diff --git a/AlgebraicDataflowArchitectureModel/src/models/algebra/Term.java b/AlgebraicDataflowArchitectureModel/src/models/algebra/Term.java
index e78c106..cd3d92d 100644
--- a/AlgebraicDataflowArchitectureModel/src/models/algebra/Term.java
+++ b/AlgebraicDataflowArchitectureModel/src/models/algebra/Term.java
@@ -73,11 +73,13 @@
 			subTerms.put(new Position(), (T) this);
 		}
 		for (int i = 0; i < children.size(); i++) {
-			HashMap<Position, T> terms = children.get(i).getSubTerms(clazz);
-			for (Entry<Position, T> term: terms.entrySet()) {
-				Position pos = term.getKey();
-				pos.addHeadOrder(i);
-				subTerms.put(pos, term.getValue());
+			if (children.get(i) != null) {
+				HashMap<Position, T> terms = children.get(i).getSubTerms(clazz);
+				for (Entry<Position, T> term: terms.entrySet()) {
+					Position pos = term.getKey();
+					pos.addHeadOrder(i);
+					subTerms.put(pos, term.getValue());
+				}
 			}
 		}
 		return subTerms;
@@ -117,6 +119,8 @@
 	@Override
 	public Expression unify(Expression another) {
 		if (another instanceof Variable) return (Expression) this.clone();
+		if (this instanceof Constant) return (Expression) this.clone();
+		if (another instanceof Constant) return (Expression) another.clone();
 		if (another instanceof Term) {
 			Term anotherTerm = (Term) another;
 			if (!symbol.equals(anotherTerm.symbol)) return null;
@@ -232,7 +236,11 @@
 	public Object clone() {
 		Term newTerm = new Term(symbol);
 		for (Expression e: children) {
-			newTerm.addChild((Expression) e.clone());
+			if (e != null) {
+				newTerm.addChild((Expression) e.clone());
+			} else {
+				newTerm.addChild(null);
+			}
 		}
 		newTerm.type = type;
 		return newTerm;
@@ -247,7 +255,11 @@
 			String delimiter = "";
 			for (int i = 1; i < children.size(); i++) {
 				Expression e = children.get(i);
-				exp += (delimiter + e.toString());
+				if (e != null) {
+					exp += (delimiter + e.toString());
+				} else {
+					exp += (delimiter + "null");
+				}
 				delimiter = ",";
 			}
 			return exp + ")";
diff --git a/AlgebraicDataflowArchitectureModel/src/models/dataConstraintModel/DataConstraintModel.java b/AlgebraicDataflowArchitectureModel/src/models/dataConstraintModel/DataConstraintModel.java
index 6d8e176..0b5f65b 100644
--- a/AlgebraicDataflowArchitectureModel/src/models/dataConstraintModel/DataConstraintModel.java
+++ b/AlgebraicDataflowArchitectureModel/src/models/dataConstraintModel/DataConstraintModel.java
@@ -262,7 +262,7 @@
 		lookup.setSignature(new Type[] {null, typeMap, null});
 		addMember.setSignature(new Type[] {typeJson, typeJson, typeString, null});
 		dot.setSignature(new Type[] {null, typeJson, typeString});
-		dotParam.setSignature(new Type[] {null, null, null});
+		dotParam.setSignature(new Type[] {null, typeJson, null});
 		pi.setSignature(new Type[] {typeDouble});
 		E.setSignature(new Type[] {typeDouble});
 		sqrt.setSignature(new Type[] {typeDouble, typeDouble});
diff --git a/AlgebraicDataflowArchitectureModel/src/models/dataConstraintModel/JsonAccessor.java b/AlgebraicDataflowArchitectureModel/src/models/dataConstraintModel/JsonAccessor.java
index ea5f9f2..aea0590 100644
--- a/AlgebraicDataflowArchitectureModel/src/models/dataConstraintModel/JsonAccessor.java
+++ b/AlgebraicDataflowArchitectureModel/src/models/dataConstraintModel/JsonAccessor.java
@@ -21,20 +21,6 @@
 		super(symbol);
 	}
 	
-	public Type getType() {
-		if (symbol.equals(DataConstraintModel.dotParam)) {
-			Type valueType = null;
-			if (getChild(1) instanceof Term) {
-				valueType = ((Term) getChild(1)).getType();
-			} else if (getChild(1) instanceof Variable) {
-				valueType = ((Variable) getChild(1)).getType();
-			}
-			if (valueType != null) return valueType;
-		}
-		return super.getType();
-	}
-	
-
 	@Override
 	public Expression getInverseMap(Expression outputValue, Position targetPos) {
 		if (targetPos.isEmpty()) return outputValue;
@@ -42,15 +28,15 @@
 		int i = targetPos.removeHeadOrder();
 		Symbol[] inverseSymbols = symbol.getInverses();
 		if (i == 0) {
-			if (symbol.equals(DataConstraintModel.dot) && getChildren().size() >= 2) {
+			if (symbol == DataConstraintModel.dot && getChildren().size() >= 2) {
 				// this term is `json.key`.
