_1))
 basic_object_1 ::= ("" | ([ \n\t]* "," [ \n\t]* basic_string [ \n\t]* ":" [ \n\t]* basic_any basic_object_1))
-basic_number_7 ::= (("0") | (basic_number_1 [1-9] [0-9]*))
+basic_number_7 ::= (("0") | ([1-9] [0-9]*))
 root_1 ::= ((root))
 """,
     )
@@ -186,7 +186,7 @@ def test_json_schema_format(
 xml_any ::= ((basic_number) | (xml_string) | (basic_boolean) | (basic_null) | (basic_array) | (basic_object))
 basic_any ::= ((basic_number) | (basic_string) | (basic_boolean) | (basic_null) | (basic_array) | (basic_object))
 basic_integer ::= (("0") | (basic_integer_1 [1-9] [0-9]*))
-basic_number ::= ((basic_number_7 basic_number_3 basic_number_6))
+basic_number ::= ((basic_number_1 basic_number_7 basic_number_3 basic_number_6))
 basic_string ::= (("\"" basic_string_sub))
 basic_boolean ::= (("true") | ("false"))
 basic_null ::= (("null"))
@@ -205,7 +205,7 @@ def test_json_schema_format(
 basic_array_1 ::= ("" | ([ \n\t]* "," [ \n\t]* basic_any basic_array_1))
 basic_object_1 ::= ("" | ([ \n\t]* "," [ \n\t]* basic_string [ \n\t]* ":" [ \n\t]* basic_any basic_object_1))
 root_prop_1_1 ::= ("" | ("-"))
-basic_number_7 ::= (("0") | (basic_number_1 [1-9] [0-9]*))
+basic_number_7 ::= (("0") | ([1-9] [0-9]*))
 root_1 ::= ((root))
 """,
     )
@@ -304,7 +304,7 @@ def test_regex_format(
 basic_string_sub ::= (("\"") | ([^\0-\x1f\"\\\r\n] basic_string_sub) | ("\\" basic_escape basic_string_sub)) (=([ \n\t]* [,}\]:]))
 basic_any ::= ((basic_number) | (basic_string) | (basic_boolean) | (basic_null) | (basic_array) | (basic_object))
 basic_integer ::= (("0") | (basic_integer_1 [1-9] [0-9]*))
-basic_number ::= ((basic_number_7 basic_number_3 basic_number_6))
+basic_number ::= ((basic_number_1 basic_number_7 basic_number_3 basic_number_6))
 basic_string ::= (("\"" basic_string_sub))
 basic_boolean ::= (("true") | ("false"))
 basic_null ::= (("null"))
@@ -320,7 +320,7 @@ def test_regex_format(
 basic_number_6 ::= ("" | ([eE] basic_number_4 basic_number_5))
 basic_array_1 ::= ("" | ([ \n\t]* "," [ \n\t]* basic_any basic_array_1))
 basic_object_1 ::= ("" | ([ \n\t]* "," [ \n\t]* basic_string [ \n\t]* ":" [ \n\t]* basic_any basic_object_1))
-basic_number_7 ::= (("0") | (basic_number_1 [1-9] [0-9]*))
+basic_number_7 ::= (("0") | ([1-9] [0-9]*))
 root_1 ::= ("" | ([\-+*/]))
 root_2 ::= ((root_1_1))
 root_1_1 ::= ("" | ([simple]))
@@ -369,7 +369,7 @@ def test_sequence_format(
 basic_string_sub ::= (("\"") | ([^\0-\x1f\"\\\r\n] basic_string_sub) | ("\\" basic_escape basic_string_sub)) (=([ \n\t]* [,}\]:]))
 basic_any ::= ((basic_number) | (basic_string) | (basic_boolean) | (basic_null) | (basic_array) | (basic_object))
 basic_integer ::= (("0") | (basic_integer_1 [1-9] [0-9]*))
-basic_number ::= ((basic_number_7 basic_number_3 basic_number_6))
+basic_number ::= ((basic_number_1 basic_number_7 basic_number_3 basic_number_6))
 basic_string ::= (("\"" basic_string_sub))
 basic_boolean ::= (("true") | ("false"))
 basic_null ::= (("null"))
@@ -385,7 +385,7 @@ def test_sequence_format(
 basic_number_6 ::= ("" | ([eE] basic_number_4 basic_number_5))
 basic_array_1 ::= ("" | ([ \n\t]* "," [ \n\t]* basic_any basic_array_1))
 basic_object_1 ::= ("" | ([ \n\t]* "," [ \n\t]* basic_string [ \n\t]* ":" [ \n\t]* basic_any basic_object_1))
-basic_number_7 ::= (("0") | (basic_number_1 [1-9] [0-9]*))
+basic_number_7 ::= (("0") | ([1-9] [0-9]*))
 or ::= ((const_string) | (root))
 root_1 ::= ((or))
 """,
@@ -423,7 +423,7 @@ def test_or_format(
 basic_string_sub ::= (("\"") | ([^\0-\x1f\"\\\r\n] basic_string_sub) | ("\\" basic_escape basic_string_sub)) (=([ \n\t]* [,}\]:]))
 basic_any ::= ((basic_number) | (basic_string) | (basic_boolean) | (basic_null) | (basic_array) | (basic_object))
 basic_integer ::= (("0") | (basic_integer_1 [1-9] [0-9]*))
-basic_number ::= ((basic_number_7 basic_number_3 basic_number_6))
+basic_number ::= ((basic_number_1 basic_number_7 basic_number_3 basic_number_6))
 basic_string ::= (("\"" basic_string_sub))
 basic_boolean ::= (("true") | ("false"))
 basic_null ::= (("null"))
@@ -439,7 +439,7 @@ def test_or_format(
 basic_number_6 ::= ("" | ([eE] basic_number_4 basic_number_5))
 basic_array_1 ::= ("" | ([ \n\t]* "," [ \n\t]* basic_any basic_array_1))
 basic_object_1 ::= ("" | ([ \n\t]* "," [ \n\t]* basic_string [ \n\t]* ":" [ \n\t]* basic_any basic_object_1))
-basic_number_7 ::= (("0") | (basic_number_1 [1-9] [0-9]*))
+basic_number_7 ::= (("0") | ([1-9] [0-9]*))
 tag ::= (("BEG" root "END"))
 root_1 ::= ((tag))
 """,