Description
A Rank-Based Type System for Stack Language 2
We extend Stack Language 2 from Homework 4 by adding the following three operations
DEC – decrements the topmost element (must be Int) on the stack SWAP exchanges the two topmost elements on the stack
POP k – pops k elements off the stack
The abstract syntax of the extended language is as follows.
type Prog = [Cmd]
data Cmd = LDI Int | LDB Bool | LEQ | ADD | MULT | DUP | IFELSE Prog Prog | DEC | SWAP | POP Int deriving Show
Again the types of the values on the Stack are Bool and Int. You should dynamically type check all operations during run time. Instead of using Either as in homework 4 you can define
data Val = I Int | B Bool
You can assume that the initial contents of the stack are all integers or empty.
We will also define a rank type system for this language that assigns ranks to stacks and operations and ensures that a program does not result in a rank mismatch. A rank mismatch occurs when there are not enough elements on the stack to perform an operation in a program. For example, executing the program p1 = [ ADD] with the stack = [1] will result in a rank mismatch “RankError” also known as a stack underflow.
The rank of a stack is given by the number of its elements. The rank of a single stack operation is given by a pair of numbers (n, m) where n indicates the number of elements operation takes off the top of the stack and m is the number of elements the operation puts onto the stack. The rank for a stack program is defined to be the rank of the stack that would be obtained if the program was run on an empty stack. The rank of a stack program p1 run with stack s1 is the rank of the stack s1 after the execution of program p1. A rank error occurs in a stack program when an operation with rank (n, m) is executed on a stack with rank k < n. For example:
The program p2 = [ LDI 5, LDI 10, ADD] has rank 1 when run with the stack s1 = [] and rank 2 when run with s2 = [5]. That is run p s2 results in [15, 5].
Use the following types to represent stack and operation ranks.
type Rank = Int
First, define a function rankC that maps each stack operation to its rank.
rankC ∷ Cmd → CmdRank
For example:
rankC (ADD) = (2, 1) since ADD removes two values from the stack and places one on. rankC (DEC) = (1, 1) since DEC removes one element decrements and places on the stack …..
The rank of Cmd (IFELSE Prog1 Prog2) will need to be calculated differently. Since we do not know which branch is executed we will need to verify that there is enough stack to execute either branch on the current stack contents. If either Prog1 or Prog2 produces a rank error then the entire IFELSE produces an error. If the IFELSE command does not produce a rank error then set the rank of the current stack used for subsequent command to the min {rank(Prog1) , rank (Prog2)}.
For example, p2 = [IFELSE [ADD] [LDI 2, DUP] ] ran with stack s2 = [B True, I 5] of rank 2 would result in a rank error. However, if p2 ran with stack s3 = [B False, I 10] the result would be [I 4, I 10]. Since we don’t know which branch the IFELSE statement will take we will assign a Rank Error if the statck is rank 2.
Next define a function rankP that computes the rank of a program when ran with stack with rank r. The Maybe data type is used to capture rank error, that is, a program that contains a rank error should be mapped to Nothing whereas ranks of other programs are wrapped by the Just constructor.
rankP ∷ Prog → Rank → Maybe Rank
Hint: You may need an auxiliary function rank ∷ Prog → Rank → Maybe Rank and define rankP using rank.
Following the example of the function evalStatTC (defined in the file TypeCheck.hs), define a function semStatTC for evaluating the stack program that first calls the function rankP to check whether the stack program is rank correct. If the program with a given stack is rank correct then semStatTC calls semCmd to execute the program otherwise SemStatTC produces a RankError. Your run function can call semStatTC.
However, note that he function semCmd called by semStatTC can be simplied since we know there will never be a stack underflow.
You may work in groups of up to three people on this assignment. You will need to sign-up for a group in Canvas at least 24 hours before the due date of the assignment.
Below are some simple examples. The result is A Stack or a RankError.