 				Expression expJson = getChild(0);
 				Expression expKey = getChild(1);
-				JsonType jsonType = null;
+				Type jsonType = null;
 				if (expJson instanceof Variable) {
-					jsonType = (JsonType) ((Variable) expJson).getType();
+					jsonType = ((Variable) expJson).getType();
 				} else if (expJson instanceof Term) {
-					jsonType = (JsonType) ((Term) expJson).getType();
+					jsonType = ((Term) expJson).getType();
 				}
 				String keyName = null;
 				if (expKey instanceof Constant) {
@@ -62,8 +48,11 @@
 					Set<String> keySet = new HashSet<>();
 					if (jsonType == null || jsonType == DataConstraintModel.typeJson) {
 						keySet.add(keyName);
-					} else {
-						keySet.addAll(jsonType.getKeys());
+					} else if (jsonType instanceof JsonType) {
+						keySet.addAll(((JsonType) jsonType).getKeys());
+						if (keySet.size() == 0) {
+							keySet.add(keyName);
+						}
 					}
 					for (String key: keySet) {
 						Term addMemberTerm = new Term(DataConstraintModel.addMember);	// addMember(jsonTerm, key, v)
@@ -79,7 +68,7 @@
 					LambdaAbstraction lambdaAbstraction = new LambdaAbstraction(var, jsonTerm);	// v -> addMember(jsonTerm, key, v)
 					inverseSymbols = new Symbol[] { lambdaAbstraction };
 				}
-			} else if (symbol.equals(DataConstraintModel.dotParam) && getChildren().size() >= 2) {
+			} else if (symbol == DataConstraintModel.dotParam && getChildren().size() >= 2) {
 				// this term is `json.{param}`.
 				Expression expListOrMap = getChild(0);
 				Expression expKey = getChild(1);
@@ -91,9 +80,9 @@
 				}
 				Type keyType = null;
 				if (expKey instanceof Variable) {
-					keyType = (JsonType) ((Variable) expKey).getType();
+					keyType = ((Variable) expKey).getType();
 				} else if (expKey instanceof Term) {
-					keyType = (JsonType) ((Term) expKey).getType();
+					keyType = ((Term) expKey).getType();
 				}
 				if (jsonType != null && keyType != null) { 
 					if (DataConstraintModel.typeList.isAncestorOf(jsonType) || keyType.equals(DataConstraintModel.typeInt)) {
diff --git a/AlgebraicDataflowArchitectureModel/src/parser/Parser.java b/AlgebraicDataflowArchitectureModel/src/parser/Parser.java
index 0c980bc..852fa34 100644
--- a/AlgebraicDataflowArchitectureModel/src/parser/Parser.java
+++ b/AlgebraicDataflowArchitectureModel/src/parser/Parser.java
@@ -424,7 +424,7 @@
 					paramType = ((Term) paramTerm).getType();
 				}
 				Term term = null;
-				if (paramType != null && DataTransferModel.typeInt.isAncestorOf(paramType)) {
+				if (literalOrLeftCurlyBracket.equals(LEFT_CURLY_BRACKET)) {
 					term = new JsonAccessor(DataTransferModel.dotParam);
 				} else {
 					term = new JsonAccessor(DataTransferModel.dot);