controlling reasoning 1.logical view on rules: cnf 2.inference: td, bu, resolution 3.control in...

Controlling Reasoning

1. Logical view on rules: CNF2. Inference: TD, BU, Resolution3. Control in PLANNER, MBASE, PRESS4. Conclusions

Notations in slides for logical connectors: &, |, ¬, , A, E.

Expert Systemen 5 2

Rules as logical implications

• A person admires a philosopher if he beats him in a race• Rule: IF x is Philosopher

AND x beats y in raceTHEN y admires x

• Logic: A x,y: Ph(x) & Be(x,y) Ad(y,x)

• PROLOG: Ad(y,x) :- Ph(x), Be(x,y).• CLIPS: (defrule (Ph x) (Be x y) => (assert (Ad y x)))

CLIPS and PROLOG: Program is list of implications.

Each is valid individually: conjunction of implications.

Implication is disjunction: p q is the same as ¬p | q

Expert Systemen 5 3

Conjunctive Normal Form

• Atom: Relation symbol Be(x,y)• Literal: Atom or negated atom Be(x,y) ¬Be(x,y)• Clause: Disjunction of Literals ¬Be(x,y) | ¬Ph(x) |

Ad(y,x)

• Conjunctive Normal Form: Conjunction of clauses.CNF is Universal: Every statement is equivalent to a CNF.

• Special case: Horn clause has ONE positive literal.Our programs have Horn clauses but…

• Horn clauses are NOT universal!Influences expressibility and efficiency of programs.

Expert Systemen 5 4

CNF is Universal

Statement: Every student uses some computer and some computer is used by every student.

Formalize in logic:(A x: St(x) (E y: Co(y) & Use(x,y)) ) &(E z: Co(z) & (A t: St(t) Use(t,z)) )

Eliminate implications:(A x: ¬St(x) | (E y: Co(y) & Use(x,y)) ) &(E z: Co(z) & (A t: ¬St(t) | Use(t,z)) )

Replace existentials by Skolem functions:(A x: ¬St(x) | (Co(Y(x)) & Use(x,Y(x))) ) &(Co(Z) & (A t: ¬St(t) | Use(t,Z)) )

Expert Systemen 5 5

Transformation to CNF, continued

Replace existentials by Skolem functions:(A x: ¬St(x) | (Co(Y(x)) & Use(x,Y(x))) ) &(Co(Z) & (A t: ¬St(t) | Use(t,Z)) )

Move universal quantifiers up front:A x, A t: ¬St(x) | (Co(Y(x)) & Use(x,Y(x))) & (Co(Z) & (¬St(t) | Use(t,Z)) )

Distribute | over & and drop quantifiers:(¬St(x) | Co(Y(x)) & // St(x) Co(Y(x))(¬St(x) | Use(x,Y(x))) & // St(x) Use(x,Y(x))Co(Z) & // Co(Z)(¬St(t) | Use(t,Z)) // St(t) Use(t,Z)

Expert Systemen 5 6

Clause notation and meaning

• { } means False• {p} means p• {¬p} means ¬p• {¬p, q} means p q• {¬p, ¬q, r} means p & q r• {p} {q} means p & q

• {¬p, q, r} means p (q | r) or (p & ¬q) r

This is a non-Horn clause!

Expert Systemen 5 7

I have a problem with my TV . . .

Experts tell me that• IF powerfailure

THEN blackscreen• IF unitfailure

THEN blackscreenDeclarative (causal)

domain knowledge

I want to express this knowledge as• IF blackscreen

THEN powerfailure OR unitfailureProcedural domain knowledge

1. Why do I want that?2. Is the conclusion justified?3. How can I do it?

(assert (fact1 | fact2)) ??

• IF blackscreen & ¬unitfailureTHEN powerfailure

1. Symmetry between failures lost2. Mixes symptom with cause3. Negated conditions are costly

Expert Systemen 5 8

Conjunctions and Disjunctions in Implications

• p | q rConditions p and q each imply r by themselves:(p r) & (q r)Horn clause: {¬p,r} {¬q,r}

• p & q rHorn clause: {¬p,¬q,r}

• p q & rCondition p implies both q and r separately:(p q) & (p r)Horn clause: {¬p,q} {¬p,r}

Disjunction in the conclusion is the only problematic case

Expert Systemen 5 9

Inference: Bottom-Up

• Facts in the database express positive literals

• MP and MMP express how to extend the fact set.

