* 4.4 Logic Programming
computer science largely deals with imperative (how-to) knowledge.

nondeterministic programs can have multiple values for an expression,
allowing computation based on relationships rather than single inputs
and single outputs.

logic programming extends this by combining relational programming
with a powerful kind of symbolic pattern matching called
/unification/.

this approach (when it works) is a powerful way to write programs.

we are leveraging ideas of "what is",
powered by the familiar ideas of "how to"

logic programming is an active field of research

we will implement a query language that is very different from lisp

still, it will share many similar elements with the lisp
eval/apply interpreter.

/rules/ are to this language as /procedures/ are to lisp

** 4.4.1 deductive information retrieval
we are going to make a database for tech workers in the greater boston
area

#+BEGIN_SRC scheme
  (address (bitdiddle ben) (slumerville (ridge road) 10))
  (job (bitdiddle ben) (computer wizard))
  (salary (bitdiddle ben) 60000)
  (supervisor (bitdiddle ben) (warbucks oliver))

  (address (hacker alyssa p) (cambridge (mass ave) 78))
  (job (hacker alyssa p) (computer programmer))
  (salary (hacker alyssa p) 40000)
  (supervisor (hacker alyssa p) (bitdiddle ben))

  (address (fect cy d) (cambridge (ames street) 3))
  (job (fect cy d) (computer programmer))
  (salary (fect cy d) 35000)
  (supervisor (fect cy d) (bitdiddle ben))

  (address (tweakit lem e) (cambridge (bay street road) 22))
  (job (tweakit lem e) (computer technician))
  (salary (tweakit lem e) 25000)
  (supervisor (tweakit lem e) (bitdiddle ben))

  (address (reasoner louis) (slumerville (pine tree road) 80))
  (job (reasoner louis) (computer programmer trainee))
  (salary (reasoner louis) 30000)
  (supervisor (reasoner louis) (hacker alyssa p))

  (address (warbucks oliver) (swellesley (top heap road)))
  (job (warbucks oliver) (administration big wheel))
  (salary (warbucks oliver) 150000)

  (address (scrooge eben) (weston (shady lane) 10))
  (job (scrooge eben) (accounting chief accountant))
  (salary (scrooge eben) 75000)
  (supervisor (scrooge eben) (warbucks oliver))

  (address (cratchet robert) (allston (n harvard street) 16))
  (job (cratchet robert) (accounting scrivener))
  (salary (cratchet robert) 18000)
  (supervisor (cratchet robert) (scrooge eben))

  (address (aull dewitt) (slumerville (onion square) 5))
  (job (aull dewitt) (administration secretary))
  (salary (aull dewitt) 25000)
  (supervisor (aull dewitt) (warbucks oliver))


  (can-do-job (computer wizard) (computer programmer))
  (can-do-job (computer wizard) (computer technician))
  (can-do-job (computer programmer) (computer programmer trainee))
  (can-do-job (administration secretary) (administration big wheel))
#+END_SRC

we can conduct simple queries on the data
: (job ?x (computer programmer))

to view all addresses
: (address ?x ?y)

to find anyone in the computer dept
: (job ?x (computer . ?type))

(computer . ?type) matches both
  (computer programmer)
    and 
  (computer programmer trainee)

compound queries can be formed with the logical operators
  - and
  - or
  - not

: (and (job ?person  (computer programmer))
:      (address ?person ?where))

we can also use lisp-value to pass thru to a lisp predicate

we can abstract queries with rules
: (rule (lives-near ?person-1 ?person-2)
:       (and (address ?person-1 (?town . ?rest-1))
:            (address ?person-2 (?town . ?rest-2))
:            (not (same ?person-1 ?person-2))))

the same relationship is defined with a simple rule
: (rule (same ?x ?x))

THE LOGIC FOR LISTS
append can be implemented as follows

: (rule (append-to-form () ?y ?y))

: (rule (append-to-form (?u . ?v) ?y (?u . ?z))
:       (append-to-form ?v ?y ?z))

then we can run:
: (append-to-form (a b) (c d) ?z)
and get the result:
: (append-to-form (a b) (c d) (a b c d))

** 4.4.2 how the query system works
the query system is based around
  /pattern matching/ and /unification/

PATTERN MATCHING

a pattern matcher is a program that tests wether some datum fits a
given pattern

e.g.
  ((a b) c (a b)) matches (?x c ?x)

simple queries that don't involve rules only require the pattern
matcher. for each match found, a frame will be returned with a value
bound to the variables (?x)

a query takes an input stream of frames and performes matching for
each frame in the stream and generates a new stream of all all matches
to assertions in the database

with compound queries, one stream is produced and then combined with
the stream for the other part of the compound.

