Brief%20Introduction%20to%20Logic - PowerPoint PPT Presentation

View by Category
About This Presentation
Title:

Brief%20Introduction%20to%20Logic

Description:

p,q Form (p o q) Form. Formulas (3) Examples: (A (B C)) (A (B C) ... {A,B} |= A B. Iff {A,B} |= A and {A,B} |= B. Iff A {A,B} and B {A,B} Electrical Engineer's view ... – PowerPoint PPT presentation

Number of Views:34
Avg rating:3.0/5.0
Slides: 45
Provided by: cisU
Learn more at: http://www.cis.upenn.edu
Category:

less

Write a Comment
User Comments (0)
Transcript and Presenter's Notes

Title: Brief%20Introduction%20to%20Logic


1
Brief Introduction to Logic
2
Outline
  • Historical View
  • Propositional Logic Syntax
  • Propositional Logic Semantics
  • Satisfiability
  • Natural Deduction Proofs.

3
Historical view
  • Philosophical Logic
  • 500 BC to 19th Century
  • Symbolic Logic
  • Mid to late 19th Century
  • Mathematical Logic
  • Late 19th to mid 20th Century
  • Logic in Computer Science

4
Philosophical Logic
  • 500 B.C 19th Century
  • Logic dealt with arguments in the natural
    language used by humans.
  • Example
  • All men are motal.
  • Socrates is a man
  • Therefore, Socrates is mortal.

5
Philosophical Logic
  • Natural language is very ambiguous.
  • Eric does not believe that Mary can pass any
    test.
  • I only borrowed your car.
  • Tom hates Jim and he likes Mary.
  • It led to many paradoxes.
  • This sentence is a lie. (The Liars Paradox)

6
The Sophists Paradox
  • A Sophist is sued for his tuition by the school
    that educated him. He argued that he must win,
    since, if he loses, the school didnt educated
    him well enough, and doesnt deserve the money.
    The school argue that he must loss, since, if he
    win, he was educated well enough therefore should
    pay for it.

7
Symbolic Logic
  • Mid to late 19th Century.
  • Attempted to formulate logic in terms of a
    mathematical language
  • Rules of inference were modeled after various
    laws for manipulating algebraic expressions.

8
Mathematical Logic
  • Late 19th to mid 20th Century
  • Frege proposed logic as a language for
    mathematics in 1879.
  • With the rigor of this new foundation, Cantor was
    able to analyze the notion of infinity in ways
    that were previously impossible. (2N is strictly
    larger than N)
  • Russells Paradox
  • T S S ? S

9
Logic in Computer Science
  • In computer science, we design and study systems
    through the use of formal languages that can
    themselves be interpreted by a formal system.
  • Boolean circuits
  • Programming languages
  • Design Validation and verification
  • AI, Security. Etc.

10
Logics in Computer Science
  • Propositional Logic
  • First Order Logic
  • Higher Order Logic
  • Theory of Construction
  • Real-time Logic, Temporal Logic
  • Process Algebras
  • Linear Logic

11
Syntax
  • The symbol of the language.
  • Propositional symbols A, B, C,
  • Prop set of propositional symbols
  • Connectives ? (and), ? (or), ? (not), ?
    (implies), ? (is equivalent to), ? (false).
  • Parenthesis (, ).

12
Formulas
  • Backus-Naur Form
  • Form Prop (?Form) (Form o Form).
  • Context-Free Grammar
  • Form ? Prop,
  • Form ? (? Form),
  • Form ? (Form o Form)

13
Formulas (2)
  • The set of formulas, Form, is defined as the
    smallest set of expressions such that
  • Prop ? Form
  • p?Form ? (?p)?Form
  • p,q ?Form ? (p o q) ? Form

14
Formulas (3)
  • Examples
  • (?A)
  • (?(?A))
  • (A ? (B ? C))
  • (A? (B? C))
  • Correct expressions of Propositional Logic are
    full of unnecessary parenthesis.

15
Formulas (4)
  • Abbreviations. Let o?, ?, ?. We write
  • A o B o C o
  • in the place of
  • (A o (B o (C o )))
  • Thus, we write
  • A ? B ? C, A?B?C,
  • in the place of
  • (A ? (B ? C)), (A? (B? C))

16
Formulas (5)
  • We omit parenthesis whenever we may restore them
    through operator precedence
  • ? binds more strictly than ?, ?, and ?, ? bind
    more strictly than ?, ?.
  • Thus, we write
  • ??A for (?(?A)),
  • ?A ?B for ((?A ) ?B)
  • A ?B? C for ((A?B) ? C),

17
Semantics
  • Def) A truth assignment, ?, is an elements of
    2Prop(I.e., ? ? 2Prop).
  • Two ways to think of truth assignment
  • 1) X ? Prop
  • 2) ? Prop ? 0,1
  • Note These notions are equivalence.

18
Philosophers view
  • ? p means
  • ? satisfies p or
  • ? is true of p or
  • p holds at ? or
  • ? is a model of p

19
Satisfaction Relation
  • Def 1) ? (2Prop x Form)
  • ? A if ? (A) 1 (or, A ? ?)
  • ? ?p if it is not the case ? p.
  • ? p?q if ? p and ? q
  • ? p ? q if ? p or ? q
  • ? p ? q if ? p implies ? q
  • ? p ? q if ? p iff ? q

20
Satisfaction Relation
  • A,B A ? B
  • Iff A,B A and A,B B
  • Iff A ? A,B and B ? A,B

21
Electrical Engineers view
  • A mapping of voltages on a wire ? Prop ? 0,1
  • ? 0,1 ? 0,1
  • ?(0) 1 and ?(1) 0
  • ? 0,12 ? 0,1
  • ?(0,0) ?(0,1) ?(1,0)0 and ?(1,1)1
  • ? 0,12 ? 0,1
  • ?(1,1) ?(0,1) ?(1,0)1 and ?(0,0)0

22
Semantics
  • Def 2)
  • A(?) ?(A)
  • (?p)(?) ?(p(?))
  • (p o q)(?) o(p(?), q(?))
  • Lemma) Let p ? Form and ? ? 2Prop, then ? p
    iff p(?) 1.

