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diff --git a/res/notes/design/grammar.md b/res/notes/design/grammar.md
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+# Grammar
+
+```
+SCALLOP_PROGRAM ::= ITEM*
+
+ITEM ::= TYPE_DECL
+       | RELATION_DECL
+       | CONST_DECL
+       | QUERY_DECL
+
+TYPE ::= u8 | u16 | u32 | u64 | u128 | usize
+       | i8 | i16 | i32 | i64 | i128 | isize
+       | f32 | f64 | char | bool
+       | String
+       | CUSTOM_TYPE_NAME
+
+TYPE_DECL ::= type CUSTOM_TYPE_NAME = TYPE
+            | type CUSTOM_TYPE_NAME <: TYPE
+            | type ENUM_TYPE_NAME = VARIANT1 [= VAL1] | VARIANT2 [= VAL2] | ...
+            | type RELATION_NAME(TYPE*)
+            | type RELATION_NAME(VAR1: TYPE1, VAR2: TYPE2, ...)
+
+CONST_DECL ::= const CONSTANT_NAME : TYPE = CONSTANT
+             | const CONSTANT_NAME1 [: TYPE1] = CONSTANT1, CONSTANT_NAME2 [: TYPE2] = CONSTANT2, ...
+
+RELATION_DECL ::= FACT_DECL
+                | FACTS_SET_DECL
+                | RULE_DECL
+
+CONSTANT ::= true | false | NUMBER_LITERAL | STRING_LITERAL
+
+CONST_TUPLE ::= CONSTANT | (CONSTANT1, CONSTANT2, ...)
+
+FOREIGN_FN ::= hash | string_length | string_concat | substring | abs
+
+BIN_OP ::= + | - | * | / | % | == | != | <= | < | >= | > | && | || | ^
+
+UNARY_OP ::= ! | -
+
+CONST_EXPR ::= CONSTANT
+             | CONST_EXPR BIN_OP CONST_EXPR | UNARY_OP CONST_EXPR
+             | $ FOREIGN_FN(CONST_EXPR*)
+             | if CONST_EXPR then CONST_EXPR else CONST_EXPR
+             | ( CONST_EXPR )
+
+TAG ::= true | false | NUMBER_LITERAL  // true/false is for boolean tags; NUMBER_LITERAL is used for probabilities
+
+FACT_DECL ::= rel RELATION_NAME(CONST_EXPR*)         // Untagged fact
+            | rel TAG :: RELATION_NAME(CONST_EXPR*)  // Tagged fact
+
+FACTS_SET_DECL ::= rel RELATION_NAME = {CONST_TUPLE1, CONST_TUPLE2, ...}                  // Untagged tuples
+                 | rel RELATION_NAME = {TAG1 :: CONST_TUPLE1, TAG2 :: CONST_TUPLE2, ...}  // Tagged tuples
+                 | rel RELATION_NAME = {TAG1 :: CONST_TUPLE1; TAG2 :: CONST_TUPLE2; ...}  // Tagged tuples forming annotated disjunction
+
+EXPR ::= VARIABLE | CONSTANT
+       | EXPR BIN_OP EXPR | UNARY_OP EXPR
+       | $ FOREIGN_FN(EXPR*)
+       | if EXPR then EXPR else EXPR
+       | ( EXPR )
+
+ATOM ::= RELATION_NAME(EXPR*)
+
+RULE_DECL ::= rel ATOM :- FORMULA | rel ATOM = FORMULA                // Normal rule
+            | rel TAG :: ATOM :- FORMULA | rel TAG :: ATOM = FORMULA  // Tagged rule
+
+FORMULA ::= ATOM
+          | not ATOM | ~ ATOM                                                   // negation
+          | FORMULA1, FORMULA2, ... | FORMULA and FORMULA | FORMULA /\ FORMULA  // conjunction
+          | FORMULA or FORMULA | FORMULA \/ FORMULA                             // disjunction
+          | FORMULA implies FORMULA | FORMULA => FORMULA                        // implies
+          | CONSTRAINT | AGGREGATION
+          | ( FORMULA )
+
+CONSTRAINT ::= EXPR // When expression returns a boolean value
+
+AGGREGATOR ::= count | sum | prod | min | max | exists | forall | unique
+
+AGGREGATION ::= VAR* = AGGREGATOR(VAR* : FORMULA)                             // Normal aggregation
+              | VAR* = AGGREGATOR(VAR* : FORMULA where VAR* : FORMULA)        // Aggregation with group-by condition
+              | VAR* = AGGREGATOR[VAR*](VAR* : FORMULA)                       // Aggregation with arg (only applied to AGGREGATOR = min or max)
+              | VAR* = AGGREGATOR[VAR*](VAR* : FORMULA where VAR* : FORMULA)  // Aggregation with arg and group-by condition (only applied to AGGREGATOR = min or max)
+
+QUERY_DECL ::= query RELATION_NAME
+             | query ATOM
+```
diff --git a/res/notes/design/group_by.md b/res/notes/design/group_by.md
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+# Design of Group By
+
+## Examples
+
+### Example 1
+
+The following count does not have a group by variable
+
+``` scl
+rel num_cars(n) :- n = count(o: is_a(o, "car"))
+```
+
+### Example 2
+
+The following count does have a group by variable `c`.
+The body of the rule `is_a(o, "car"), color(o, c)` bounds two variables: `o` and `c`.
+We want the variables that occur in the head that is not "to-aggregate" or "argument" values.
+
+``` scl
+rel num_cars_of_color(c, n) :- n = count(o: is_a(o, "car"), color(o, c))
+```
+
+### Example 3
+
+The following count does have a group by variable `c`, note that `s` is not a group-by variable:
+
+``` scl
+rel num_cars_of_color(c, n) :- n = count(o: is_a(o, "car"), color(o, c), shape(o, s))
+```
+
+Although we have `shape(o, s)`, but we are not storing `s` in the head.
+Therefore we do not treat `s` as a group-by variable.
+
+### Example 4
+
+``` scl
+rel num_cars_of_color(c, n) :- n = count(o: is_a(o, "car"), color(o, c) where c: all_colors(c))
+```
+
+body_bounded_vars: o, c
+group_by_bounded_vars: c
+group_by_vars: c
+
+### Example 5
+
+``` scl
+rel eval_yn(e, b) :- b = exists(o: eval_obj(f, o) where e: exists_expr(e, f))
+```
+
+body_bounded_vars: f, o
+group_by_bounded_vars: e, f
+group_by_vars: e
+to_agg_vars: o