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
• Inference rules for quantifiers
• In KB, we have

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• 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)

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• 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)

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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

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• 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)

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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

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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)

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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

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Generalized Modus Ponens
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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

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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

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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

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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

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• 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

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Example

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

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• For forward chaining, we will have two iterations

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• 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

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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)

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Backward-chaining algorithm

• This is better to illustrate with a proof tree

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• 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.

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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

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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.,

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• Conversion to CNF
• 6 steps

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• 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

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

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Resolution inference rule
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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)

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Example proof

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

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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
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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

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• 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

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• 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

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

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

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• 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