inference in first- order logic. outline reducing first-order inference to propositional inference...
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Inference in first-order logic
Outline
Reducing first-order inference to propositional inference
Unification Generalized Modus Ponens Forward chaining Backward chaining Resolution
Universal instantiation (UI)
Every instantiation of a universally quantified sentence is entailed by it:v α
Subst({v/g}, α)
for any variable v and ground term g
E.g., x King(x) Greedy(x) Evil(x) yields:King(John) Greedy(John) Evil(John)
King(Richard) Greedy(Richard) Evil(Richard)
King(Father(John)) Greedy(Father(John)) Evil(Father(John))
.
.
.
Existential instantiation (EI)
For any sentence α, variable v, and constant symbol k that does not appear elsewhere in the knowledge base:
v αSubst({v/k}, α)
E.g., x Crown(x) OnHead(x,John) yields:
Crown(C1) OnHead(C1,John)
provided C1 is a new constant symbol, called a Skolem constant
Reduction to propositional inference Suppose the KB contains just the following:
x King(x) Greedy(x) Evil(x) King(John) Greedy(John) Brother(Richard,John)
Instantiating the universal sentence in all possible ways, we have: King(John) Greedy(John) Evil(John) King(Richard) Greedy(Richard) Evil(Richard) King(John) Greedy(John) Brother(Richard,John)
The new KB is propositionalized: proposition symbols are King(John), Greedy(John), Evil(John), King(Richard), etc.
Reduction contd.
Every FOL KB can be propositionalized so as to preserve entailment A ground sentence is entailed by new KB iff entailed by original
KB
Idea: propositionalize KB and query, apply resolution, return result
Problem: with function symbols, there are infinitely many ground terms, e.g., Father(Father(Father(John)))
Reduction contd.
Theorem: Herbrand (1930). If a sentence α is entailed by an FOL KB, it is entailed by a finite subset of the propositionalized KB
Idea: For n = 0 to ∞ do create a propositional KB by instantiating with depth-$n$ terms see if α is entailed by this KB
Problem: works if α is entailed, loops if α is not entailed Theorem: Turing (1936), Church (1936) Entailment for FOL is
semidecidable algorithms exist that say yes to every entailed sentence no algorithm exists that also says no to every nonentailed sentence.
Problems with propositionalization
Propositionalization seems to generate lots of irrelevant sentences. Example
from: x King(x) Greedy(x) Evil(x) King(John) y Greedy(y) Brother(Richard,John)
it seems obvious that Evil(John), but propositionalization produces lots of facts such as Greedy(Richard) that are irrelevant
With p k-ary predicates and n constants, there are p·nk instantiations.
Unification
We can get the inference immediately if we can find a substitution θ such that King(x) and Greedy(x) match King(John) and Greedy(y) θ = {x/John,y/John} works
Unify(α,β) = θ if αθ = βθ p q θ Knows(John,x) Knows(John,Jane) Knows(John,x) Knows(y,OJ) Knows(John,x) Knows(y,Mother(y)) Knows(John,x) Knows(x,OJ)
Standardizing apart eliminates overlap of variables, e.g., Knows(John,z27) Knows(z17,OJ)
{x/Jane}
{x/OJ, y/John}
{x/Mother(John),y/John}
No substitution possible yet.
