Mechanical Theorem Proving____ The Intellectual Excitement of Computer Science

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Mechanical Theorem Proving____The Intellectual
Excitement of Computer Science
Group Members Elita Cheung Lily Irani Paul
Tenney
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Introduction

• Mechanical theorem proving is an important
  subject in artificial intelligence
• Even though Turing showed that there is no
  general decision procedure to check the validity
  of formulas of the first-order logic, there are
  proof procedures which can verify that a formula
  is valid if indeed it is valid...

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Our Research Journey

• Journals about automated theorem proving
• Difficult and technical material required
  background we lacked
• Talked with professors, read about basic logic

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Overview of Automated Theorem Proving

• Philosophical issues regarding a mechanical
  theorem prover
• Theory and history of the field -- lesson in
  logic
• Applications of automated theorem provers

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Quick History and Theory

• Principles of Automated Theorem Proving heavily
  based on symbolic logic
• Learning the basic vocabulary and concepts was
  essential to understanding those principles
• The history of this field can be easier
  understood along with theories
• Quick lesson in symbolic logic J

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Different sorts of logic...

• Higher Order
• First Order
• Propositional

More interactive
More Expressive
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Propositional Logic

• A proposition is a declarative sentence that is
  either true or false (it cannot be both).
• Examples of propositions Stuff at Stanford
  Shopping Mall is expensive", Elita is a bargain
  hunter", Elita is shop-aholic at Stanford mall".

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Propositional Logic

• B º Stuff at Stanford Shopping Mall is expensive
  C º Elita is a bargain hunter D º Elita is a
  shop-aholic at Stanford Mall
• Symbols, such as B, C, D, that are used to denote
  propositions are called atoms

Simple symbols...
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Propositional Logic

• Example The sentence "If stuff at Stanford
  Shopping mall is expensive and Elita is a bargain
  hunter, then Elita is not a shop-aholic at
  Stanford Mall" can be represented by
• (( B Ù C) (ØD))
• As we see, this compound proposition can
  represent a complicated idea that we deal with in
  everyday life.

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Propositional Logic

• Truth Table

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Propositional Logic

• The assignment of truth values T,F to G, H is
  one of four interpretations of formula F º (G Ù
  H)
• Equivalent formulas
• Example Suppose that bike accidents increase if
  there are more freshmen on campus. Also, suppose
  that students will start building their own
  impact airbags for their bikes when bike
  accidents increase. Assume that there are more
  freshmen on campus. Show that you can conclude
  that students will starting building their own
  airbags.

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Propositional Logic Example...

• The four following statements correspond to this
  example
• 1. If there are more freshmen on campus, the bike
  accidents increase2. If bike accidents increase,
  students start building bike airbags3. More
  freshmen on campus4. Students will start
  building bike airbags

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First Order Logic

• First order logic is a more expressive logic than
  propositional logic. For example, propositional
  logic cannot denote the following
• P Every man is mortalQ Confucius is a manR
  Confucius is mortal

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First Order Logic

• First order logic has three more logical notions
  than propositional logic
• terms, predicates, and quantifiers
• Most of mathematical and everyday language can be
  symbolized by the first-order logic.

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First Order Logic - New Terms

• Predicate
• Quantifier
• Interpretation -- different from propositional
• "An interpretation of a formula F in the
  first-order logic consists of a nonempty domain
  D, and an assignment of 'values' to each
  constant, function symbol, and predicate symbol
  occurring in F as follows
• To each constant, we assign an element in D.
• To each n-place function symbol, we assign a
  mapping from Dn to D.
• To each n-place predicate symbol, we assign a
  mapping from Dn to T, F."

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First Order Logic - New Terms

• Satisfiable- A formula P is satisfiable
  (consistent) if and only if there exists an
  interpretation I such that P has a truth value of
  True in I.
• Unsatisfiable

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Herbrands theorem and a little history

• Leibniz (1646-1716) tried to prove validity of
  formula
• Turing and Church (1936)
• Herbrands contribution
• Robinsons Resolution

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Resolution

• Herbrands procedures problem amount of time
  needed to implement increase exponentially (too
  many interpretations to generate!)
• Resolution decreases the number of interpretations

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Resolution

• The basic idea of the resolution principle is to
  check rather any set S of clauses contains the
  empty clause . If S contains , then S is
  unsatisfiable. If S does not contain , then
  check to see if can be derived from S. If it
  can, then it is also unsatisfiable.
• Example in propositional logic
• Example in first order logic

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Propositional Resolution

• For propositional logic, the principle can be
  roughly described as the following combine the
  literal that are complementary to each other so
  that they cancel out (e.g. P and P are
  complementary).
• Example in propositional logic

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First Order Resolution

• substitution and unification
• Example in first order logic

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First Order Resolution

• S ØT(x,y,u,v) v P(x,y,u,v), ØP(x,y,u,v) v
  E(x,y,v,u,v,y), T(a,b,c,d), ØE(a,b,d,c,d,b)
• 1. ØT(x,y,u,v) v P(x,y,u,v)
• 2. ØP(x,y,u,v) v E(x,y,v,u,v,y)
• 3. T(a,b,c,d)
• 4. ØE(a,b,d,c,d,b)
• 5. P(a,b,c,d)
• 6.T(a,b,c,d)
• 7.

a resolvent of 2 and 4 a resolvent of a and 5 a
resolvent of 3 and 6
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Applied Theory

• First order specifications
• Boyer and Moores Induction

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Intel Pentium Chip Specification - IEEE level 74

• when rounding towards negative infinity, the
  result shall be the formats value ... closest to
  and no greater than the infinitely precise result

Informal
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Intel Pentium Chip Specification - IEEE level 74

• round(toNegInf, R, V)
• (R
• R result, V value to be rounded,
• ulp smallest representable increment

Formal (First Order)
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Induction Algorithm
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Applications

• Mathematical proof checking
• The QED Project
• Computer chip verifications
• Software verification

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Mathematical Proof Checking

• Automated theorem provers do not automate math
• Debugs proofs
• Hard to use many proof checkers

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The QED Project

• Effort of scientists from many laboratories and
  institutions

The development of mathematics towards a
greater appreciation has led... to the
formalization of large tracts of it, so that one
can prove any theorem using nothing but a few
mechanical rules. -K.Gödel

• Will represent mathematical knowledge, technique
• Based on a few pages of math
• Still in early stages

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The QED Project- Hoped Benefits

• Reduce mathematical noise pollution.
• Speed publication of papers by taking focus off
  of proof checking. Referees can focus on
  relevance.
• Cultural monument to mathematics.

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Chip Verification

• Formal vs. testbench
• Comparison verification
• NP-Complete
• IBM, Intel, AMD successes

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Software Verification

• Hardware is more economically viable
• More design effort put into software
• Software verification is viable
• Especially useful for critical applications
  safety, e-commerce, military

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Software Verification Paradox

• What will verify the verification program?
• Pragmatism does not demand ideal accuracy
• Significant improvement enough

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More Information

• Our website
• demonstrations of theorem proving tools online
• additional research

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Credits

• Thank you to Professor David Dill for information
  and support through e-mail and in person.