Automated reasoning and theorem proving

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Automated reasoning and theorem proving

• Introduction logic in AI

Automated reasoning Resolution Unification
Normalization
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Introduction

• Motivating example

Automated reasoning
Logic Syntax Model semantics Logica
l entailment
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The AI dream in the 60s

• Logic allows to express almost everything
  formally.

• Logic also allows to prove theorems based on
  the information given.

• Can we exploit this to build automated reasoning
  systems ??

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Underlying premises
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Example

• The following knowledge is given

2. Marcus was a Pompeian. 3. All Pompeians were
Romans. 4. Caesar was a ruler. 5. All Romans were
either loyal to Caesar or hated him. 6. Everyone
is loyal to someone. 7. People only try to
assassinate rulers to whom they are not loyal. 8.
Marcus tried to assassinate Caesar.

• Can we automatically answer the following
  questions?

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Conversion to the First Order Logic

• Representation of facts

1. Marcus was a man. man(Marcus)
2. Marcus was a Pompeian. Pompeian(Marcus)
4. Caesar was a ruler. ruler(Caesar)
8. Marcus tried to assassinate Caesar. try_assa
ssinate(Marcus, Caesar)
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Conversion tothe First Order Logic (2)

• General representation (representation of rules)

3. All Pompeians were Romans.
?x Pompeian(x) ? Roman(x)
5. All Romans were either loyal to Caesar or
hated him.
?x Roman(x) ? loyal_to(x,Caesar) ?
hates(x,Caesar)
6. Everyone is loyal to someone.
?x ?y loyal_to(x,y)
7. People only try to assassinate rulers to whom
they are not loyal.
?x?y person(x) ? ruler(y) ? try_assassinate(x,y)
? loyal_to(x,y)
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The theorem ?
Was Marcus loyal to Caesar?
loyal_to(Marcus,Caesar)
Did Marcus hate Caesar?
hates(Marcus,Caesar)
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A proof using backward-reasoning
problem-reduction
loyal_to(Marcus,Caesar)
Modus ponens
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Problems1) knowledge representation

• Natural language is imprecise / ambiguous
• see People only try

• Obvious information is easily forgotten.
• see man lt-gt person

• Some information is more difficult to represent
  in logic.
• Vb. perhaps , possibly, probably,
  the chance of is 45,

• Logic is inconvenient from a software
  engineering perspective.
• too fine-grained (like an assembly language)

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Problems2) Problem solving

• All trade-offs that we had with search methods
  based on states space representation
• backward/forward, tree/graph, OR-tree/AND-OR, con
  trol aspects, ...

• What deduction rules are needed in general?
• Example prove hates(Marcus,Caesar)

• How do we handle ?x and ?y ?

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Problems2) Problem solving (2)

• How to compute substitutions in the general case ?

• Which theorem do we try to prove?
• Ex. loyal_to(Marcus,Caesar) or
  loyal_to(Marcus,Caesar)

• How to handle equality of objects?
• Problem combinatorial explosion of the derived
  equalities (reflexivity, symmetry, transitivity,
  )

• How to guarantee correctness/completeness?

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The formal model semantics of Logic

• The meaning of Logical entailment

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Propositional logic
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Basic concepts

• In propositional logic

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Semantics (meaning)

• In general (for all knowledge representation
  formalisms)

• 2 approaches to define semantics

• Describe the meaning by means of a natural
  language
• Exs. (propositional logic)
• ? implies
• not true that
• ? or
• p ? q p if and only if q
• p ? r not p and r
• every symbol and every well-formed formula gets
  meaning through the associated natural language

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Semantics (2)

• Describe the meaning by converting to an
  associated mathematical object
• In propositional logic
• the set of all propositional symbols that are
  logically entailed by the given formulas

Logically entailed
p
q
r
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But how to define logical entailment ?

• NOT as
• Everything that we can derive from the formulas

• SINCE
• At this moment we do not know yet
• what is a complete set of the derivation rules
• This is exactly what Automated Reasoning aims
  to find out!

• BUT by
• Interpretations
• Models

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Interpretation

• a function that assigns a truth value to each
  atomic formula

Truth table
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Model

• Given a set of formulas S a model is an
  interpretation that makes all formulas in S
  true

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Logical entailment

• Given a set of formulas S and a formula F F
  is logically entailed by S ( S F ), if all
  models of S also make F true.

