Chapter 8 Inference in first-order logic - PowerPoint PPT Presentation

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Chapter 8 Inference in first-order logic

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Chapter 8 Inference in first-order logic Inference in FOL, removing the quantifiers i.e., converting KB to PL then use Propositional inference which is easy to do – PowerPoint PPT presentation

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Title: Chapter 8 Inference in first-order logic


1
Chapter 8 Inference in first-order logic
  • Inference in FOL,
  • removing the quantifiers
  • i.e., converting KB to PL
  • then use Propositional inference
  • which is easy to do
  • Inference rules for quantifiers
  • In KB, we have

2
  • A rule called Universal Instantiation (UI)
  • substituting a ground term for the variable
  • SUBST(? , ?) denotes
  • the result of applying the substitution ? to the
    sentence ?
  • Here the examples are
  • x / John, x / Richard, x / Father(John)

3
  • Existential Instantiation
  • For any sentence ?, variable v, and constant k
    that does not appear elsewhere in the KB
  • ?x Crown(x) ? OnHead(x, John)
  • If C1 does not appear elsewhere in the KB
  • then we can infer
  • Crown(C1) ? OnHead(C1, John)

4
Reduction to Propositional Inference
  • Main idea
  • for existential quantifiers
  • find a ground term to replace the variable
  • remove the quantifier
  • add this new sentence to the KB
  • for universal quantifiers
  • find all possible ground terms to replace the
    variable
  • add the set of new sentences to the KB

5
  • apply UI to the first sentence
  • from the vocabulary of the KB
  • x / John, x / Richard two objects
  • we then obtain
  • view other facts as propositional variables
  • use inference to induce Evil(John)

6
Propositionalization
  • The previous techniques
  • applying this technique to
  • every quantified sentence in the KB
  • we can obtain a KB consisting of propositional
    sentences only
  • however, this technique is very inefficient
  • in inference
  • it generates other useless sentences

7
Unification and Lifting
  • Unification
  • a substitution ? such that applying on two
    sentences will make them look the same
  • e.g., ? x / John is a unification
  • applying on
  • it becomes
  • and we can conclude the implication
  • using King(John) and Greedy(John)

8
Generalized Modus Ponens (GMP)
  • The process capturing the previous steps
  • A generalization of the Modus Ponens
  • also called the lifted version of M.P.
  • For atomic sentences pi , pi', and q,
  • there is a substitution ? such that
  • SUBST(?, pi) SUBST(?, pi'), for all i

9
Generalized Modus Ponens
10
Unification
  • UNIFY, a routine which
  • takes two atomic sentences p and q
  • return a substitution ?
  • that would make p and q look the same
  • it returns fail if no such substitution ? exists
  • Formally, UNIFY(p, q)?
  • where SUBST(?, p) SUBST(?, q)
  • ? is called the unifier of the two sentences

11
Standardizing apart
  • UNIFY failed on the last sentence
  • in finding a unifier
  • reason?
  • two sentences use the same variable name
  • even they are having different meanings
  • so, assign them with different names
  • internally in the procedure of UNIFY
  • standardizing apart

12
MGU
  • Most Generalized Unifier
  • there may be many unifiers for two sentences
  • which one is the best?
  • the one with less constraints
  • e.g., UNIFY(Knows(John, x), Knows(y, z))
  • one unifier y/John, x/John, z/John
  • another y/John, z/x the best
  • even if z and x are not yet found

13
Forward and backward chaining
  • Forward chaining
  • start with the sentences in KB
  • generate new conclusions that
  • in turn allow more inferences to be made
  • usually used
  • when a new fact is added to the KB
  • and we want to generate its consequences

14
  • First-order Definite clauses
  • the normal form of sentences for FC
  • can contain variables
  • either atomic or
  • an implication whose
  • antecedent is a conjunction of positive literals
  • consequent a single positive literal
  • Not every KB can be converted into a set of
    definite clauses, but many can
  • why? the restriction on single-positive-literal