• Modus Ponens:

p p s

s

• Multi Modus Ponens:

p q r p&q&r s

s

Initial Facts

Goal

Bottom-Up inference produces many irrelevant facts.

Expert Systemen 5 10

Inference: Top-Down

• Focus on goal avoids irrelevant facts

• Goal expansion:

Goal s p s

Goal p

• Goal expansion:

Goal s p & q & r s

Goals p, q, r

• Achieve empty goal list

Initial Facts

Goal

Top-Down inference persues many unreachable goals.


Resolution generalizes BU and TD inference

• Proving G is deriving a contradiction from ¬G.

• Goal list: negated statements

Goal expansion:

Goal s p s

Goal p

becomes ¬s p s

¬p

or: Modus Tollens

Achieve contradiction:prove False.

Initial Facts ¬G

False

Resolution may still derive too many facts.


The Resolution Principle

Horn clauses generalizefacts and rules

Resolution generalizes Modus Ponens and Modus Tollens:

From { p, s1, s2, …}and {¬p, t1, t2, …},conclude {s1, s2, …, t1, t2, …}Clauses clash on literal p

Justification:In both clauses a literal is TRUE.The p literal is TRUE in one.In at least one clause, another

literal is TRUE

Examples of Resolution:• Modus Ponens:

{¬p, q} {p} give {q}• Modus Tollens:

{¬p, q} {¬q} give {¬p}• Multi Modus Ponens:

{¬p, ¬q, ¬r, s} {p} {q} {r}give in three steps

• {¬q, ¬r, s}

• {¬r, s}

• {s}

• Contradiction:{¬p} {p} give {}


More proof rules subsumed by Resolution

Chaining:• p q and q r

imply p r

• If p is known, r is derived with 2x Modus Ponens

• Natural deduction:assume and then eliminate p

• Resolution:{¬p, q} and {¬q, r}clash on q and give{¬p, r}

Case distinction:• p q and ¬p q

imply q

• Resolution:{¬p, q} and {p, q}clash on p and give{q}


Resolution Conflict strategy

• Backward (Set-of-support):Use ¬G (or clause derived from it) in every step.

• Forward reasoning:Ignore ¬G until you can resolve it with G.

• Input Resolution:In each step use an IF or ¬G

• Unit preference:Prefer clauses with one literal

• Recency, Specificity, …

Initial Facts ¬G

False

TD-side

BU-side

Can we model and use Human Problem Solving

Knowledge here?


PLANNER (Hewitt, 1971)

Distinguish antecedent andconsequent theorems• A block is on the table

(ante (block x) (assert (on x table)))is triggered by antecedent(Bottom-Up theorem)

• Men are mortal(conse (mortal x) (goal (man x)))is triggered by consequent(Top-Down theorem)

Pros:• Employs some HE insights• Some efficiency gain

Cons:• Sacrifice completeness:

All true statements can be derived.Counterexample: (ante p (assert q)) (ante t (assert G)) (conse t (goal q)) p ¬G // Goal: G

• Efficiency gain is small


MECHO System Architecture

MECHO: Solver for highschool mechanics problemsImplemented in PROLOG, but with extended inference

control: MBASE

languageinterpreter

MECHO:physics

PRESS:mathematics

MBASE

PROLOG


MBASE: A Prolog extension

PROLOG inference control:rule ordering, clause ordering, !

First order conflict resolution:try rules in order in program.Put grandpa (abraham, jacob).before grandpa (x, y) :- fath(x, z), fath(z, y).

Second order conflict resolution:try clauses in listed order:Better write grandpa (x, y) :- fath(z, y), fath(x, z).

In both cases the two programs are logically equivalent.


The Use of ! (cut)

If x is found to be a woman, he cannot be anybodies grandpa: grandpa(x, y) :- female(x), fail.

Not enough; PROLOG will still try alternative clauses: grandpa(x, y) :- female(x), !, fail.

To prevent the penguin from flying: flies(X) :- penguin(X), !, fail. flies(X) :- bird(X).

Makes the program logically inconsistent.

This clause is not successful

Don’t try anything else


MBASE additional control: DBC

Knowledge that brothers share grandpa may find grandpa more efficiently: grandpa(x, y) :- brother (y, z), grandpa (x, z).

Want to use known facts only: grandpa(x, y) :- brother (y, z), DBC (grandpa (x, z) ).

