Logic and Computability

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Logic and Computability

• 1 Intro Propositional Logic

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Administrative

• Class Tue 12-2 c.t.
• Material
• Huth and Ryan, Logic in Computer Science,
  Cambridge University Press, 2004. (Buy the book)
• Slides will be online after class
• Lectures
• Content
• Propositional Logic
• Predicate Logic, computability
• Temporal logic

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Uebung
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Why Logic?
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Why Logic?

• Rules for discussions
• Understand thinking
• Understand reasoning
• Learn how to write proofs
• Describe questions unambiguously
• Learn how to automate proofs
• Understand what is and is not provable
• Use a prover to solve a question
• We will be precise, even pedantic in our proof
  rules why?

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Types of Logic

• Three important types
• Propositional logic
• If it rains, I get wet. I am not wet. Therefore,
  it does not rain.
• Simple statements about facts (propositions)
• Can be proven true or false automatically
• Very powerful tools are available (SAT solvers)
• Predicate logic
• For every lock there is a key, but there is no
  key for every lock.
• More complicated statements involving quantifiers
• Cannot in general be proven correct or incorrect
• Tools (theorem provers ) help you write proofs
• Temporal logic
• Whenever it rains, eventually the sun will shine
• Statements involving time

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Use of Logic

• Propositional logic has strong ties to computer
  circuits (also called logic circuits)
• Check equivalence of two circuits
• Automatic Test Pattern Generation
• Check if circuit adheres to spec
• But also search questions over finite domains

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Use of Logic

• Predicate logic
• Very general
• Subsumes many other restricted logics in common
  use
• Program correctness
• Quantifiers help solve game-style questions.
• Not necessarily finite domain
• Temporal Logic
• Verification of reactive systems
• Whenever a request arrives, eventually the
  request is granted
• The request is never granted to two requestors at
  the same time

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Classical Questions in Logic

• Does one formula entail another?
• If x then y. x is true. Therefore, y is true
• If x then y. y is false. Therefore, x is false
• Is this formula always true? (validity)
• ((x implies y) and x) implies y
• Can this formula ever be true? (satisfiability)
• (x implies y) and x and (not y)
• (x implies y) and (not x) and y
• Is the formula true in this case? (model
  checking)
• x is true, y is true. Can we conclude (x and y)?
• (These questions are closely related)
• Can I write a program that answers these
  questions automatically?

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Contents of Class

• Propositional Logic
• The language and its meaning, proof by search for
  a model
• Deductive proofs
• Connection between model-based and deductive
  proofs
• Questions is this formula always true? Can it be
  true at all?
• Predicate Logic
• The language and its meaning
• Deductive proofs
• Why we cannot prove everything
• Questions is this formula always true? Can it be
  true at all?
• Temporal Logic
• The language and its meaning
• Model checking
• Question Does a given model fulfill a formula?

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Today

• The syntax of propositional logic
• The semantics of propositional logic
• Proving validity and satisfiability
• Complexity
• An example
• (HR 1.4)

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Syntax of Propositional Logic

• We have
• proposition symbols p, q, r. They stand for
  statements that we do not analyze any further.
  it rains, I am wet, I have an umbrella.
• We have Boolean connectives
• ? (and, conjunction) p ? q, it rains and I
  have an umbrella
• ? (or, disjunction) p ? q, it rains or I
  have an umbrella (inclusive disjunction both may
  be true)
• ? (not, negation) ?p it does not rain
• ? (implication) p ? q if it rains, I am wet
• In BNF ? p ? ? ? ? ? ? ?? ? ? ?

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Terminology

• An environment gives a value for each
  propositional symbols.
• Example (ptrue, qfalse, rfalse) it rains, I
  am not wet, I do not have an umbrella
• We will write T for true and F for false. (Also
  usual 1,0)
• A model of a formula is an environment that makes
  the model true.
• Example (ptrue, qfalse, rfalse) is a model of
  p ? ?q
• We write (ptrue, qfalse, rfalse) p ? ?q
• A formula is satisfiable if it has a model.
• p ? ?q is satisfiable
• p ? ?p is not satisfiable
• A formula is valid if all its environments are
  models
• (p ? q) ? q is valid

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Semantic of Propositional Logic Truth Tables

• Compositional if you know whether the parts are
  true, you know whether the whole is true
• Perhaps ? is not what you expect.
• It does not rain. Therefore, If it rains, I am
  wet.

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Parsing a Formula

• What is the meaning of (((?p) ? q) ? (p ? (q ?
  (?r))))?
• It is a tree, represented by
• ?
• ?
• ?
• p
• q
• ?
• p
• ?
• q
• ?
• r

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Semantics of Propositional Logic

• Take the model (p T, qF, r F). What is the
  value of (((?p) ? q) ? (p ? (q ? (?r))))?
• ? 1
• ? 0
• ? 0
• p 1
• q 0
• ? 0
• p 1
• ? 0
• q 0
• ? 1
• R 0

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Semantics of Propoositional Logic
Is p ? (q ? ? r) valid? Satisfiable?
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A Proof rule

• Suppose ? has variables p1,, pk. How doe we
  check satisfiability of ??
• How about validity?
• What is the complexity?

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A Proof rule

• For each environment e
• If e ? then ? is satisfiable
• If not (e ?), then print not valid
• Complexity of solution there are 2k possible
  environments. Checking one environment takes
  time linear in the size of the formula. Thus
  exponential time.
• In fact, satisfiability is NP complete
• Checking an answer can be done in polynomial time
• Any polynomial time Turing machine can be encoded
  as a propositional-logic formula of polynomial
  size
• What is the complexity of checking validity?

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Example Magic Square

• Every element is 1,2, , or 9
• Each row, each column, and each diagonal sums to
  15
• Can I present this requirement as a propositional
  logic formula?

? 15
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First Steps
15 159 168 249 258 267
348 357 456

• Introduce variables x11x19, x21x29,,x19x99
• xij is supposed to mean xi j
• Let r1 (x11?x25?x39) ? (x11?x26?x38) ?
  (x11?x28?x36) ? row 1 sums to 15
• Similar for r2, r3, c1, c2, c3, d1, d2 rows,
  columns, and diagonals
• A position holds exactly one number
• Let vij be an abbreviation for xij ? ?k ?j ? xik
• Let vi ?j vij ? ?j xij exactly one number for
  square i
• Let v ?i vi exactly one number in every
  position
• Let ui ?j ?k ?j ? (xji ? xki) number i occurs
  once in square
• Let u ?i ui Each number occurs once in square
• Your formula is
• v ? u ? r1 ? r2 ? r3 ? c1 ? c2 ? c3 ? d1 ? d2

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Next

• Write a script to generate the formula
• Download limboole and a SAT solver like Precosat
  from Armin Bieres home page
• Run the formula through precosat to get an answer
• Then, try the same for Sudoku