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Controlling Reasoning

Controlling Reasoning. Logical view on rules: CNF Inference: TD, BU, Resolution Control in PLANNER, MBASE, PRESS Conclusions Notations in slides for logical connectors: &, |, ¬, , A, E. Rules as logical implications. A person admires a philosopher if he beats him in a race

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Controlling Reasoning

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  1. Controlling Reasoning Logical view on rules: CNF Inference: TD, BU, Resolution Control in PLANNER, MBASE, PRESS Conclusions Notations in slides for logical connectors: &, |, ¬, , A, E.

  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 race THEN 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)  rThis is a non-Horn clause! Expert Systemen 5

  7. Experts tell me that IF powerfailureTHEN blackscreen IF unitfailureTHEN blackscreen Declarative (causal) domain knowledge I want to express this knowledge as IF blackscreenTHEN powerfailure OR unitfailure Procedural domain knowledge Why do I want that? Is the conclusion justified? How can I do it?(assert (fact1 | fact2)) ?? IF blackscreen & ¬unitfailureTHEN powerfailure Symmetry between failures lost Mixes symptom with cause Negated conditions are costly I have a problem with my TV . . . Expert Systemen 5

  8. 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 Conjunctions and Disjunctions in Implications Expert Systemen 5

  9. 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 Goal Initial Facts Inference: Bottom-Up Bottom-Up inference produces many irrelevant facts. Expert Systemen 5

  10. 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 Inference: Top-Down Top-Down inference persues many unreachable goals. Goal Initial Facts Expert Systemen 5

  11. 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. False Initial Facts ¬G Resolution generalizes BU and TD inference Resolution may still derive too many facts. Expert Systemen 5

  12. 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 {} The Resolution Principle Expert Systemen 5

  13. Chaining: p  q and q  rimply 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} More proof rules subsumed by Resolution Expert Systemen 5

  14. 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, … False Initial Facts ¬G Resolution Conflict strategy Can we model and use Human Problem Solving Knowledge here? BU-side TD-side Expert Systemen 5

  15. 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 PLANNER (Hewitt, 1971) Distinguish antecedent and consequent 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) Expert Systemen 5

  16. MECHO System Architecture MECHO: Solver for highschool mechanics problems Implemented in PROLOG, but with extended inference control: MBASE language interpreter MECHO: physics PRESS: mathematics MBASE PROLOG Expert Systemen 5

  17. 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. Expert Systemen 5

  18. 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. The Use of ! (cut) This clause is not successful Don’t try anything else Expert Systemen 5

  19. 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. MBASE additional control: DBC Expert Systemen 5

  20. 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 Expert Systemen 5

  21. 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. Expert Systemen 5

  22. 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 ), !. Expert Systemen 5

  23. 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. Expert Systemen 5

  24. 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 Expert Systemen 5

  25. 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) ). Expert Systemen 5

  26. 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 as • a demonstration of what was possible in AI / PROLOG • a guide for later research in strategic knowledge representation Expert Systemen 5

  27. Experts tell me that IF powerfailureTHEN blackscreen IF unitfailureTHEN blackscreen I want to express this knowledge as IF blackscreenTHEN powerfailure OR unitfailure Why do I want that? 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 I still have a problem with my TV . . . Expert Systemen 5

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