Logic in Computer Science: A Historical and Practical Overview, Slides of Logic

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Logic in Computer Science

Frank Wolter

Meta Information

Slides, exercises, and other relevant information are available at:

http://www.liv.ac.uk/~frank/teaching/comp118/comp118.html

The module has

• 18 lectures.
• 5 tutorials (Thursday groups start in week 3, Monday groups in week 4).
• Participation and reasonable attempts to solve problems before tutorials is worth 6 percent of final mark).
• Two class tests (25 minutes each, worth 14 percent of final mark).
• Exam (90 minutes, worth 80 percent of final mark).

If you need to see me individually I will be available in my office (room 114, Ashton Building) on Mondays 4–6pm.

Learning Outcomes

At the end of the module the student should be able to:

• translate natural language descriptions and reasoning processes to and from logical equivalents in the propositional and predicate logic.
• evaluate first-order predicate logic formulae in relational stuctures and un- derstand the relationship to relational databases.
• state and apply a proof system (either tableaux or sequent) for proposi- tional and predicate logic.

The Unusual Effectiveness of Logic in

Computer Science

Title refers to a symposium and article (by the same title) held at the 1999 Meet- ing of the American Association for the Advancement of Science. The paper is co-authored by J. Halpern, R. Harper, N. Immerman. P. Kolaitis, M. Vardi, and V. Vianu. It refers to

• the article On the unreasonable effectiveness of mathematics in the natu- ral sciences, by E.P. Wigner (1960), a joint winner of the 1963 Nobel Prize for Physics. http://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.html.

Very Brief History of Logic: mid 19th century

George Boole (1847): “The Mathematical Analysis of Logic” attempts to for- malise logic in the same way as mathematics formalises the manipulation of equations (and other expressions) with numbers.

For example, the distributive law for numbers:

(x + y) × z = (x × z) + (y × z)

corresponds to a distributive law in propositional logic (or boolean algebra):

(p ∨ q) ∧ r ≡ (p ∧ r) ∨ (q ∧ r)

George Boolos: “The design of the following treatise is to investigate the funda- mental laws of the operations of mind by which reasoning is performed.”

Logic and Foundations of Mathematics: from late 19th century

In mid 19th century, Mathematics (Geometry, Calculus) had rather shaky foun- dations. For example, no clear answer could be given to:

• What does it mean that

i=1 ai^ exists?

• Is Euclidean geometry the only possible geometry?

In “Begriffsschrift” (1879), Frege proposed logic as a foundation for mathemat- ics. He invented, among other things, the basics for first-order predicate logic:

• Constants such as π;
• Predicates such as < to assert 4 < 8 ;
• Functions such as + to form 1 + 1;
• Logical connectives such as ∧ from Boole;
• Quantifiers such as “for all”: ∀.

Logic and Foundations of Mathematics: from late 19th century

In Principia Mathematica (1913, more than 2000 pages), Russell and Whitehead attempt to repair Frege’s system and develop logic as a foundation for mathe- matics.

The following problems where then formulated by Hilbert:

• Can we prove that Principia Mathematica (mathematics) is consistent?
• Can we develop a complete formal system for mathematics?
• Is mathematics decidable (is there a mechanical way to determine whether a given mathematical statement is true)?

Logic and Foundations of Mathematics: from late 19th century

The answers are negative (proofs use again self reference):

• G ¨odel (1930th): one cannot axiomatize arithmetic;
• G ¨odel (1930th): one cannot prove the consistency of mathematics;
• Church and Turing (1930th): there is no mechanical procedure that can decide whether a first-order predicate logic sentence is a tautology.

Birth of Computer Science: To prove the results above one has to answer the following questions: what is a mechanical procedure? What is a solvable prob- lem? What is an algorithm?

Try Logicomix, a graphic novel dealing with the Foundations of Mathematics and figures such as Russell, Frege, Hilbert, Cantor, Wittgenstein, Goedel, etc.

Syllabus

• Introduction: the unusual effectiveness of logic in computer science;
• Propositional logic (5 lectures):
  • Reminder: syntax and semantics of propositional logic,
  • SAT, logical consequence, logical equivalence, and normal forms,
  • a proof system for propositional logic.
• Introduction to First-order Predicate Logic (11 lectures):
  • syntax of first-order predicate logic,
  • semantics of first-order predicate logic,
  • evaluating first-order predicate logic and relational databases,
  • a proof system for first-order predicate logic,
  • undecidability of first-order predicate logic.
• Outlook: the unusual effectiveness of logic in computer science (1 lecture)