= PIL: Parallel Intermediate Language

== Basic Properties
1. PIL programs are target of translators from C, Fortran, MPI/OpenMP/etc. They can also be written by humans.
1. PIL should be a high-level language like CIVL-C.
1. PIL programs can be easily translated to PFG.
1. PIL must also have a totally well-defined semantics and syntax.
1. Every operation should have a well-defined outcome, even division by zero or an illegal pointer dereference.  However, when analyzing a PIL program, a user can specify exactly which of these operations should be considered erroneous and reported.   A reasonable default might be to make all of them erroneous.
1. PIL programs can have nested funtion definitions.
1. PIL programs can use preprocessor directives like in C
1. There are no automatic conversions.  There is no "array-pointer pun".  All conversions must be done by explicit casts or other expressions.  To convert an array `a` to a pointer to element 0 of `a`, write `&a[0]`.
1. PIL supports `$input`/`$output` variables in global scope (like CIVL-C)
1. PIL identifiers are the same as C's, e.g., `x`, `f10`, ...
1. Keywords, built-in or library functions, constants, and types not in C start with `$` (like CIVL-C) 
1. PIL supports libraries: similar to C, `#include <stdlib.h>`, but these name PIL libraries and there are additional standard PIL libraries not in the C standard library
1. Program order:
 1. A program is a set of variable, function, and type definitions.
 1. The order is totally irrelevant.
 1. A variable can be used anywhere it is in scope.
 1. A function can be called anywhere it is in scope.
1. PIL programs can be divided into multiple files (translation units)
 1. One TU can refer to a variable, function, or type, in another TU.
 1. The variable just needs to be declared somewhere in the TU.
 1. The function just needs a prototype somewhere in the TU.
 1. The use of `static` in a variable declaration or function definition makes it private to that TU (so two can have same name), as in C.
 1. There is no need for `extern` so this is not in PIL.
 1. At most one of the TUs declaring the variable can have an initializer.
 1. At most one of the TUs can have a definition for a function.

== Types
- '''type names''' are used for all declarations.  There are no C declarators.  Examples:
 * `$int[] a`: declares `a` to be an array of integer
 * `$int* p`: pointer to integer
 * `$int*[] b`: array of pointer to integer
 * `$int[]* q`: pointer to array of integer
 * `$int*[]($real) f`: function from Real to array of pointer to integer
- basic types: 
 * `$bool` : the set consisting of `$true` and `$false`
 * `$int`: the set of integers
 * `$real`: the set of Reals
 * `$int<lo,hi>`: set of integers between `lo` and `hi`, inclusive. `lo` and `hi` are constant expressions so are known at compile time.
 * `$uint<hi>`: nonnegative integers less than the constant expression `hi`.  Arithmetic is performed modulo `hi` (like C's unsigned integer types). 
 * `$float<p,emax>`: the set of IEEE binary floating point numbers with precision `p` and emax `emax`.  These are also constant expressions.
- `T*`: pointer to `T`
- `struct TAG { T1 f1; ... Tn fn; }`: a C struct
- `union`: similar
- `T[]`: sequence of `T`.  Note: there is no "T[n]".  Sequences are first-class values: they can be assigned, returned, passed as arguments, etc.  
- `R(T1, ..., Tn)`: the type of a procedure which consumes values in `T1`, ..., `Tn` and returns a value in `R`.   This is basically C's function type.
- `$fun<T1,T2>` :  "logic functions":  deterministic, total, side-effect free functions from `T1` to `T2`.  Note however the function may depend on the state (i.e., the state should be considered a hidden input).   A `$pure` function is a function that does not depend on the state.
- `$set<T>` : finite set of `T`
- `$map<T1,T2>` : finite map from `T1` to `T2`.  A map is a set of ordered pairs `(x,y)` with the property that if `(x,y)` and `(x,z)` are in the map, then `y`=`z`.
- `$tuple<T,...>` : tuples of specified type.  This is similar to `struct`, but there is no tag and the fields do not have names.
- Type definitions have the form: `typedef typename ID;`

== Functions

There are two kinds of functions in PIL:
1. Imperative functions = "procedures".  A procedure has a type of the form `R(T1,...,Tn)` where `R` is the return type and the `Ti` are the input types.
1. Logic functions.  One of these has a type of the form `$fun<R,T1,...,Tn>`.