Unification
We can get the inference immediately if we can find a substitution θ such that King(x) and Greedy(x) match King(John) and Greedy(y) θ = {x/John,y/John} works
Unification finds substitutions that make different logical expressions look identical UNIFY takes two sentences and returns a unifier for them, if one
exists UNIFY(p,q) = where SUBST(,p) = SUBST (,q)
Basically, find a that makes the two clauses look alike
Unification
ExamplesUNIFY(Knows(John,x), Knows(John,Jane)) = {x/Jane}
UNIFY(Knows(John,x), Knows(y,Bill)) = {x/Bill, y/John}
UNIFY(Knows(John,x), Knows(y,Mother(y))= {y/John, x/Mother(John)
UNIFY(Knows(John,x), Knows(x,Elizabeth)) = fail Last example fails because x would have to be both John and Elizabeth We can avoid this problem by standardizing:
The two statements now read UNIFY(Knows(John,x), Knows(z,Elizabeth))
This is solvable: UNIFY(Knows(John,x), Knows(z,Elizabeth)) = {x/Elizabeth,z/John}
Unification
To unify Knows(John,x) and Knows(y,z) Can use θ = {y/John, x/z } Or θ = {y/John, x/John, z/John}
The first unifier is more general than the second. There is a single most general unifier (MGU) that
is unique up to renaming of variables.MGU = { y/John, x/z }
Unification
Unification algorithm:Recursively explore the two expressions side
by sideBuild up a unifier along the wayFail if two corresponding points do not match
The unification algorithm
The unification algorithm
Simple Example
Brother(x,John)Father(Henry,y)Mother(z,John)
Brother(Richard,x)Father(y,Richard)Mother(Eleanore,x)
Generalized Modus Ponens (GMP)
p1', p2', … , pn', ( p1 p2 … pn q) qθ
p1' is King(John) p1 is King(x)
p2' is Greedy(y) p2 is Greedy(x) θ is {x/John,y/John} q is Evil(x) q θ is Evil(John)
GMP used with KB of definite clauses (exactly one positive literal) All variables assumed universally quantified
where pi'θ = pi θ for all i
Soundness of GMP Need to show that
p1', …, pn', (p1 … pn q) ╞ qθprovided that pi'θ = piθ for all I
Lemma: For any sentence p, we have p ╞ pθ by UI
• (p1 … pn q) ╞ (p1 … pn q)θ = (p1θ … pnθ qθ)• p1', \; …, \;pn' ╞ p1' … pn' ╞ p1'θ … pn'θ • From 1 and 2, qθ follows by ordinary Modus Ponens
Storage and Retrieval
Use TELL and ASK to interact with Inference Engine Implemented with STORE and FETCH
STORE(s) stores sentence s FETCH(q) returns all unifiers that the query q unifies with
Example: q = Knows(John,x) KB is:
Knows(John,Jane), Knows(y,Bill), Knows(y,Mother(y)) Result is
1={x/Jane}, 2=, 3= {John/y,x/Mother(y)}
Storage and Retrieval
First approach: Create a long list of all propositions in Knowledge Base Attempt unification with all propositions in KB
Works, but is inefficient Need to restrict unification attempts to sentences that
have some chance of unifying Index facts in KB
Predicate Indexing Index predicates:
All “Knows” sentences in one bucket All “Loves” sentences in another
Use Subsumption Lattice (see below)
Storing and Retrieval Subsumption Lattice
Child is obtained from parent through a single substitution Lattice contains all possible queries that can be unified with it.
Works well for small lattices Predicate with n arguments has a 2n lattice
Structure of lattice depends on whether the base contains repeated variables
Knows(John,John)
Knows(x,John) Knows(x,x) Knows(John,x)
Knows(x,y)
Forward Chaining
Forward Chaining Idea:
Start with atomic sentences in the KB Apply Modus Ponens
Add new atomic sentences until no further inferences can be made
Works well for a KB consisting of Situation Response clauses when processing newly arrived data
Forward Chaining
First Order Definite Clauses Disjunctions of literals of which exactly one is positive: Example:
King(x) Greedy(x) Evil(x) King(John) Greedy(y)
First Order Definite Clauses can include variables Variables are assumed to be universally quantified
Greedy(y) means y Greedy(y) Not every KB can be converted into first definite
clauses
Example knowledge base
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 missiles, and all of its missiles were sold to it by Colonel West, who is American.