15
Example
  • We will prove that West is a Criminal
  • First step
  • translate these facts as first-order definite
    clauses
  • next figure shows the details

16
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17
  • For forward chaining, we will have two iterations

18
  • The above is the proof tree generated
  • No new inferences can be made at this point using
    the current KB
  • such a KB is called a fixed point of the
    inference process

19
Backward chaining
  • start with something we want to prove
  • (goal/query)
  • find implication sentences
  • that would allow to conclude
  • attempt to establish their premises in turn
  • normally used
  • when there is a goal to prove (query)

20
Backward-chaining algorithm
  • This is better to illustrate with a proof tree

21
  • One remark
  • backward chaining algorithm uses
  • composition of substitutions
  • SUBST(COMPOSE(?1, ?2), p)
  • SUBST(?2, SUBST(?1, p))
  • its used because
  • different unification are found for different
    goals
  • we have to combine them.

22
Resolution
  • Modus Ponens rule
  • can only allow us to derive atomic conclusions
  • A, AgtB B
  • However, it is more natural
  • to allow us derive new implication
  • A gt B, B gt C AgtC, the transitivity
  • a more powerful tool resolution rule

23
CNF for FOL
  • Conjunctive Normal Form
  • a conjunction (AND) of clauses
  • each of them is a disjunction (OR) of literals
  • the literals can contain variables
  • e.g.,

24
  • Conversion to CNF
  • 6 steps

25
  • Skolemize
  • process of removing ?
  • i.e., translate ?x P(x) into P(A), A is a new
    constant
  • If we apply this rule in our sample, we have
  • which is completely wrong
  • since A is a certain animal
  • To overcome it, we use a function to represent
    any animal, these functions Skolem functions

26
  • Drop universal quantifiers
  • all variables now are assumed to be universally
    quantified
  • Fortunately, all the above steps can be automated

27
Resolution inference rule
28
Example proof
  • Resolution proves that KB by
  • proving KB ? ?a unsatisfiable, i.e., empty clause
  • First convert the sentences into CNF
  • next figure, empty clause
  • so, we include the negated goal ? Criminal(West)

29
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30
Example proof
  • This example involves
  • skolemization
  • non-definite clauses
  • hence making inference more complex
  • Informal description

31
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32
The following answers Did Curiosity kill the
cat? First assume Curiosity didnt kill the
cat. Add it into KB
Empty clause, so the assumption is false
33
Resolution strategies
  • Resolution is effective
  • but inefficient
  • because it is like forward chaining
  • the reasoning is randomly tried
  • There are four general guidelines in applying
    resolution

34
  • Unit preference
  • When using resolution on two sentences
  • one of the sentences must be a unit clause
  • (P, Q, R, etc.)
  • The idea is to produce a shorter sentence
  • e.g., P ? Q gt R and P
  • will produce Q gt R
  • hence reduce the complexity of the clauses

35
  • Set of support
  • Identifying a subset of sentences from the KB
  • Every resolution combines a sentence
  • from the subset
  • and another sentence from the KB
  • The resolvent (conclusion) of the resolution
  • is added to the subset
  • and continue the resolution process
  • How to choose this set?
  • a common approach the negated query
  • to prove the query, assume negative
  • and prove the contradiction
  • advantage goal-directed

36
  • Input resolution
  • Every resolution combines
  • one of the input sentences (facts)
  • from the query
  • or the KB
  • with some other sentence
  • Next fig

37
  • For each resolution,
  • at least one of the sentences from the query or KB

38
  • Subsumption (inclusion, ??)
  • eliminates all sentences
  • that are subsumed by (i.e., more specific than)
    an existing sentence in the KB
  • If P(x) is in KB, x means all arguments
  • then we dont need to store the specific
    instances of P(x) P(A), P(B), P(C) ,
  • Subsumption helps keep the KB small
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