=== Procedures

A procedure is similar to a C function.  It consumes some values of a specified type and possibly returns a value of a specified type.  Procedure definitions look like C function definitions:
{{{
  R f(T1 x1, ..., Tn xn) { stmts }
}}}
defines a procedure named `f` which consumes inputs of types `T1`, ..., `Tn` and returns a value of type `R`.    `R` can be `void` if the procedure does not return a value.

The definition above defines a '''constant''' `f` of type '''`R(T1, ..., Tn)`'''.   Procedures are first-class values.    One may declare a variable of type `R(T1, ..., Tn)`, a procedure may return a value of that type, a procedure may consume a value of that type, a value of that type may be assigned to a variable, etc.   Hence the procedure type is just like any other type, and procedure definitions define new constants of that type, just as `1` is a constant of type `$int`.   Note this is different from C in that C uses function pointers; PIL dispenses with the need for function pointers and just uses functions.

A '''procedure call expression'''  has the usual form `g(e1, ..., en)`.  This is an expression that can be used anywhere an expression with side-effects is allowed.  Here, `g` is an expression of functional type, say `R(T1, ..., Tn)`, and `ei` is an expression of type `Ti` (for i=1, ..., n).  The procedure call expression has type `R`.  

Procedure calls can have side-effects, be nondeterministic, and the behavior can depend on non-local state; they may access any variable in scope, the statements may dereference pointers, etc.

{{{
  int f(int x) { return x+1; }  // f is a constant of type int(int)
  int callon1( (int)int g ) {
    return g(1);
  }
  ...
  int y = callon1(f); // y is now 2
}}}

Procedure definitions can be nested.   It is an error to call a procedure `f` when `f` is not in scope.  (This is similar to GNU C.)   In other words, if the call takes place in dyscope d, then the definition of `f` must be in d's static scope, or in the parent of that scope, or its parent, etc.

There is a second way to specify a procedure, using a lambda expression, which is described below.
     
=== Logic functions

Logic functions are a certain class of functions that have no side-effects, and are deterministic total functions of their arguments and the current state.  A logic function has a type of the form `$fun<R,T1,...,Tn>`, which signifies the set of logical functions which consume inputs of type `T1`, ..., `Tn` and return a value of type `R`.  

Logic functions are also first-class objects in PIL.   An application of a logic function `f(x1,... ,xn)` is a side-effect-free expression that can be used anywhere an expression is allowed.  A logic function is not necessarily pure, i.e., the value of an application may depend on any part of the state, not just the arguments.

Despite the apparent similarity with procedures, logic functions and procedures are clearly distinguished and one cannot be converted to another.

A logic function can be defined as follows:
{{{
  $logic R f(T1 x1, ..., xn) = expr;
}}}
where `expr` is a side-effect-free expression of type `R` and can refer to any variables in scope.

=== Misc.

Both procedure and logic function definitions can be templated, e.g.,
{{{     
       <T1,T2> int f($map<T1,T2> f, T1 x) { ... }
       <T1,T2> $logic int g($map<T1,T2> f, T1 x) { ... }
}}}   
This defines one procedure or logic function for each assignment of types to the `Ti`.

Both kinds of functions can be declared without providing definitions, indicating that the definition can be found in a different translation unit:
{{{
  int f(int x);
  $logic int g(int x);
}}}

=== Lambda expressions

Lambda expressions can be used to define functions that are anonymous and that are '''closures''', i.e., have an associated environment that persists for the life of the function.

