COSC 4350 and 5350 Artificial Intelligence

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COSC 4350 and 5350 Artificial Intelligence

• Propositional Logic and Resolution (Part I)
• Knowledge Representation, Propositional Logic,
  and Inference
• Dr. Lappoon R. Tang

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What is logic?
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Overview

• History of Logic
• Knowledge Representation What is it? Why?
• Using logic for representing knowledge
• Propositional logic
• Syntax
• Semantic
• Rules of inference

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Readings

• Section 7.1 (skim thru)
• Section 7.2 (skim thru)
• Section 7.3
• Section 7.4
• Skim thru equivalence, validity, and
  satisfiability

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What is Logic?

• The formal mathematical study of the methods,
  structure, and validity of mathematical deduction
  and proof MathWorld
• Study of deduction and proof systems
• A formalism that captures rational thinking
• A formalism in which truths are derived through a
  consistent standardized procedure deduction
• Rational thinking a kind of thinking that some
  of us are naturally capable of
• Presumably ?
• All of us are taught to reason rationally in our
  education
• If only everyone treats education seriously ?

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Brief history of Logic

• 1st age of logic (500 B.C. to 19th Century)
  Symbolic Logic
• Developed by Aristotle (384 322 B.C.)
• Used by the Sophist (a Greek philosopher who
  speculated on a wide range of subjects back in
  500 B.C.)

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Brief history of Logic (Contd)

• 2nd age of logic (Mid to late 19th Century)
  Algebraic Logic or Boolean Algebra
• Developed by George Boole in 1847
• Attempted to formulate logic in terms of algebra
  rules of inference were modeled after various
  laws for manipulating algebraic expressions (e.g.
  commutative law of )

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Brief history of Logic (Contd)

• 3rd age of logic (late 19th to mid 20th century)
  Mathematical Logic
• Frege proposed logic to be used as a language for
  mathematics in 1879
• Mathematical statements should be expressed in
  logic (instead of in imprecise and ambiguous
  languages like English)
• Helps to get rid of nasty paradoxes constructed
  because of flawed reasoning
• Strengthen the rigor of mathematical proofs

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Brief history of Logic (Contd)

• 4th age of logic (Mid 20th to present) Logic for
  Computer Sciences
• Application of logic in solving CS problems
• Logic circuits (design of CPU)
• Logic programming
• Formal verification for program correctness
• Development of more sophisticated forms of logic
• Non-monotonic logic (learning new things can
  reduce what is known)
• Modal logic (a kind of logic that handles
  concepts like possibility, impossibility,
  necessity)

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What is Knowledge Representation?

• Incomplete Definition The subfield of AI
  concerned with designing and using systems for
  storing knowledge facts and rules about some
  subject (Hyper-dictionary)
• Problem 1 How about utilizing the knowledge for
  drawing inference?
• Problem 2 It assumes that knowledge can be
  represented as facts and rules only
• Real Definition Representation of knowledge in a
  formalism that can be manipulated by a machine
• Idea We dont add to that body of knowledge,
  rather, we just translate it for the machines

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Why use Logic for Knowledge Representation?

• Logic is not the only feasible language for
  knowledge representation (i.e. there are others)
• It is commonly chosen
• It has well defined semantics
• Historical background
• Problem computational complexity with drawing
  inference, especially in first-order logic (FOL)
• A solution Horn clause logic (first-order or
  not)
• Not as powerful as FOL or propositional logic
• Reasonably efficient
• Most popular choice as a KR language

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Propositional Logic The Syntax

• Idea Syntax concerns how a logical sentence
  (aka well-founded
• formula WFF) is constructed

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Propositional Logic The Semantics

• Idea Semantics concerns how one can determine
  the truth value of
• a WFF (i.e how do I know when a WFF is true?)

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Propositional Logic The Rules of Inference

• Q Is there a systematic way by which one
• can prove that a certain statement is true?
• A Yes, one can prove that a statement is
• true by something called rules of inference

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Propositional Logic The Rules of Inference
(Modus Ponens)

• A valid rule of inference

If A then B (This statement is TRUE) A
(And, this statement is TRUE) Therefore, B
(This statement is also TRUE)
If it is raining, then the ground is wet It is
raining
. Therefore, the ground is wet
If it is raining, then the ground is wet The
ground is wet
. Therefore, ??
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Propositional Logic The Rules of Inference
(Modus Tollens)

• Another valid rule of inference (kind of like the
  inverse of Modus Ponens)

If A then B not(B) . Therefore,
not(A)
If it is raining, then the ground is wet The
ground is dry
. Therefore, it is not raining
If it is raining, then the ground is wet It is
not raining
. Therefore, ??
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Propositional Logic The Rules of Inference
(Examples of Invalid Inference)
Some As are Bs Some Bs are Cs Some As are Cs
If A then B If not(A) then not(B)
?
Some women are vegetarians Some vegetarians are
men Some women are men
If you eat an ice-cream, you can taste something
sweet If you dont eat an ice-cream, then you
cannot taste something sweet
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Propositional Logic The Rules of Inference

• Q We have seen two rules of inference that
• resemble logical reasoning performed by a
• human, is there a rule of inference more
• suitable for a machine to use?

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Propositional Logic The Rules of Inference
(Resolution Proof by Refutation)

• This is the rule of inference used by the machine
  the one most unlike those used by the humans ?
• Idea To prove X, assume X is NOT true (i.e.
  not(X) is true), and show that it leads to a
  contradiction
• This kind of proof is based on a rule of
  inference called resolution Sometime, it is
  much easier to construct a successful proof this
  way ?
• Example prove that there is no greatest integer
• Proof
• Assume the greatest integer is N
• Since N1 is also an integer
• But N1 gt N
• !! (Contradiction)

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Conclusion

• Logic is a formalism for deduction
• Logic has long history of development and has
  become a branch in Mathematics
• In AI, we can use logic as a language for
  knowledge representation
• The two rules of inference Modus Ponens and
  Modus Tollens resemble human reasoning
• Machine usually uses a different kind of rule of
  inference called resolution