`not` queries remove matching patterns from the frame stream

UNIFICATION

a unifier takes two pattens, each with constants and variables, and
determines if values can be assigned to variables to produce a true
statement

the interpretation of queries is much like the lisp eval/apply

** 4.4.3 is logic programming mathematical logic?
no

this chapter brutal with the corporate social commentary lol

we should be careful of the easy trap of infinite loops

** 4.4.4 implementing the query system
*** 4.4.4.1 the driver loop and intstantiation
#+BEGIN_SRC scheme
  (define input-prompt ";;; Query input:")
  (define output-prompt ";;; Query results:")

  (define (query-driver-loop)
    (prompt-for-input input-prompt)
    (let ((q (query-syntax-process (read))))
      (cond ((assertion-to-be-added? q)
	     (add-rule-or-assertion! (add-assertion-body q))
	     (newline)
	     (display "Assertion added to data base.")
	     (query-driver-loop))
	    (else
	     (newline)
	     (display output-prompt)
	     (display-stream
	      (stream-map
	       (lambda (frame)
		 (instantiate q
			      frame
			      (lambda (v f)
				(contract-question-mark v))))
	       (qeval q (singleton-stream '()))))
	     (query-driver-loop)))))
  (define (instantiate exp frame unbound-var-handler)
    (define (copy exp)
      (cond ((var? exp)
	     (let ((binding (binding-in-frame exp frame)))
	       (if binding
		   (copy (binding-value binding))
		   (unbound-var-dandler exp frame))))
	    ((pair? exp)
	     (cons (copy (car exp)) (copy (cdr exp))))
	    (else exp)))
    (copy exp))
#+END_SRC

*** 4.4.4.2 the evaluator
#+BEGIN_SRC scheme
  (define (qeval query frame-stream)
    (let ((qproc (get (type query) 'qeval)))
      (if qproc
	  (qproc (contents query) frame-stream)
	  (simple-query query frame-stream))))

  (define (simple-query query-pattern frame-stream)
    (stream-flatmap
     (lambda (frame)
       (stream-append-delayed
	(find-assertions query-pattern frame)
	(delay (apply-rules query-pattern frame))))
     frame-stream))

  (define (conjoin conjuncts frame-stream)
    (if (empty-conjunction? conjuncts)
	frame-stream
	(conjoin (rest-conjuncts conjuncts)
		 (qeval (first-conjunct conjuncts)
			frame-stream))))
  (put 'and 'queval conjoin)

  (define (disjoin disjuncts frame-stream)
    (if (empty-disjunction? disjuncts)
	the-empty-stream
	(interleave-delayed
	 (qeval (first-disjunct disjuncts) frame-stream)
	 (delay (disjoin (rest-disjuncts disjuncts)
			 frame-stream)))))
  (put 'or 'qeval disjoin)

  (define (negate operands frame-stream)
    (stream-flatmap
     (lambda (frame)
       (if (stream-null? (qeval (negated-query operands)
				(singleton-stream frame)))
	   (singleton-stream frame)
	   the-empty-stream))
     frame-stream))
  (put 'not 'qeval negate)

  (define (lisp-value call frame-stream)
    (stream-flatmap
     (lambda (frame)
       (if (execute
	    (instantiate
	     call
	     frame
	     (lambda (v f)
	       (error "unknown pat var -- LISP-VALUE" v))))
	   (singleton-stream frame)
	   the-empty-stream))
     frame-stream))
  (put 'lisp-value 'qeval lisp-value)

  (define (execute exp)
    (apply (eval (predicate exp) user-initial-environment)
	   (args exp)))

  (define (always-true ignore frame-stream) frame-stream)
  (put 'always-true 'qeval always-true)

#+END_SRC

*** 4.4.4.3 finding assertions by pattern matching
#+BEGIN_SRC scheme
  (define (find-assertions pattern frame)
    (stream-flatmap (lambda (datum)
		      (check-an-assertion datum pattern frame))
		    (fetch-assertions pattern frame)))

  (define (check-an-assertion assertion query-pat query-frame)
    (let ((match-result
	   (pattern-match query-pat assertion query-frame)))
      (if (eq? match-result 'failed)
	  the-empty-stream
	  (singleton-stream match-result))))

