Lesson 1.11: Casting Terms

The purpose of this lesson is to explain how to use cast expressions in order to disambiguate terms using sort information. We also explain how the variable sort inference algorithm works in K, and how to change the default behavior by casting variables to a particular sort.

Casting in K

Sometimes the grammar you write for your rules in K can be a little bit ambiguous on purpose. While grammars for programming languages may be unambiguous when considered in their entirety, K allows you to write rules involving arbitrary fragments of that grammar, and those fragments can sometimes be ambiguous by themselves, or similar enough to other fragments of the grammar to trigger ambiguity. As a result, in addition to the tools covered in Lesson 1.4, K provides one additional powerful tool for disambiguation: cast expressions.

K provides three main types of casts: the semantic cast, the strict cast, and the projection cast. We will cover each of them, and their similarities and differences, in turn.

Semantic casts

The most basic, and most common, type of cast in K is called the semantic cast. For every sort S declared in a module, K provides the following (implicit) production for use in sentences:

  syntax S ::= S ":S"

Note that S simply represents the name of the sort. For example, if we defined a sort Exp, the actual production for that sort would be:

  syntax Exp ::= Exp ":Exp"

At runtime, this expression will not actually exist; it is merely an annotation to the compiler describing the sort of the term inside the cast. It is telling the compiler that the term inside the cast must be of sort Exp. For example, if we had the following grammar:

module LESSON-11-A imports INT syntax Exp ::= Int | Exp "+" Exp syntax Stmt ::= "if" "(" Exp ")" Stmt | "{" "}" endmodule

Then we would be able to write 1:Exp, or (1 + 2):Exp, but not {}:Exp.

You can also restrict the sort that a variable in a rule will match by casting it. For example, consider the following additional module:

module LESSON-11-B imports LESSON-11-A imports BOOL syntax Term ::= Exp | Stmt syntax Bool ::= isExpression(Term) [function] rule isExpression(_E:Exp) => true rule isExpression(_) => false [owise] endmodule

Here we have defined a very simple function that decides whether a term is an expression or a statement. It does this by casting the variable inside the isExpression rule to sort Exp. As a result, that variable will only match terms of sort Exp. Thus, isExpression(1) will return true, as will isExpression(1 + 2), but isExpression({}) will return false.


Verify this fact for yourself by running isExpression on the above examples. Then write an isStatement function, and test that it works as expected.

Strict casts

On occasion, a semantic cast is not strict enough. It might be that you want to, for disambiguation purposes, say exactly what sort a term is. For example, consider the following definition:

module LESSON-11-C imports INT syntax Exp ::= Int | "add[" Exp "," Exp "]" [group(exp)] syntax Exp2 ::= Exp | "add[" Exp2 "," Exp2 "]" [group(exp2)] endmodule

This grammar is a little ambiguous and contrived, but it serves to demonstrate how a semantic cast might be insufficient to disambiguate a term. If we were to write the term add[ I1:Int , I2:Int ]:Exp2, the term would be ambiguous, because the cast is not sufficiently strict to determine whether you mean to derive the "add" production defined in group exp or the one in group exp2.

In this situation, there is a solution: the strict cast. For every sort S in your grammar, K also defines the following production:

  syntax S ::= S "::S"

This may at first glance seem the same as the previous cast. And indeed, from the perspective of the grammar and from the perspective of rewriting, they are in fact identical. However, the second variant has a unique meaning in the type system of K: namely, the term inside the cast cannot be a subsort, i.e., a term of another sort S2 such that the production syntax S ::= S2 exists.

As a result, if we were to write in the above grammar the term add[ I1:Int , I2:Int ]::Exp2, then we would know that the second derivation above should be chosen, whereas if we want the first derivation, we could write add[ I1:Int , I2:Int ]::Exp.

Care must be taken when using a strict cast with brackets. For example, consider a similar grammar but using an infix "+":

module LESSON-11-D imports INT syntax Exp ::= Int | Exp "+" Exp [group(exp)] syntax Exp2 ::= Exp | Exp2 "+" Exp2 [group(exp2)] | "(" Exp2 ")" [bracket] endmodule

The term I1:Int + I2:Int is ambiguous and could refer to either the production in group exp or the one in group exp2. To differentiate, you might try to write (I1:Int + I2:Int)::Exp2 similarly to the previous example.

Unfortunately though, this is still ambiguous. Here, the strict cast ::Exp2 applies directly to the brackets themselves rather than the underlying term within those brackets. As a result, it enforces that (I1:Int + I2:Int) cannot be a strict subsort of Exp2, but it has no effect on the sort of the subterm I1:Int + I2:Int.

For cases like this, K provides an alternative syntax for strict casts:

  syntax S ::= "{" S "}::S"

The ambiguity can then be resolved with {I1:Int + I2:Int}::Exp or {I1:Int + I2:Int}::Exp2.

Projection casts

Thus far we have focused entirely on casts which exist solely to inform the compiler about the sort of terms. However, sometimes when dealing with grammars containing subsorts, it can be desirable to reason with the subsort production itself, which injects one sort into another. Remember from above that such a production looks like syntax S ::= S2. This type of production, called a subsort production, can be thought of as a type of inheritance involving constructors. If we have the above production in our grammar, we say that S2 is a subsort of S, or that any S2 is also an S. K implicitly maintains a symbol at runtime which keeps track of where such subsortings occur; this symbol is called an injection.

Sometimes, when one sort is a subsort of another, it can be the case that a function returns one sort, but you actually want to cast the result of calling that function to another sort which is a subsort of the first sort. This is similar to what happens with inheritance in an object-oriented language, where you might cast a superclass to a subclass if you know for sure the object at runtime is in fact an instance of that class.

K provides something similar for subsorts: the projection cast.

For each pair of sorts S and S2, K provides the following production:

  syntax S ::= "{" S2 "}" ":>S"

What this means is that you take any term of sort S2 and cast it to sort S. If the term of sort S2 consists of an injection containing a term of sort S, then this will return that term. Otherwise, an error occurs and rewriting fails, returning the projection function which failed to apply. The sort is not actually checked at compilation time; rather, it is a runtime check inserted into the code that runs when the rule applies.

For example, here is a module that makes use of projection casts:

module LESSON-11-E imports INT imports BOOL syntax Exp ::= Int | Bool | Exp "+" Exp | Exp "&&" Exp syntax Exp ::= eval(Exp) [function] rule eval(I:Int) => I rule eval(B:Bool) => B rule eval(E1 + E2) => {eval(E1)}:>Int +Int {eval(E2)}:>Int rule eval(E1 && E2) => {eval(E1)}:>Bool andBool {eval(E2)}:>Bool endmodule

Here we have defined constructors for a simple expression language over Booleans and integers, as well as a function eval that evaluates these expressions to a value. Because that value could be an integer or a Boolean, we need the casts in the last two rules in order to meet the type signature of +Int and andBool. Of course, the user can write ill-formed expressions like 1 && true or false + true, but these will cause errors at runtime, because the projection cast will fail.


  1. Extend the eval function in LESSON-11-E to include Strings and add a . operator which concatenates them.

  2. Modify your solution from Lesson 1.9, Exercise 2 by using an Exp sort to express the integer and Boolean expressions that it supports, in the same style as LESSON-11-E. Then write an eval function that evaluates all terms of sort Exp to either a Bool or an Int.

Next lesson

Once you have completed the above exercises, you can continue to Lesson 1.12: Syntactic Lists.