sandbox/uc/cl/meaning.lisp

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2018-01-16 17:46:44 +00:00
;;; # 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))