;;; # The Meaning of Programs

;;; What is computation? what a computer does.

;;; computation environment:
;;;
;;; * machine
;;; * language
;;; * program
;;; 
;;; Fundamentally, programming is about ideas: a program is a snapshot of
;;; an idea, a codification of the mental structures of the programmer(s)
;;; involved.
;;;  
;;; ## The Meaning of "Meaning"
;;;  
;;; *semantics* is the connection between words and meanings: while "dog"
;;; is an arrangment of shapes on a page, an actual dog is a very separate
;;; things. Semantics relates concrete signifiers and abstract meanings,
;;; and attempts to study the fundamental nature of the abstract meanings.
;;;  
;;; *Formal semantics* is the attempt at formalising the meanings of
;;; programmings, and using this formalisation to reason about
;;; languages. In order to specify a programming language, we need to
;;; define both its *syntax* (the representation) and its *semantics* (the
;;; meaning).
;;;  
;;; Most languages lack a formal specification and opt to use a canonical
;;; reference implementation. An alternative is to write a prose
;;; specification, which is the approach of C++. A third approach is to
;;; mathematically specify the language such that automated mathematical
;;; analysis can be done.
;;;  
;;; ### Syntax
;;;  
;;; The language's syntax is what differentiates valid examples of code like

(defun y (x) (+ x 1))

;;; from nonsense like `$%EHI`. In general, a parser reads a string
;;; (like "(defun y (x) (+ x 1)") and turns it into an *abstract
;;; syntax tree*. Syntax is ultimately only concerned with the surface
;;; appearance of the program

;;; ### Operational Semantics

;;; A practical means of thinking about the meaning of a program is
;;; *what it does*. *operational semantics* defines rules for how
;;; programs run on some machine (often an *abstract machine*). 

;;; ### Small-Step Semantics
;;;
;;; Let's imagine an abstract machine that evaluates by directly
;;; operating on the syntax, reducing it iteratively to bring about the
;;; final result.
;;;
;;; For example, to evaluate (1 * 2) + (3 * 4):

;;; 0. Compute the left-hand multiplication (1 * 2 -> 2), simplifying
;;; the expression to 2 + (3 * 4).

;;; 0. Compute the right-hand multiplication (3 * 14 -> 12),
;;; simplifying the expression to 2 + 12.

;;; 0. Carry out the addition, resulting in 14.

;;; We determine that 14 is the result because it cannot be simplified
;;; any further. It is a special type of algebraic expression (a
;;; *value*) with its own meaning.

;;; We can write down formal rules like these describing how to proceed
;;; with each small reduction step; these rules are written in a
;;; *metalanguage* (which is often mathematical notation).

;;; Let's explore the semantics of a toy language called SIMPLE. There
;;; is a formal mathematical language to this, but we'll use a
;;; programming language to make it more clear.

(defclass ast-node ()
  ()
  (:documentation "The superclass of all AST nodes."))

(defclass ast-number (ast-node)
  ((value :initarg :value :accessor nvalue))
  (:documentation "A discrete, irreducible value."))

(defclass ast-add (ast-node)
  ((left :initarg :left :accessor ast-left)
   (right :initarg :right :accessor ast-right))
  (:documentation "AST node that adds its arguments together."))

(defclass ast-mult (ast-node)
  ((left :initarg :left :accessor ast-left)
   (right :initarg :right :accessor ast-right))
  (:documentation "AST node that multiplies its arguments together."))

(defun make-node (&key (type :number) (values (list 0 0)))
  (cond
    ((equal type :number) (make-instance 'ast-number :value (car values)))
    ((equal type :add) (make-instance 'ast-add :left (car values) :right (cadr values)))
    ((equal type :mult) (make-instance 'ast-mult :left (car values) :right (cadr values)))))

(make-node :type :add
           :values (list (make-node :type :mult
                                    :values (list (make-node :values (list 1))
                                                  (make-node :values (list 2))))
                         (make-node :type :mult
                                    :values (list (make-node :values (list 3))
                                                  (make-node :values (list 4))))))

;;; We can use these to build an AST by hand:

;;; print Add(Multiply (Number 2) Multiply (Number 3))