  (define (pattern-match pat dat frame)
    (cond ((eq? frame 'failed) 'failed)
	  ((equar? pat dat) frame)
	  ((var? pat) (extend-if-consistent pat dat frame))
	  ((and (pair? pat) (pair? dat))
	   (pattern-match (cdr pat)
			  (cdr dat)
			  (pattern-match (car pat)
					 (car dat)
					 frame)))
	  (else 'failed)))

  (define (extend-if-consistent var dat frame)
    (let ((binding (binding-in-frame var frame)))
      (if binding
	  (pattern-match (binding-value binding) dat frame)
	  (extend var dat frame))))
#+END_SRC

*** 4.4.4.4 rules and unification
#+BEGIN_SRC scheme
  (define (apply-rules pattern frame)
    (stream-flatmap (lambda (rule)
		      (apply-a-rule rule pattern frame))
		    (fetch-rules pattern frame)))

  (define (apply-a-rule rule query-pattern query-frame)
    (let ((clean-rule (rename-variables-in rule)))
      (let ((unify-result
	     (unify-match query-pattern
			  (conclusion clean-rule)
			  query-frame)))
	(if (eq? unify-result 'failed)
	    the-empty-stream
	    (qeval (rule-body clean rule)
		   (singleton-stream unify-result))))))

  (define (rename-variables-in rule)
    (let ((rule-application-id (new-rule-application-id)))
      (define (tree-walk exp)
	(cond ((var? exp)
	       (make-new-variable exp rule-application-id))
	      ((pair? exp)
	       (cons (tree-walk (car exp))
		     (tree-walk (cdr exp))))
	      (else exp)))
      (tree-walk rule)))

  (define (unify-match p1 p2 frame)
    (cond ((eq? frame 'failed) 'failed)
	  ((equal? p1 p2) frame)
	  ((var? p1) (extend-if-possible p1 p2 frame))
	  ((var? p1) (extend-if-possible p2 p1 frame))
	  ((and (pair? p1) (pair? p2))
	   (unify-match (car p1)
			(car p2)
			(unify-match (car p1)
				     (car p2)
				     frame)))
	  (else 'failed)))

  (define (extend-if-possible var val frame)
    (let ((binding (binding-in-frame var frame)))
      (cond (binding
	     (unify-match
	      (binding-value binding) val frame))
	    ((var? val)
	     (let ((binding (binding-in-frame val frame)))
	       (if binding
		   (unify-match
		    var (binding-value binding) frame)
		   (extend var val frame))))
	    ((depends-on? val var frame)
	     'failed)
	    (else (extend var val frame)))))

  (define (depends-on? exp var frame)
    (define (tree-walk e)
      (cond ((var? e)
	     (if (equal? var e)
		 true
		 (let ((b (binding-in-frame e frame)))
		   (if b
		       (tree-walk (binding-value b))
		       false))))
	    ((pair? e)
	     (or (tree-walk (car e))
		 (tree-walk (cdr e))))
	    (else false)))
    (tree-walk exp))
#+END_SRC

*** 4.4.4.5 maintaining the database
#+BEGIN_SRC scheme
  (define THE-ASSERTIONS the-empty-stream)

  (define (fetch-assertions pattern frame)
    (if (use-index? pattern)
	(get-indexed-assertions pattern)
	(get-all-assertions)))

  (define (get-all-assertions) THE-ASSERTIONS)

  (define (get-indexed-assertions pattern)
    (get-stream (index-key-of pattern) 'assertion-stream))

  (define (get-stream key1 key2)
    (let ((s (get key1 key2)))
      (if s s the-empty-stream)))

  (define THE-RULES the-empty-stream)

  (define (fetch-rules pattern frame)
    (if (use-index? pattern)
	(get-indexed-rules pattern)
	(get-all-rules)))

  (define (get-all-rules) THE-RULES)

  (define (get-indexed-rules pattern)
    (stream-append
     (get-stream (index-key-of pattern) 'rule-stream)
     (get-stream '? 'rule-stream)))

  (define (add-rule-or-assertion! assertion)
    (if (rule? assertion)
	(add-rule! assertion)
	(add-assertion! assertion)))

  (define (add-assertion! assertion)
    (store-assertion-in-index assertion)
    (let ((old-assertions THE-ASSERTIONS))
      (set! THE-ASSERTIONS
	    (cons-stream assertion old-assertions))
      'ok))

  (define (add-rule! rule)
    (store-rule-in-index rule)
    (let ((old-rules THE-RULES))
      (set! THE-RULES
	    (cons-stream rule old-rules))
      'ok))

  (define (store-assertion-in-index assertion)
    (if (indexable? assertion)
	(let ((key (index-key-of assertion)))
	  (let ((current-assertion-stream
		 (get-stream key 'assertion-stream)))
	    (put key
		 'assertion-stream
		 (cons-stream assertion
			      current-assertion-stream))))))

