Automated Deduction

1
Automated Deduction

• resolution (Otter)
• backward-chaining (Prolog)
• forward-chaining (Rete, Clips, Jess)
• also called theorem provers or inference
  engines
• example application
• think about using rules to infer right-of-way
  in a driving simulation

2
Resolution

• first-order (with unification)
• (a?b)?(?a?c)?(b?c)
• where qunifier(a,a) and bapply(b,q),
  capply(c,q)
• example
• ?X ?works_for(X,government) v receives_pension(X)
• works_for(kate,government) v works_for(kate,walmar
  t)
• works_for(kate,walmart) v receives_pension(kate)
• qX/kate

3

• resolution is refutation-complete for FOL
• must convert all sentences to CNF
• a ground instance of a clause is formed by
  instantiating all variables with some constant
• P(X,Y)vQ(Y,Y) P(sam,bill)vQ(bill,bill)
  P(sam,joe)vQ(joe,joe)...
• Herbrands Theorem if a set of sentences is
  unsatisfiable, then there exists a set of ground
  (propositional) instances that is unsatisfiable
• Ground Resolution Theorem if a set of
  propositional clauses is unsatisfiable, then
  there is a finite derivation of the empty clause
• Lifting Lemma If there is proof of the empty
  clause using ground instances, then there is
  parallel proof using the original clauses with
  variables (using unification)

4
Converting FOL sentences to CNF

• eliminate implications (a?b??a?b)
• move neg. inwards (DeMorgans Laws, ??x P??x ?P)
• standardize variables apart (if X is used in
  multiple quantifiers, replace instance with X1,
  X2...)
• skolemization (replace ? vars with new
  constants)
• drop universal quantifiers ?
• distribute ? over ?
• break conjunctions into separate clauses
• Everyone who loves all animals is loved by
  someone.
• ?x (?y animal(y)?loves(x,y))??y loves(y,x)

5
Example

• The law says that it is a crime for an American
  to sell weapons to hostile nations. The country
  Nono, an enemy of America, has some missles, and
  all of its missles were sold to it by Colonel
  West, who is an American.
• Prove that Colonel West is a criminal.
• ?x,y,z american(x)weapon(y)sells(x,y,z)hostile(
  z) ?criminal(x)
• ?x owns(nono,x)missle(x)
• ?x owns(nono,x)missle(x)?sells(west,x,nono)
• ?x missle(x)?weapon(x)
• ?x enemy(x,america)?hostile(x)
• american(west), enemy(nono,america)
• negation of query ?criminal(west)

6
convert to CNF

• ?x,y,z american(x)weapon(y)sells(x,y,z)hostile(
  z) ?criminal(x)
• ?american(x1),?weapon(y1),?sells(x1,y1,z1),?hosti
  le(z1), criminal(x1)
• ?x missle(x)?weapon(x)
• ?missle(x2),weapon(x2)
• ?x owns(nono,x)missle(x)?sells(west,x,nono)
• ?owns(nono,x3), ?missle(x3),sells(west,x3,nono)
• ?x owns(nono,x)missle(x)
• owns(nono,M),missle(M) M is a skolem constant
• ?x enemy(x,america)?hostile(x)
• ?enemy(x4,america),hostile(x4)
• american(west), enemy(nono,america),?criminal(west
  )

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8

• maybe thousands of clauses difficult search
• search strategies which clauses to resolve?
• unit preference
• one clause must be of length 1
• guarantees to reduce length of other clause
  (toward 0)
• set of support
• one clause must be from the sos, e.g. negated
  query
• source of the unsatisfiability
• input resolution
• one clause must be from the input (KB or query)
• linear resolution
• one clause must be from negated query OR a
  successor derived from it
• complete
• Otter a real-world resolution theorem prover

9
Backward-chaining

• recall subgoal stack
• try to reduce to facts might have to back-track
• KB must be in Horn-clause form
• in FOL, use unification
• for popped subgoal, try to unify with fact or
  head of some clause
• Prolog a practical implementation of a
  back-chaining theorem prover
• funky syntax
• can be used for many solving many problems
• learn how to use back-chaining as computational
  model
• widely employed for expert systems, intelligent
  agents, control applications...

10

• Prolog syntax
• facts predicate(args,...).
• rules
• no quantifiers, variables in capital letters,
• written backwards, - means ? , read as if
• , means conjunction no disjunction
• ?X dog(X) ? mammal(X)
• mammal(X) - dog(X).
• canPlay(Child) - hasEaten(Child,dinner),finished(
  Child,homework).
• also support for strings, lists, numbers...