Prove that Col. West is a criminal
Example knowledge base contd.... it is a crime for an American to sell weapons to hostile nations:
American(x) Weapon(y) Sells(x,y,z) Hostile(z) Criminal(x)Nono … has some missiles, i.e., x Owns(Nono,x) Missile(x):
Owns(Nono,M1) and Missile(M1)… all of its missiles were sold to it by Colonel West
Missile(x) Owns(Nono,x) Sells(West,x,Nono)Missiles are weapons:
Missile(x) Weapon(x)An enemy of America counts as "hostile“:
Enemy(x,America) Hostile(x)West, who is American …
American(West)The country Nono, an enemy of America …Enemy(Nono,America)
Forward chaining algorithm
Forward chaining proof
Forward chaining proof
Forward chaining proof
Properties of forward chaining
Sound and complete for first-order definite clauses
Datalog = first-order definite clauses + no functions
FC terminates for Datalog in finite number of iterations
May not terminate in general if α is not entailed This is unavoidable: entailment with definite clauses is
semidecidable
Efficiency of forward chaining
Incremental forward chaining: no need to match a rule on iteration k if a premise wasn't added on iteration k-1 match each rule whose premise contains a newly added positive
literal
Matching itself can be expensive:Database indexing allows O(1) retrieval of known facts
e.g., query Missile(x) retrieves Missile(M1)
Forward chaining is widely used in deductive databases
Hard matching example
Colorable() is inferred iff the CSP has a solution CSPs include 3SAT as a special case, hence
matching is NP-hard
Diff(wa,nt) Diff(wa,sa) Diff(nt,q) Diff(nt,sa) Diff(q,nsw) Diff(q,sa) Diff(nsw,v) Diff(nsw,sa) Diff(v,sa) Colorable()
Diff(Red,Blue) Diff (Red,Green) Diff(Green,Red) Diff(Green,Blue) Diff(Blue,Red) Diff(Blue,Green)
Backward Chaining
Improves on Forward Chaining by not making irrelevant conclusions Alternatives to backward chaining:
restrict forward chaining to a relevant set of forward rules Rewrite rules so that only relevant variable bindings are
made: Use a magic set Example:
Rewrite rule: Magic(x)American(x) Weapon(x) Sells(x,y,z) Hostile(z)Criminal(x)
Add fact: Magic(West)
Backward Chaining
Idea:Given a query, find all substitutions that
satisfy the query.Algorithm:
Work on lists of goals, starting with original query Algo finds every clause in the KB that unifies with
the positive literal (head) and adds remainder (body) to list of goals
Backward chaining algorithm
SUBST(COMPOSE(θ1, θ2), p) = SUBST(θ2, SUBST(θ1, p))
Backward chaining example
Backward chaining example
Backward chaining example
Backward chaining example
Backward chaining example
Backward chaining example
Backward chaining example
Backward chaining example
Properties of backward chaining
Depth-first recursive proof search: space is linear in size of proof
Incomplete due to infinite loops fix by checking current goal against every goal on stack
Inefficient due to repeated subgoals (both success and failure) fix using caching of previous results (extra space)
Widely used for logic programming
Logic programming: Prolog Algorithm = Logic + Control Basis: backward chaining with Horn clauses + bells & whistles
Widely used in Europe, Japan (basis of 5th Generation project) Program = set of clauses:
head :- literal1, … literaln. criminal(X) :- american(X), weapon(Y), sells(X,Y,Z),
hostile(Z).
Depth-first, left-to-right backward chaining Built-in predicates for arithmetic etc., e.g., X is Y*Z+3 Built-in predicates that have side effects (e.g., input and output
predicates, assert/retract predicates) Closed-world assumption ("negation as failure")
e.g., given alive(X) :- not dead(X). alive(joe) succeeds if dead(joe) fails
Prolog
Appending two lists to produce a third:append([],Y,Y).