23
Software Engineers view
  • Intuition a formula specifies a set of truth
    assignments.
  • Def 3) Function Models From ? 22Prop
  • models(A) ? ?(A) 1, A ? Prop
  • models(?p) 2Prop models(p)
  • models(p?q) models(p) ? models(q)
  • models(p?q) models(p) ? models(q)
  • models(p?q) (2Prop models(p)) ? models(q)

24
Theorem
  • Let p ? Form and ? ? 2Prop, then the following
    statements are all true
  • 1. ? p
  • 2. p(?) 1
  • 3. ? ? models(p)

25
Relevance Lemma
  • Lets use AP(p) to denote the set of all
    propositional symbols occurred in p. Let ?1, ?2 ?
    2Prop, p?Form.
  • Lemma) if ?1AP(p) ?2AP(p) , then
  • ?1 p iff ?2 p
  • Corollary) ? p iff ?AP(p) p

26
Algorithmic Perspective
  • Truth Evaluation Problem
  • Given p?Form and ? ? 2AP(p), does
    ? p ? Does p(?) 1 ?
  • Eval(p, ?)
  • If p ? A, return ?(A).
  • If p ? (?q), return ?(Eval(q, ?))
  • If p ? (q o r), return o(Eval(p), Eval(q))
  • Eval uses polynomial time and space.

27
Extension of
  • Let T ? 2Prop, ? ? Form
  • Def) T p if T ? models(p)
  • i.e., ? 22Prop X Form
  • Def) T ? if T ? models(?)
  • models(?) ?p?? models(p)
  • I.e., ? 22Prop X 2Form

28
Extension of
  • ? 2Form x 2Form
  • Def) ?1 ?2
  • iff models(?1) ? models(?2)
  • Iff for all ? ? 2Prop
  • if ? ?1 then ?
    ?2

29
Semantic Classification
  • A formula p is called valid if models(p) 2Prop.
    We denote validity of the formula p by p
  • A formula p is called satisfiable if models(p) ?
    ?.
  • A formula is not satisfiable is called
    unsatisfiable or contradiction.

30
Semantic Classification(II)
  • Lemma
  • A formula p is valid iff ?p is unsatifiable
  • p is satisfiable iff ?p is not valid
  • Lemma
  • p q iff (p ? q)

31
Satisfiability Problem
  • Given a p, is p satisfiable?
  • SAT(p)
  • B0
  • for all ? ? 2AP(p)
  • B B ? Eval(p,?)
  • end
  • return B
  • NP-Complete

32
Proofs
  • Formal Proofs. We introduce a notion of formal
    proof of a formula p Natural Deduction.
  • A formal proof of p is a tree whose root is
    labeled p and whose children are assumptions p1,
    p2, p3, of the rule r we used to conclude p.

33
Proofs
  • Natural Deduction Rules. For each logical symbol
    o?, ?, ?, ?, and each formula p with outermost
    connective o, we give
  • A set of Introduction rules for o, describing
    under which conditions p is true
  • A set of Elimination rules for o, describing what
    we may infer from the truth of p.

34
Proofs
  • Natural Deduction notations for proofs.
  • Let p be any formula, and ? be a set of formulas.
    We use the notation
  • ?
  • p
  • abbreviated by ?- p, for
  • there is a proof of p whose assumptions are
    included in ?.

35
Proofs
  • Natural Deduction assumptions of a proof
  • p1 p2 p3
  • r --------------------------------
  • p
  • are inductively defined as
  • all assumptions of proofs of p1, p2, p3, , minus
    all assumptions we crossed.

36
Proofs
  • Identity Principle The simplest proof is
  • p
  • -----
  • p
  • having 1 assumption, p, and conclusion the same
    p.
  • We may express it by ?-p, for all p??
  • We call this proof The Identity Principle (from
    p we derive p).

37
Proofs
  • Rules for ?
  • Introduction rules none (? is always false).
  • Elimination rules from the truth of ? (a
    contradiction) we derive everything
  • ?
  • ----
  • p
  • If ?- ?, then ?-p, for all p

38
Proofs
  • Rules for ?
  • Introduction rules
  • p q
  • --------
  • p ? q
  • If ?- p and ?- q then ?- p ? q

39
Proofs
  • Elimination rules
  • p ? q p ? q
  • -------- -------
  • p q
  • If ?- p ? q, then ?- p and ?- q

40
Proofs
  • Rules for ? Introduction rule
  • p
  • q
  • --------
  • p?q
  • If ?,p - q, then ?-p?q
  • We may drop any number of assumptions equal to p
    from the proof of q.

41
Proofs
  • Elimination rule
  • p?q p
  • ----------------
  • q
  • If ?-p?q and ?-p, then ? - q.

42
Proofs
  • The only axiom not associated to a connective,
    nor justified by some Introduction rule, is
    Double Negation
  • ?p
  • .
  • ?
  • ---
  • p
  • If ?, ?p- ?, then ?-p
  • We may drop any number of assumptions equal to ?p
    from the proof of q.

43
Soundness
  • - p then p

44
Completeness
  • p then - p
About PowerShow.com