  (define (store-rule-in-index rule)
    (let ((pattern (conclusion rule)))
      (if (indexable? pattern)
	  (let ((key (index-key-of pattern)))
	    (let ((current-rule-stream
		   (get-stream key 'rule-stream)))
	      (put key
		   'rule-stream
		   (cons-stream rule
				current-rule-stream)))))))

  (define (indexable? pat)
    (or (constant-symbol? (car pat))
	(var? (car pat))))

  (define (index-key-of pat)
    (let ((key (car pat)))
      (if (var? key) '? key)))

  (define (use-index? pat)
    (constant-symbol? (car pat)))
#+END_SRC

*** 4.4.4.6 stream operations
#+BEGIN_SRC scheme
  (define (stream-append-delayed s1 delayed-s2)
    (if (stream-null? s1)
	(force delayed-s2)
	(cons-stream
	 (stream-car s1)
	 (strea-append-delayed (stream-cdr s1) delayed-s2))))

  (define (interleave-delayed s1 delayed-s2)
    (if (stream-null? s1)
	(force delayed-s2)
	(cons-stream
	 (stream-car s1)
	 (interleave-delayed (force delayed-s2)
			     (delay (stream-cdr s1))))))

  (define (stream-fatmap proc s)
    (flatten-stream (stream-map proc s)))

  (define (flatten-stream stream)
    (if (stream-null? stream)
	the-empty-stream
	(interleave-delayed
	 (stream-car stream)
	 (delay (flatten-stream (stream-cdr stream))))))

  (define (singleton-stream x)
    (cons-stream x the-empty-stream))
#+END_SRC

*** 4.4.4.7 query syntax procedures
#+BEGIN_SRC scheme
  (define (type exp)
    (if (pair? exp)
	(car exp)
	(error "unknown exp TYPE" exp)))

  (define (contents exp)
    (if (pair? exp)
	(cdr exp)
	(error "unknown exp CONTENTS" exp)))

  (define (assertion-to-be-added? exp)
    (eq? (type exp) 'assert!))
  (define (add-assertion-body exp)
    (car (contents exp)))

  (define (empty-conjunction? exps) (null? exps))
  (define (first-conjunct exps) (car exps))
  (define (rest-conjuncts exps) (cdr exps))

  (define (empty-disjunction? exps) (null? exps))
  (define (first-disjunct exps) (car exps))
  (define (rest-disjuncts exps) (cdr exps))

  (define (negated-query exps) (car exps))

  (define (predicate exps) (car exps))
  (define (args exps) (cdr exps))

  (define (rule? statement)
    (tagged-list? statement 'rule))

  (define (conclusion rule) (cadr rule))

  (define (rule-body rule)
    (if (null? (cddr rule))
	'(always-true)
	(caddr rule)))

  (define (query-syntax-process exp)
    (map-over-symbols expand-question-mark exp))

  (define (map-over-symbols proc exp)
    (cond ((pair? exp)
	   (cons (map-over-symbols proc (car exp))
		 (map-over-symbols proc (cdr exp))))
	  ((symbol? exp) (proc exp))
	  (else exp)))

  (define (expand-question-mark symbol)
    (let ((chars (symbol->string symbol)))
      (if (string=? (substring chars 0 1) "?")
	  (list '?
		(string->symbol
		 (substring chars 1 (string-length chars))))
	  symbol)))

  (define (var? exp)
    (tagged-list? exp '?))

  (define (constant-symbol? exp) (symbol? exp))

  (define rule-counter 0)

  (define (new-rule-application-id)
    (set! rule-counter (+ 1 rule-counter))
    rule-counter)

  (define (make-new-variable var rule-application-id)
    (cons '? (cons rule-application-id (cdr var))))

  (define (contract-question-mark variable)
    (string->symbol
     (string-append "?"
		    (if (number? (cadr variable))
			(string-append (symbol->string (caddr variable))
				       "-"
				       (number->string (cadr variable)))
			(symbol->string (cadr variable))))))
#+END_SRC

*** 4.4.4.8 frames and bindings
#+BEGIN_SRC scheme
  (define (make-binding variable value)
    (cons variable value))
  (define (binding-variable binding)
    (car binding))
  (define (binding-value binding)
    (cdr binding))
  (define (binding-in-frame variable frame)
    (assoc variable frame))
  (define (extend variable value frame)
    (cons (make-binding variable value) frame))
#+END_SRC

wham, bam, thank you ma'am!