append([X|L],Y,[X|Z]) :-
append(L,Y,Z). query: append(A,B,[1,2]) ? answers: A=[] B=[1,2] A=[1] B=[2] A=[1,2] B=[]
Prolog Has problems with repeated states and infinite
paths Example: Path finding in graphs
path(X,Z) :- link(X,Z) path(X,Z) :- path(X,Y),link(Y,Z)
A B Cpath(a,c)
link(a,c)fail link(Y,c)path(a,Y)
{ Y/b}
link(a,b){Y/b }
Prolog Has problems with repeated states and infinite
paths Example: Path finding in graphs
path(X,Z) :- path(X,Y),link(Y,Z) path(X,Z) :- link(X,Z)
A B C
path(a,c)
path(a,Y)
fail
link(Y,b)
path(a,Y’) link(Y’,Y)
path(a,Y) link(Y,b)
Resolution
Resolution for propositional logic is a complete inference procedure Existence of complete proof procedures in Mathematics would
entail: All conjectures can be established mechanically All mathematics can be established as the logical consequence of a set
of fundamental axioms Gődel 1930: Completeness Theorem for first order logic
Any entailed sentence has a finite proof No algorithm given until J.A. Robinson’s resolution algorithm in 1965
Gődel 1931: Incompleteness Theorem: Any logical system with induction is necessarily incomplete There are sentences that are entailed, but not proof can be given
Resolution
First order logic requires sentences in CNF Conjunctive Normal Form: Each clause is a
disjunction of literals, but literals can contain variables, which are assumed to be universally quantified
Example: Convertx American(x) Weapon(y) Sells(x,y,z) Hostile(z)
Criminal(x)
American(x) Weapon(y) Sells(x,y,z) Hostile(z) Criminal(x)
Conversion to CNF
Everyone who loves all animals is loved by someone:x [y Animal(y) Loves(x,y)] [y Loves(y,x)]
1. Eliminate biconditionals and implicationsx [y Animal(y) Loves(x,y)] [y Loves(y,x)]
2. Move inwards: x p ≡ x p, x p ≡ x px [y (Animal(y) Loves(x,y))] [y Loves(y,x)] x [y Animal(y) Loves(x,y)] [y Loves(y,x)] x [y Animal(y) Loves(x,y)] [y Loves(y,x)]
Conversion to CNF
3. Standardize variables: each quantifier should use a different onex [y Animal(y) Loves(x,y)] [z Loves(z,x)]
4. Skolemize: a more general form of existential instantiation.Each existential variable is replaced by a Skolem function of the enclosing
universally quantified variables: x [Animal(F(x)) Loves(x,F(x))] Loves(G(x),x)
5. Drop universal quantifiers: [Animal(F(x)) Loves(x,F(x))] Loves(G(x),x)
6. Distribute over : [Animal(F(x)) Loves(G(x),x)] [Loves(x,F(x)) Loves(G(x),x)]
Resolution Inference Rule
Full first-order version:l1 ··· lk, m1 ··· mn
Subst(θ ,l1 ··· li-1 li+1 ··· lk m1 ··· mj-1 mj+1 ··· mn)θwhere Unify(li, mj) = θ.
The two clauses are assumed to be standardized apart so that they share no variables.
For example,Rich(x) Unhappy(x) Rich(Ken)
Unhappy(Ken)with θ = {x/Ken}
Apply resolution steps to CNF(KB α); complete for FOL
Resolution
Show KB ⊢ α by showing that KB α is unsatisfyable
Resolution Example
Everyone who loves all animals is loved by someone
Anyone who kills an animal is loved by no one. Jack loves all animals. Either Jack or Curiosity killed the cat, who is
named Tuna All dogs kill a cats Rintintin is a dog Question: Did Curiosity kill the cat?
Resolution Example
Everyone who loves all animals is loved by someone. Formulate in FOL
x [y [Animal(y) Loves(x,y)]] [z Loves(z,x)] Remove Implications
x [[y Animal(y) Loves(x,y)]] [z Loves(z,x)] x [[y Animal(y) Loves(x,y)]] [z Loves(z,x)]