A lambda expression that specifies a procedure closure has the form:
{{{
  $lambda [U1 v1=init1; ... Um vm=initm;] R (T1 x1, ..., Tn xn) { S1; ... }
}}}
where
* the `Ti` and `Uj` are types
* the `xi` and `vj` are variables
* the `initj` are expressions that can refer to any variables in scope
* R is a type (the return type), which may be void
* {S1; ...} is a block (same as in a procedure definition)
* if R is not void, the block must return a value of type R
* the only variables that can occur free in the block are the `xi` and `vj`.

The type of this expression is `R(T1, ..., Tn)`.   The resulting value of this is type is a procedure which can be called or assigned to a variable, etc., just like any other procedure value.

Note that the definition can only use the specified variables.  Evaluating this expression yields a closure, which is a pair consisting of a dyscope and the body of the procedure.   The dyscope has variables `vj`, which are initialized by evaluating the `initj` when the lambda expression is evaluated.  The body of the procedure may read and write to the `vj`.  That dyscope has no parent and will live as long as the procedure is around.  Hence a function may return a closure and that closure may still be called at any time, anywhere in the program, regardless of whether the original lambda expression is in scope.

When a procedure closure is called, a new dyscope is created whose parent dyscope is the dyscope of the closure.  In the new dyscope, the formal parameters are assigned the actual values and procedure is executed in that scope.  When it returns, the new dyscope is removed.

A lambda expression that specifies a logic function has the form
{{{
  $lambda [U1 v1=init1; ... Um vm=initm;] R (T1 x1, ..., Tn xn) expr
}}}
where 
* the `Ti` and `Uj` are types
* the `xi` and `vj` are variables
* the `initj` are expressions that can refer to any variables in scope
* R is a type (the return type), which cannot be void
* `expr` is a side-effect-free expression of type R 
* the only variables that can occur free in `expr` are the `xi` and `vj`.

The type of this expression is `$fun<R, T1, ..., Tn>`.   As with procedural lambdas, this yields a logic function with a dynamic scope that persists, so can be called anywhere, even after the lambda expression goes out of scope.

== Tuples

Expressions: these are all side-effect-free, assuming `expr1`, ... are side-effect-free.
- `t1 == t2` : the two tuples have the same number of components and corresponding components are equal
- `t.i` : the value of component `i` (a constant integer) of tuple `t`
- `($tuple<T1,...>){ expr1, ... }` : the literal tuple with the given list of components.
- `t.[i : expr1]` : the tuple which is the same as `t`, except for component `i` (a constant integer), which has value `expr1`

Mutating expressions:
- `t.[i] = expr` : sets component `i` of tuple `t` to `expr`

== Sets

Side-effect free expressions:
- `s1 == s2`
- `($set<T>)$empty`   // empty set of type T

Logic functions:
- `$pure $logic $bool $set_contains($set<T> s, T x);`
 * is `x` an element of `s`?
- `$pure $logic $set<T> $set_with($set<T> s, T x);`
 * `s` U {`x`}
- `$pure $logic $set<T> $set_without($set<T> s, T x);`
 * `s` - {`x`}
- `$pure $logic $set<T> $set_union($set<T> s1, $set<T> s2);`
 * `s1` U `s2` 
- `$pure $logic $set<T> $set_difference($set<T> s1, $set<T> s2);`
 * `s1`-`s2` 
- `$pure $logic $set_intersection($set<T> s1, $set<T> s2);`
 * `s1`  \cap `s2`
- `$pure $logic T[] $set_elements($set<T> s);`
 * returns the set of elements of `s` as an array in some deterministic order
- `$pure $logic $bool $set_isSubsetOf($set<T> s1, $set<T> s2);`
 * is `s1` a subset of `s2`?
- `$pure $logic $set<U> $set_map($set<T> s, $fun<T,U> f);`
 * { `f`(`x`) | `x` in `s`}
- `$pure logic $set<T> $set_comprehension($set<T>, $fun<T,$bool> p);`
 * { `x` in `s` | `p`(`x`) }

