4.1. Defining SLang 1¶
4.1.1. Syntax of SLang 1¶
So far we have examined how to program from a functional programming perspective and how the lambda calculus forms the theoretical foundation for that perspective. We will now shift our focus and consider how we could actually develop a small interpreter for a language based on the lambda calculus. We'll call that language SLang1, short for "Simple Language 1".
The development process for this interpeter will require our writing a JISON grammar to define the syntax of the language and transform programs in the language into an abstract syntax tree (AST).
How does abstract syntax differ from concrete syntax? In other words, how do parse trees differ from abstract syntax trees?
A parse tree has a leaf node for each token, and an interior node for each nonterminal. Parse trees are useful to represent the structure or syntax of the source program.
However, we are now interested in the meaning of the program, not just its superficial structure. Abstract syntax trees (or ASTs) are parse trees stripped of all nodes that are not essential for later processing (for us: the execution or interpretation process).
In an AST:
 Operators appear at interior nodes, not at leaf nodes and their operands become their children.
 Chains of unit productions (i.e., productions of the form
<X> ::= <Y>
) are collapsed.  Lists are flattened.
 Syntactic details (semicolons, parens, etc.) are omitted.
This is illustrated in the example below.
Parse Tree Abstract Syntax Tree
========== ====================
exp *
 / \
term 3 +
/\ / \
term * factor 4 2
 /\
 /  \
factor ( exp )
 /\
3 exp + term
 
term factor
 
factor 2

4
Additionally an AST often does not contain any nodes corresponding to nonterminals. Nonetheless, it contains all of the information that is needed for the interpreter to derive the correct meaning of the input program, that is, to evaluate the program and return its correct value.
Input Parse Tree
===== ==========
{ ____ methodBody _________
x = 0; / / \ \
while (x<10) { { declList stmtList }
x = x+1;  / \
} epsilon stmtList stmt___
y = x*2; / \ /  \ \
} stmtList stmt ID = exp ;
/ \ \ (y) /  \
AST stmtList stmt ... exp * term
===  /   \  
epsilon ID = exp ; term factor
methodBody (x)   
/ \ INTLITERAL factor INT
declList stmtList (0)  (2)
/  \ ID
assign while assign (x)
/ \ ... / \
ID INT ID *
(x) (0) (y) / \
ID INT
(x) (2)
The concrete syntax of Slang1 is defined by the following grammar:
<program> ::= <exp>
<exp> ::= <var_exp>  <fn_exp>  <app_exp> \red{ <papp_exp>  <int>}
<fn_exp> ::= fn '(' (<var_exp> (',' <var_exp>)*)? ')' => <exp>
<app_exp> ::= '(' <exp> <exp> ')'
<papp_exp> ::= <prim_op> '(' <args>? ')'
<args> ::= <exp> (',' <exp>)*
<prim_op> ::= +  *  add1
The SLang1 "program" (fn (a,b) => b y 3) would result in the following parse tree and AST.
parse tree AST
========== ===
program program
 
exp app_exp
 / \
_ app_exp ______ fn_exp args
/   \ \ / \ / \
( exp exp exp ) [a,b] var_exp var_exp int
_________/ \ \   
/ var_exp int b y 3
______fn_exp___________ \ \
/ / /   \ \ \ y 3 [ "Program",
/ / /   \ \ \ [ "AppExp",
fn ( var_exp , var_exp ) => exp [ "FnExp",
   ["a","b"],
a b var_exp ["VarExp","b"]],
 [ "args",
b ["VarExp","y"],
["IntExp",3]]]]
4.1.2. Concrete Syntax SLang1¶
This problem will help you master the concrete syntax of SLang 1. To earn credit for it, you must complete this randomized problem correctly three times in a row.
4.1.3. Abstract Syntax SLang1¶
This problem will help you master the abstract syntax of SLang 1.
4.1.4. Curry in SLang1¶
This problem will illustrate the semantics of SLang 1 while helping you review the definition of the curry function.
4.1.5. Semantics of SLang1¶
This problem focuses on the semantics of SLang 1.