F r ? p
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Predicate logic
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p(Yvonne) p(x) ? q(x) ?z p(z) ?x ?y p(x) ? q(y)
? p(f(Yvonne, Yvette)
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Formally

• An alphabet consists of variables, constants,
  function symbols, predicate symbols (all
  user-defined) and of connectors, punctuations en
  quantifiers.

• Terms are either
• variables,
• constants,
• function symbols provided with as many terms as
  arguments, as the function symbol expects.

• Well-formed formulas are constructed from
  predicate symbols, provided with terms as
  arguments, and from connectors, quantifiers and
  punctuation according to the rules of the
  connectors.

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Example

• Alphabet

0, x,y, s, odd,even, Con, Pun, Quan

• Terms

0, s(0), s(s(0)), s(s(s(0))), x, s(x),
s(s(x)), s(s(s(x))), y, s(y), s(s(y)),
s(s(s(y))),

• Well-formed formulas

odd(0), even(s(0)), odd(x), odd(s(y)),
odd(x) ? even(s(s(x))), ?x ( odd(x) ?
even(s(x)) ), odd(y) ? ?x (even( s(x))), ...
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Interpretation

• a set D (the domain), and

• a function that maps constants to D, and

• a function that maps function symbols to
  functions D -gt D, and

• a function that maps predicate symbols to
  predicates D -gt Booleans.

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Assigning truth values

• 1. To ground atomic formulas

• form p(f(a), g(a,b))

I( odd( s ( s( 0 ) ) ) ) ) ? I (odd) (
I(s) ( I(s) ( I(0) ) ) ) even ( succ ( succ
( 0 ) ) ) even ( succ ( 1 ) )
even ( 2 ) true
Example
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Assigning truth values (2)

• 2. For closed well-formed formulas
• ( no non-quantified variables)

• ?x F(x) is true if
• for all d ?D I( F(d) ) true
• ? x F(x) is true if
• there exists d ?D such that I( F(d) ) true
• further use the truth tables.

I(?x odd( s ( x ) ) ? odd(x) ) ? true if
for all d in N I (odd( s (d) ) ? odd(d) )
true
Example
even ( succ(d) ) ? even (d) Assume d
0, then false ? true Truth tables false !
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Semantics / Logical entailment

• Exactly as in propositional logic !

• Additional inconsistency

• Given a set of formulas S S is inconsistent
  if S has no models.

Example S p(a), p(a)
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Marcus example
A
F ?
ruler
person
P
man
try_assassinate
Pompeian
Roman
loyal_to
hates
Intended interpretation
Is a model IF ALL FORMULAS ARE CORRECT
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Marcus example
I(man) I(person) I(Roman) natural number
I(try_assassinate) gt
I(Pompeian) even number
I(loyal_to) divides
I(ruler) prime number
I(hates) doesnt divide
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Model ??
YES !
1. Marcus was a man. 4 is a natural number
2. Marcus was Pompeian. 4 is an even number
4. Caesar was a ruler. 3 is a prime number
8. Marcus tried to assassinate Caesar. 4 gt 3
3. All Pompeians were Romans.
Even numbers are naturals.
5. All Romans were either loyal to Caesar or
hated him.
A number either divides 3 or doesnt divide 3.
6. Everybody is loyal to somebody.
Each number is a divisor of some number.
7. People try to assassinate only those rulers to
whom they are not loyal.
A natural number that is greater than a prime
number doesnt divide the prime number.
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Logic is all form, no content
Only the underlying structure of a set of logical
formulas is important for the conclusions! (up
to the names isomorphism)
But from the knowledge representation perspective
also the contents is important.
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Relation with respect to other courses
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Logic as a foundation for AI
A much deeper and more formal study of Logic for
Knowledge Representation and Reasoning !
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Programming Languages and Programming
Methodologies
Logic-based programming languages
(Prolog/CLP)
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Selected Topics in Logic Programming
Formal studies of semantics and formal methods
(analysis, termination) of logic-based
programming languages
(also beyond Prolog)
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Fundamentals of Artificial Intelligence
Mostly in the exercises
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Methodology for knowledge representation?

• Very complicated. Not many simple guidelines.

• Choose an alphabet that allows us to represent
  all objects and all relations from the problem
  domain
• What are the basic objects, functions and
  relations in your problem domain ?
• Ontology represent only the RELEVANT
  information
• Choose constants, function and predicate symbols
  to represent them.
• Translate every sentence in a natural language
  in one or more corresponding logical formulas.