Move negation inward x [y Animal(y) Loves(x,y)] [z Loves(z,x)] x [y Animal(y) Loves(x,y)] [z Loves(z,x)]
Skolemize x [ Animal(F(x)) Loves(x,F(x))] [Loves(G(x),x)]
N.B.: Argument of Skolemization function are all universally qualified variables Drop universal quantifier
[ Animal(F(x)) Loves(x,F(x))] [Loves(G(x),x)] Use distributive law (and get two clauses)
Animal(F(x)) Loves(G(x),x); Loves(x,F(x)) Loves(G(x),x)
Resolution Example
Anyone who kills an animal is loved by no one. Transfer to FOL
x [y (Animal(y) Kills(x,y)] (z Loves(z,x)
Remove Implications x [y (Animal(y) Kills(x,y)] (z Loves(z,x)
Move negations inwards x [ y Animal(y) Kills(x,y)] (z Loves(z,x))
Remove quantifiers Animal(y) Kills(x,y) Loves(z,x)
Resolution Example
Jack loves all animals.FOL form
x [Animal(x) Loves(Jack, x)]
Remove implications x [Animal(x) Loves(Jack, x)]
Remove quantifier Animal(x) Loves(Jack, x)
Resolution Example
Either Jack or Curiosity killed the cat, who is named Tuna.FOL form
Kills(Jack,Tuna) Kills(Curiosity,Tuna); Cat(Tuna)
Resolution Example
All cats are animalsFOL form
x [Cat(x) Animal(x)
Remove implications x [Cat(x) Animal(x)]
Remove quantifier Cat(x) Animal(x)
Resolution Example
All dogs kill a cats FOL form
x [Dog(x) y[Cat(y) Kills(x,y)]] Remove implications
x [Dog(x) y[Cat(y) Kills(x,y)]] Skolemize
x [Dog(x) [Cat(H(x)) Kills(x,H(x))]] Drop universal quantifiers
Dog(x) [Cat(H(x)) Kills(x,H(x))] Distribute (and obtain two clauses)
Dog(x) Cat(H(x); Dog(x) Kills(x,H(x))]
Resolution Example
Rintintin is a dogFOL form
Dog(Rintintin)
Resolution Example
Animal(F(x)) Loves(G(x),x) Loves(x,F(x)) Loves(G(x),x) Animal(y) Kills(x,y) Loves(z,x) Animal(x) Loves(Jack, x) Kills(Jack,Tuna) Kills(Curiosity,Tuna) Cat(Tuna) Cat(x) Animal(x) Dog(x) Cat(H(x) Dog(x) Kills(x,H(x))] Dog(Rintintin)
Resolution Example
Animal(F(x)) Loves(G(x),x) Loves(x,F(x)) Loves(G(x),x) Animal(y) Kills(x,y) Loves(z,x) Animal(x) Loves(Jack, x) Kills(Jack,Tuna) Kills(Curiosity,Tuna) Cat(Tuna) Cat(x) Animal(x) Dog(x) Cat(H(x) Dog(x) Kills(x,H(x))] Dog(Rintintin)
Question: Did Curiosity kill the cat?
Kills(Curiosity,Tuna)]
Resolution Example Animal(F(x)) Loves(G(x),x) Loves(x,F(x)) Loves(G(x),x) Animal(y) Kills(x,y) Loves(z,x) Animal(x) Loves(Jack, x) Kills(Jack,Tuna) Kills(Curiosity,Tuna) Cat(Tuna) Cat(x) Animal(x) Dog(x) Cat(H(x) Dog(x) Kills(x,H(x))] Dog(Rintintin) Kills(Curiosity,Tuna)]
Cat(Tuna) , Cat(x) Animal(x)
Unify(Cat(Tuna), Cat(x)) = { x/Tuna }
First line thus resolves to:
Animal(Tuna)
Resolution Example Animal(F(x)) Loves(G(x),x) Loves(x,F(x)) Loves(G(x),x) Animal(y) Kills(x,y) Loves(z,x) Animal(x) Loves(Jack, x) Kills(Jack,Tuna) Kills(Curiosity,Tuna) Cat(Tuna) Cat(x) Animal(x) Dog(x) Cat(H(x) Dog(x) Kills(x,H(x))] Dog(Rintintin) Kills(Curiosity,Tuna) Animal(Tuna)
Kills(Jack,Tuna) Kills(Curiosity,Tuna), Kills(Curiosity,Tuna)
Resolves to:
Kills(Jack,Tuna)
Resolution Example Animal(F(x)) Loves(G(x),x) Loves(x,F(x)) Loves(G(x),x) Animal(y) Kills(x,y) Loves(z,x) Animal(x) Loves(Jack, x) Kills(Jack,Tuna) Kills(Curiosity,Tuna) Cat(Tuna) Cat(x) Animal(x) Dog(x) Cat(H(x) Dog(x) Kills(x,H(x))] Dog(Rintintin) Kills(Curiosity,Tuna) Animal(Tuna) Kills(Jack,Tuna)
Animal(y) Kills(x,y) Loves(z,x), Animal(Tuna)Unify(Animal(Tuna), Animal(y)) = {y/Tuna}Resolves to:
