A Brief History of Logic Steffen H olldobler International Center - - PowerPoint PPT Presentation

◮ History ◮ A Simple Example ◮ Literature ◮ Module Foundations

Steffen H¨

• lldobler

A Brief History of Logic 1

A Brief History of Logic

Steffen H¨

• lldobler

International Center for Computational Logic Technische Universit¨ at Dresden Germany

History: Basic Ideas

Formalization Aristotle (†322 B.C.) syllogisms SeP PeQ SeQ Calculization Herodot (†430 B.C.) Egyptian stones, abacus Mechanization Herodot (†430 B.C.) mechanai

Steffen H¨

• lldobler

A Brief History of Logic 2

History: Combining the Ideas (1)

Descartes (1596-1650) geometry Hobbes (1588-1679) thinking = calculating Leibnitz (1646-1719) lingua characteristica calculus ratiocinator universal encyclopedia Lullus (1232-1315) ars magna Pascal (1623-1662) Leibnitz (1646-1719)

Steffen H¨

• lldobler

A Brief History of Logic 3

History: Combining the Ideas (2)

DeMorgan (1806-1871) Boole (1815-1864) propositional logic Frege(1882) first order logic “Begriffsschrift” Whitehead, Russell (1910-1913) Principia Mathematica Javins(1869) evaluating boolean expressions Babbage (1792-1871) analytical engine

Steffen H¨

• lldobler

A Brief History of Logic 4

History: Combining the Ideas (3)

Skolem, Herbrand, G¨

• del (1930)

completeness of first order logic civil servant’s logic: F | = G iff G ∈ F higher civil servant’s logic: F | = G iff F = {G} Turing (1936) Turing machine Zuse (1936-1941) Z1, Z3

Steffen H¨

• lldobler

A Brief History of Logic 5

History: Finally, Computers Arrive

von Neumann (1946) computer Zuse (1949) Plankalk¨ ul Turing (1950) Turing test Can machines think?

Steffen H¨

• lldobler

A Brief History of Logic 6

History: Deduction Systems

◮ early 1950s: Davis: Preßburger arithmetic. ◮ 1955/6: Beth, Sch¨

utte, Hintikka: semantic tableaus.

◮ 1956: Simon, Newell: first heuristic theorem prover. ◮ late 1950s: Gilmore, Davis, Putnam:

theorem prover based on Herbrand’s “Eigenschaft B Methode”.

◮ 1960: Prawitz: unification. ◮ 1965: J.A. Robinson: resolution principle. ◮ thereafter: improved resolution rules vs. intelligent heuristics. ◮ 1996: McCune’s OTTER proves Robbin’s conjecture. ◮ today: TPTP library, yearly CASC competition.

Steffen H¨

• lldobler

A Brief History of Logic 7

History: Logic Programming

◮ 1971: A. Colmerauer: System Q Prolog.

brother-of(X, Y) ← father-of(Z, X) ∧ father-of(Z, Y) ∧ male(X).

◮ 1979: R.A. Kowalski: algorithm = logic + control. ◮ late-70s to mid-80s: theoretical foundations. ◮ 1977: D.H.D. Warren: first Prolog compiler. ◮ 1982: A. Colmerauer: Prolog II constraints.

⊲ Constraint logic programming.

Steffen H¨

• lldobler

A Brief History of Logic 8

A Simple Example

◮ Socrates is a human. All humans are mortal. Hence, Socrates is mortal.

human(socrates) (forall X) (if human(X) then mortal(X)) mortal(socrates) h(s) (∀X) (h(X) → m(X)) m(s)

◮ 5 is a natural number. All natural numbers are integers. Hence, 5 is an integer.

Steffen H¨

• lldobler

A Brief History of Logic 9

Deduction

◮ A world without deduction would be a world without science, technology, laws,

social conventions and culture (Johnson–Laird, Byrne: 1991).

◮ Think about it!

Steffen H¨

• lldobler

A Brief History of Logic 10

The Addition of Natural Numbers

◮ The sum of zero and the number Y is Y. The sum of the successor of the

number X and the number Y is the successor of the sum of X and Y. ⊲ Are you willing to conclude from these statements that the sum of one and one is two? 0 + Y = Y s(X) + Y = s(X + Y) s(0) + s(0) = s(s(0)) ⊲ Are you willing to conclude that addition is commutative? 0 + Y = Y s(X) + Y = s(X + Y) X + Y = Y + X

Steffen H¨

• lldobler

A Brief History of Logic 11

Applications

◮ Functional equivalence of two chips ◮ Verification of hard- and software ◮ Year 2000 problem ◮ Eliminating redundancies in group communication systems ◮ Designing the layout of yellow pages ◮ Managing a tunnel project ◮ Natural language processing ◮ Cognitive Robotics ◮ Semantic web (description logics) ◮ Law ◮ Optimization Problems

Logic is Everywhere

Steffen H¨

• lldobler

A Brief History of Logic 12

Some Background Literature

◮ L. Chang and R.C.T. Lee: Symbolic Logic and Mechanical Theorem Proving.

Academic Press, New York (1973).

◮ M. Fitting: First–Order Logic and Automated Theorem Proving.

Springer Verlag. Berlin, second edition (1996).

◮ J. Gallier:

Logic for Computer Science: Foundations of Automated Theorem Proving. Harper and Row. New York (1986).

◮ S. H¨

• lldobler: Logik und Logikprogrammierung.

Synchron Publishers GmbH, Heidelberg (2009).

◮ D. Poole and A. Mackworth and R. Goebel:

Computational Intelligence: A Logical Approach. Oxford University Press, New York, Oxford (1998).

◮ S. Russell and P. Norvig: Artificial Intelligence.

Prentice Hall, Englewood Cliffs (1995).

◮ U. Sch¨

• ning: Logik f¨

ur Informatiker. Spektrum Akademischer Verlag (1995).

Steffen H¨

• lldobler

A Brief History of Logic 13

Module Foundations

◮ Two lectures

⊲ Logic ⊲ Science of Computational Logic

◮ Logic is offered from now until the end of November. ◮ Science of Computational Logic is offered from beginning of December

until end of the lecturing period.

◮ Exact dates will be announced later. ◮ Exams:

⊲ Logic: written exam (Dec 21, 2014) ⊲ Science of Computational Logic: oral exam

Steffen H¨

• lldobler

A Brief History of Logic 14

Logic

◮ Agenda

⊲ Introduction ⊲ Propositional Logic ⊲ First Order Logic

◮ Exercises

⊲ Exercises are announced each week. ⊲ We expect students to discuss their solutions.

◮ Tests

⊲ Their will be two written tests. ⊲ 10% of the final mark will be given based on performance in the tests.

◮ See our web pages for more detail. ◮ Ask questions as soon as they arise, anywhere and at anytime. ◮ Don’t accept a situation, where you do not understand everything.

Steffen H¨

• lldobler

A Brief History of Logic 15