DBC: Use fact if in database, but do not derive.

Knowledge that anybody has just one grandpa may stop superfluous search for another: grandpa(x, y) :- DBC (grandpa (z, y) ), different(z, x), !, fail.


MECHO additional control: Frames

• Hierarchical grouping of concepts:• a1 isa centrifugal

• centrifugal isa acceleration

• acceleration isa quantity

• Knowledge is associated to concepts.

Frame concept is NOT supported by MECHO/PROLOG,but implemented within MECHO


How PRESS solves an equation

Human problem solving strategy• Equation deduced from text (by MECHO):

log(x+1) = c – log(x - 1)• Rewrite to bring x on one side:

log(x+1) + log(x-1) = c• Rewrite to collect (reduce no. of occurrences of x):

log(x2 – 1) = c• Rewrite to isolate the value x:

x = √(ec +1)

Insufficient: algebraic rewriting rules, domain knowledge. (x+1)(x-1) = x2 - 1

We must supply them together with an application context.


PRESS solution strategies:

/* Isolate a single occurrence of x */solution( Lhs=Rhs, SolvedEquation ) :- occur( x, Lhs, 1 ), occur( x, Rhs, 0 ), isolate( Lhs=Rhs, SolvedEquation ), !.

/* Try to combine two occurrences */solution( Equation, SolvedEquation ) :- occur( x, Equation, N ), N > 1, collect( Equation, Equation1 ), solution( Equation1, SolvedEquation ), !.

/* Try two move two occurrences closer together */solution( Equation, SolvedEquation ) :- occur( x, Equation, N ), N > 1, attract( Equation, Equation1 ), solution( Equation1, SolvedEquation ), !.


Guide attraction by occurrences of x

/* Attraction can be done in subexpressions */attract( Equation, Equation1 ) :- subexpr( E, Equation ), attract_rewrite( E, E1 ), replace_sub( E, Equation, E1, Equation1 ).

/* Combine sum of logs, both containing x */attract_rewrite( log(U)+log(V), log(U*V) ) :- occur( x, U, NU ), NU > 0, occur( x, V, NV ), NV > 0.


Collection is the ultimate attraction:

/* Allow collection on subexpressions */collect( Equation, Equation1 ) :- sub_expr( E, Equation ), collect_rewrite( E, E1 ), replace_sub( E, Equation, E1, Equation1 ).

/* Formula’s that vanish one x */collect_rewrite( (x+N)*(x-N), (x^2-Nsq) ) :- Nsq is N^2.

collect_rewrite( A*x + B*x , C * x ) :- C is A + B.

Familiar product rule, but now has a direction and a goal

context


Isolation: Peel the Onion to the kernel

/* isolate brings an equation of the form LHS=RHS, with a single x in LHS, in the form x=Rhs */

/* What does that kernel look like */isolate( x=Rhs, x=Rhs ) :- !.

/* Otherwise, apply recursion */isolate( Equation, SolvedEquation ) :- isolate_rewrite( Equation, Equation1 ), isolate( Equation1, SolvedEquation ).

/* A is the subexpr of LHS that contains x */isolate_rewrite( sin(A)=B, A=arcsin(B) ).isolate_rewrite( log(A)=B, A=exp(B) ).isolate_rewrite( A^N=B, A=B^(1/N) ).


Conclusion about Mbase/Press

• MECHO was successful because it workedwhich is what we want

• Rules are not just laws of algebra,but solution strategies of Human Expertwhich is what we want

• Solution strategies are coded together with algebra,which is not what we want

• Many are represented implicitly (eg., by clause order)which is not what we want

MECHO was influential as1. a demonstration of what was possible in AI / PROLOG2. a guide for later research in strategic knowledge

representation


I still have a problem with my TV . . .

Experts tell me that• IF powerfailure

THEN blackscreen• IF unitfailure

THEN blackscreen

I want to express this knowledge as• IF blackscreen

THEN powerfailure OR unitfailure

1. Why do I want that?2. Is the conclusion justified?

Both TD and BU inference only draw logically valid conclusions

Closed World Assumption:If a statement is true, it can be proved

CWA in PROLOG: as long as blackscreen cannot be derived, your TV is fine.

CWA: The expert knowledge is complete wrt. possible causes

controlling reasoning 1.logical view on rules: cnf 2.inference: td, bu, resolution 3.control in...

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