Kills(x,Tuna) Loves(z,x),
Resolution Example Animal(F(x)) Loves(G(x),x) Loves(x,F(x)) Loves(G(x),x) Animal(y) Kills(x,y) Loves(z,x) Animal(x) Loves(Jack, x) Kills(Jack,Tuna) Kills(Curiosity,Tuna) Cat(Tuna) Cat(x) Animal(x) Dog(x) Cat(H(x) Dog(x) Kills(x,H(x))] Dog(Rintintin) Kills(Curiosity,Tuna) Animal(Tuna) Kills(Jack,Tuna) Kills(x,Tuna) Loves(z,x),
Loves(x,F(x)) Loves(G(x),x), Animal(z) Loves(Jack, z)
Unify( Loves(x,F(x)) , Loves(Jack, z)) = { x / Jack, z / F(x)}
Resolvent clause is obtained by substituting the unification ruleLoves(G(Jack),Jack) Animal(F(Jack))
Resolution Example Animal(F(x)) Loves(G(x),x) Loves(x,F(x)) Loves(G(x),x) Animal(y) Kills(x,y) Loves(z,x) Animal(x) Loves(Jack, x) Kills(Jack,Tuna) Kills(Curiosity,Tuna) Cat(Tuna) Cat(x) Animal(x) Dog(x) Cat(H(x) Dog(x) Kills(x,H(x))] Dog(Rintintin) Kills(Curiosity,Tuna) Animal(Tuna) Kills(Jack,Tuna) Kills(x,Tuna) Loves(z,x) Loves(G(Jack),Jack) Animal(F(Jack))
Animal(F(x)) Loves(G(x),x), Loves(G(Jack),Jack) Animal(F(Jack))
Unify(Animal(F(x)) , Animal(F(Jack)))= { x / Jack}
Resolvent clause is obtained by substituting the unification ruleLoves(G(Jack),Jack)
Resolution Example Animal(F(x)) Loves(G(x),x) Loves(x,F(x)) Loves(G(x),x) Animal(y) Kills(x,y) Loves(z,x) Animal(x) Loves(Jack, x) Kills(Jack,Tuna) Kills(Curiosity,Tuna) Cat(Tuna) Cat(x) Animal(x) Dog(x) Cat(H(x) Dog(x) Kills(x,H(x))] Dog(Rintintin) Kills(Curiosity,Tuna) Animal(Tuna) Kills(Jack,Tuna) Kills(x,Tuna) Loves(z,x) Loves(G(Jack),Jack) Animal(F(Jack)) Loves(G(Jack),Jack)
Kills(x,Tuna) Loves(z,x), Loves(G(Jack),Jack)
Unify( Loves(z,x), Loves(G(Jack),Jack) ) = { x / Jack, z / G(Jack)}
Resolvent clause is obtained by substituting the unification rule Loves(G(Jack),Jack)
Resolution Example Animal(F(x)) Loves(G(x),x) Loves(x,F(x)) Loves(G(x),x) Animal(y) Kills(x,y) Loves(z,x) Animal(x) Loves(Jack, x) Kills(Jack,Tuna) Kills(Curiosity,Tuna) Cat(Tuna) Cat(x) Animal(x) Dog(x) Cat(H(x) Dog(x) Kills(x,H(x))] Dog(Rintintin) Kills(Curiosity,Tuna) Animal(Tuna) Kills(Jack,Tuna) Kills(x,Tuna) Loves(z,x) Loves(G(Jack),Jack) Animal(F(Jack)) Loves(G(Jack),Jack) Loves(G(Jack),Jack)
Loves(G(Jack),Jack), Loves(G(Jack),Jack)
Resolvent clause is empty. Proof succeeded
Resolution Resolution is refutation-complete
If a set of sentences is unsatisfiable, then resolution will be able to produce a contradiction
Theorem provers Use control in order to be more efficient
Focus of most research effort Separate control from knowledge base
Example: Otter (Organized Technique for Theorem proving and Effective Research) A set of clauses known as the SoS - Set of Support
The important facts about a problem Search if focused on resolving SoS with another axiom
A set of usable axioms Background knowledge about problem field
Rewrites / demodulators Rules to transform expressions into a canonical form
Set of parameters or clauses that defines the control strategy to allow user to control search and filtering functions to eliminate useless subgoals
Theorem Prover Successes
First formal proof of Gődel’s Incompleteness Theorem (1986)
Robbins algebra (a simple set of axioms) is Boolean algebra (1996)
Software verification:Remote agent spacecraft control program
(2000)
Resolution proof: definite clauses
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