Mutating procedures:
- `$bool $set_add($set<T> * this, T x);`
- `$bool $set_remove($set<T> * this, T x);`
- `void $set_addAll($set<T> * this, $set<T> that);`
- `void $set_removeAll($set<T> * this, $set<T> that);`
- `void $set_keepOnly($set<T> * this, $set<T> that);`

==  Sequences (arrays)

Expressions:
- `a1 == a2`
- `a[i]`
- `(T[]){ expr1, ... }`

Logic functions:
- `$pure $logic T[] $seq_fun($int len, $fun<$int,T> f);`
- `$pure $logic T[] $seq_uniform($int n, T val);`
- `$pure $logic $int $length(T[] a);` // length of a
- `$pure $logic T[] $seq_write(T[] a, int i, T x);`  // a[i:=x]
- `$pure $logic T[] $seq_subseq(T[] a, int i, int n);`  // a[i..i+n-1]
- `$pure $logic T[] $seq_without(T[] a, int i);` // a with position i removed
- `$pure $logic T[] $seq_with(T[] a, int i, T x);`
- `$pure $logic T[] $seq_concat(T[] a1, T[] a2);`
- `$pure $logic U[] $seq_map(T[] a, $fun<T,U> f);`
- `$pure $logic T[] $seq_filter(T[] a, $fun<T,$bool> f);`
- `$pure $logic U $seq_foldl(T[] a, $fun<$tuple<T,U>,U> f, U init);`
- `$pure $logic U $seq_foldr(T[] a, $fun<$tuple<T,U>,U> f, U init);`

Mutating expressions:
- `a[i]=x;`

Mutating procedures:
- `T $seq_remove(T[] * this, int i);`
- `void $seq_insert(T[] * this, int i, T x);`
- `void $seq_append(T[] * this, T[] that);`

==  Maps

Non-mutating expressions:
- `m1 == m2`
- `($map<K,V>)$empty`
- `$logic V $map_get($map<K,V> K key);`
- `$logic $bool $map_containsKey($map<K,V> map, K key);`
- `$logic $bool $map_containsValue($map<K,V> map, V val);`
- `$logic $map<K,V> $map_with($map<K,V> map, K key, V val);`
- `$logic $map<K,V> $map_without($map<K,V> map, K key);`
- `$logic $set<K> $map_keys($map<K,V> map);`
- `$logic $set<$tuple<K,V>> $map_entries($map<K,V> map);`

Mutating procedures:
- `V $map_put($map<K,V> * this, K key, V val);`
- `V $map_remove($map<K,V> * this, K key);`
- `void $map_removeAll($map<K,V> * this, $set<K> keys);`
- `void $map_addAll($map<K,V> * this, $map<K,V> that);`


== Heaps

There is a single heap for dynamic allocation.  The following built-in procedures are provided:

- `T* $new<T>()`: creates an object `A` of type `T` in heap and returns `&A`.
- `T* $alloc<T>(int n)`: creates an array `A` of length `n` of `T` in heap and returns `&A[0]`.
- `void $free<T>(T* p)`: frees the object referred to by `p`, the pointer returned by an earlier call to `$new` or `$alloc`.


== Questions

* how to deal with "undefined" values?
* can there be nondeterministic expressions, i.e., expressions that evaluate to a set of values instead of one value?
* how to implement C's malloc?

Ideas on malloc:

Union of all types occurring in program:
{{{
typedef union {
  T1 t1;
  T2 t2;
  ...
} BigUnion;
}}}

A call to malloc returns a memory block:
  - size (in bytes) (int)
  - num_elements (int)
  - sequence of tuples:
    - offset (int)
    - size (int)
    - value (type `BigUnion`)