11

• Problem solving by back-chaining
• combinatorial enumeration/search (unbound
  variables)
• often interested in variable binding of solution

father(X,Y) - parent(X,Y),male(X). mother(X,Y)
- parent(X,Y),female(X). sibling(X,Y) -
parent(Z,X),parent(Z,Y). grandparent(X,Y) -
parent(X,Z),parent(Z,Y). uncle(X,Y) -
parent(Z,Y),sibling(X,Z),male(X). male(john).
male(sam). male(joe). male(bill). female(sue).
female(ellen). parent(sam,john),
parent(ellen,john). parent(ellen,joe).
parent(sam,joe). parent(sue,bill).
parent(al,sam). parent(sue,ellen). query ?-
uncle(bill,john). yes. query ?-
uncle(sue,john). fail. query ?-
grandparent(X,john). Xal Xsue
12
Map-Coloring in Prolog
color(red). color(green). color(blue). color(yello
w). valid_coloring(A,B,C,D,E) -
color(A),color(B),color(C),color(D),color(E),
not AB, not AC, not AD, not AE, not BC,
not BD, not CD, not DE.

• effectively enumerates all combinations of colors
  and tests them for consistency.
• will try Ared, Bred, Cred, Dred, Ered first
• fail, because doesnt satisfy not AB.
• then back-track to Ared, Bred, Cred, Dred,
  Egreen, which fails not AC.
• and so on, until reach Ared, Bgreen, Cblue,
  Dyellow, Egreen
• not very efficient, but illustrative of the kind
  of combinatorial problem-solving that can be
  simulated via back-chaining

13
Negation in Prolog

• Can have negative literals in antecedents.
• not strict FOL semantics
• negation-as-failure
• ...,not p(X),... means try to prove p(X) (by
  back-chaining with current variable bindings)
• if cannot prove it, then proceed
• if can prove it, then fail and back-track
• very handy allows default inference, compact KB
• canFly(X) - bird(X),not broken(wings(X)).
• bird(X) - canary (X).
• bird(X) - penguin(X).
• bird(X) - eagle(X).
• canary(tweety). penguin(opus). eagle(sam).
• broken(wings(opus)).

?- canFly(B). Bsam. Btweety.
14
Forward-Chaining

• requires Horn-clause KB
• combining universally-quantified rules with
  ground facts can generate many inferences
• how to do this efficiently
• Rete algorithm
• generates a graph structure
• firing rules to create nodes
• production system, basis of many expert systems
• e.g. XCON or R1, for configuring computers, or
  MYCIN for diagnosing blood diseases
• also the basis of cognitive architectures like
  ACT and SOAR (Allan Newell and Herb Simon)
• based on theory that brain does symbolic pattern
  matching, which triggers associations that
  activate other concepts...

15

• Rete algorithm
• until quiescence...
• find all rules that could fire (i.e. are
  activated by a combination of input nodes, see
  colors below)
• pick rule with highest priority (conflict
  resolution)
• unify antecedent with incoming edges
• apply unifier to consequent create new node
• uses hash tables and many other optimizations for
  efficiency

father(X,Y) - parent(X,Y),male(X). mother(X,Y)
- parent(X,Y),female(X). sibling(X,Y) -
parent(Z,X),parent(Z,Y). grandparent(X,Y) -
parent(X,Z),parent(Z,Y). uncle(X,Y) -
parent(Z,Y),sibling(X,Z),male(X). male(john).
male(sam). male(joe). male(bill). female(sue).
female(ellen). parent(sam,john),
parent(ellen,john). parent(ellen,joe).
parent(sam,joe). parent(sue,bill).
parent(al,sam). parent(sue,ellen).
uncle(bill,john)
sibling(bill,ellen)
mother(sue)
grandparent(sue,john)
16

• practical implementations of Rete
  (forward-chaining inference engines)
• CLIPS
• invented at NASA in 1980s
• used for many applications, especially discrete
  simulations, e.g. of traffic flow, shuttle
  operation, clock mechanisms, agents
• encode rules in KB executes forward-chaining by
  Rete
• http//www.siliconvalleyone.com/clips.htm
• JESS
• re-implementation in Java at Sandia National Lab
• http//www.jessrules.com/

17

• syntax (defrule name PATTERN gt ACTION)
• (defrule welcome-toddlers
• (person ?X) (age ?X ?Y) (lt ?Y 3))
• gt
• (assert (toddler ?X))
• (printout t "Hello, little one!" crlf))
• (defrule library-rule-1
• (book (name ?X) (status late) (borrower ?Y))
• (borrower (name ?Y) (address ?Z))
• gt (send-late-notice ?X ?Y ?Z))
• pattern-match left-hand side
• execute right-hand side
• actions assert, retract, printout...
• can have structured objects with slots
• can define functions...
• can call invoke external procedures (e.g.
  graphics)