A Brief History of Logic ¨ Steffen Holldobler International Center for Computational Logic ¨ Dresden Technische Universitat Germany
I History I A Simple Example I Literature I Module Foundations
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History: Basic Ideas
Aristotle (†322 B.C.) Formalization
syllogisms SeP PeQ SeQ
Herodot (†430 B.C.) Calculization
Egyptian stones, abacus
Herodot (†430 B.C.) Mechanization mechanai
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History: Combining the Ideas (1)
Descartes (1596-1650)
Hobbes (1588-1679)
Leibnitz (1646-1719)
geometry
thinking = calculating
lingua characteristica calculus ratiocinator universal encyclopedia
Lullus (1232-1315)
Pascal (1623-1662)
Leibnitz (1646-1719)
ars magna
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History: Combining the Ideas (2)
DeMorgan (1806-1871) Boole (1815-1864) propositional logic
Frege(1882)
Whitehead, Russell (1910-1913)
first order logic “Begriffsschrift”
Principia Mathematica
Javins(1869)
Babbage (1792-1871)
evaluating boolean expressions
analytical engine
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History: Combining the Ideas (3)
civil servant’s logic: F |= G iff G ∈ F higher civil servant’s logic: F |= G iff F = {G} ¨ Skolem, Herbrand, Godel (1930) completeness of first order logic Zuse (1936-1941) Z1, Z3 Turing (1936) Turing machine
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History: Finally, Computers Arrive
von Neumann (1946)
Zuse (1949)
Turing (1950)
computer
Plankalkul ¨
Turing test Can machines think?
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History: Deduction Systems
I early 1950s: Davis: Preßburger arithmetic. I 1955/6: Beth, Schutte, ¨ Hintikka: semantic tableaus. I 1956: Simon, Newell: first heuristic theorem prover. I late 1950s: Gilmore, Davis, Putnam:
theorem prover based on Herbrand’s “Eigenschaft B Methode”. I 1960: Prawitz: unification. I 1965: J.A. Robinson: resolution principle. I thereafter: improved resolution rules vs. intelligent heuristics. I 1996: McCune’s OTTER proves Robbin’s conjecture. I today: TPTP library, yearly CASC competition.
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History: Logic Programming
I 1971: A. Colmerauer: System Q
Prolog.
brother-of (X , Y ) ← father-of (Z , X ) ∧ father-of (Z , Y ) ∧ male(X ). I 1979: R.A. Kowalski: algorithm = logic + control. I late-70s to mid-80s: theoretical foundations. I 1977: D.H.D. Warren: first Prolog compiler. I 1982: A. Colmerauer: Prolog II
constraints.
. Constraint logic programming.
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A Simple Example
I 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)
I 5 is a natural number. All natural numbers are integers. Hence, 5 is an integer.
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Deduction
I A world without deduction would be a world without science, technology, laws,
social conventions and culture (Johnson–Laird, Byrne: 1991). I Think about it!
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The Addition of Natural Numbers I 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
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Applications I Functional equivalence of two chips I Verification of hard- and software I Year 2000 problem I Eliminating redundancies in group communication systems I Designing the layout of yellow pages I Managing a tunnel project I Natural language processing I Cognitive Robotics I Semantic web (description logics) I Law I Optimization Problems
Logic is Everywhere
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Some Background Literature I L. Chang and R.C.T. Lee: Symbolic Logic and Mechanical Theorem Proving.
Academic Press, New York (1973). I M. Fitting: First–Order Logic and Automated Theorem Proving.
Springer Verlag. Berlin, second edition (1996). I J. Gallier:
Logic for Computer Science: Foundations of Automated Theorem Proving. Harper and Row. New York (1986). ¨ I S. Holldobler: Logik und Logikprogrammierung. Synchron Publishers GmbH, Heidelberg (2009). I D. Poole and A. Mackworth and R. Goebel:
Computational Intelligence: A Logical Approach. Oxford University Press, New York, Oxford (1998). I S. Russell and P. Norvig: Artificial Intelligence.
Prentice Hall, Englewood Cliffs (1995). ¨ I U. Schoning: Logik fur ¨ Informatiker. Spektrum Akademischer Verlag (1995).
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Module Foundations
I Two lectures
. Logic . Science of Computational Logic I Logic is offered from now until the end of November. I Science of Computational Logic is offered from beginning of December
until end of the lecturing period. I Exact dates will be announced later. I Exams:
. Logic: written exam (Dec 21, 2014) . Science of Computational Logic: oral exam
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Logic I Agenda
. Introduction . Propositional Logic . First Order Logic I Exercises
. Exercises are announced each week. . We expect students to discuss their solutions. I Tests
. Their will be two written tests. . 10% of the final mark will be given based on performance in the tests. I See our web pages for more detail. I Ask questions as soon as they arise, anywhere and at anytime. I Don’t accept a situation, where you do not understand everything.
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