Computability and Logic students briefing on the bachelor year and - PowerPoint PPT Presentation

Bachelor Information Meeting • Department of Computer Science offers an meeting for Computability and Logic students briefing on the bachelor year and the master’s studies dBerLog • Time: 15-17, November 1, 2007 Q1 2007 • Place: Store Auditorium dBerLog 2007 1 dBerLog 2007 2 dBerLog - final lecture! Universal Computational Models • Summary Turing: Turing machines (1930s) – Universality Goedel: recursive functions (1930s) – Duality Church: λ -calculus (1930s) – Self-reference Chomsky: Language grammars (1950s) – Program Verification • Life stories All these have been shown to be “equivalent” wrt • About the exam expressiveness! dBerLog 2007 3 dBerLog 2007 4

Church’s λ -calculus Chomsky type 0 grammars • Context dependent rules: β • ( λ x. E) (F) → E [F/x] α A β → α γ β where A ∈ V and α , β , γ ∈ (V ∪ T)* α • ( λ x. E) → λ y. (E [y/x]) • Theorem The class of languages generated by Chomsky type 0 • Theorem grammars is exactly the class of (Turing) recursively A (partial) function is computable in the λ -calculus iff it is enumerable languages! computable by a Turing machine! dBerLog 2007 5 dBerLog 2007 6 Gödels µ -recursive functions N k → N dBerLog - final lecture! • Successor, zero-test, projections, function composition, and • Summary primitive recursion: – Universality f(0, x) = h(x) – Duality f(n+1, x) = g(n, x, f(n, x)) – Self-reference unbounded minimization: – Program Verification f(n) = min y. (g(y,n) = 0) • Life stories • Theorem • About the exam A (partial) function is definable as a µ -recusive function iff it is computable by a Turing machine. dBerLog 2007 7 dBerLog 2007 8

Hækleopskrift: en dug... ...og en hækler Opskrift/program: Materialer/data: Hækl 21 lm 1. række: * 1 stm, 1 lm, spring over Perlebomuld # 5 næste lm *; Gentag fra * til * 9 En luftmaske gange til; 1 stm i sidste lm (10 Hæklenål 1,9 mm mlmrum), vend 2.-10. række: * 1 stm på stm, 1 lm, spring over mlmrum * ; Gentag fra * til * 9 gange til; ......... dBerLog 2007 9 dBerLog 2007 10 Programming ENIAC 1943-45 Duality between programs and data John von Neumann 1903-1957 First draft on a report on EDVAC 1945 dBerLog 2007 11 dBerLog 2007 12

✞ ✏ ✍ ✎ ✡✄ ✏ ✝✑ ✒ ✞ ✓ ✓✞ ☛✔ ☛ ☛ ✄ ✝ ✌ ✑ ✒ ✞ ✓ ✓ ✏ ✒ ✠ ✒ ✞ ✡ ✞ ✞ ✌ ✆☞ ✞ ✓✞ ✗ ✞ ✒ ✁ ✕ ✄ ✂ ✑ ✓ ✂ ✖ ✕ ✞ ✏ ✠ ✕ � ✁ ✂ ✄ ☎ ✆ ✂ ✝✞ ✟ ✠ ✡ ☛ ☛ Duality between programs and data Bootstrapping Alan Turing 1912-1954 On Computable Numbers with an application to the Entscheidungsproblem 1936 dBerLog 2007 13 dBerLog 2007 14 Duality Rene Magritte (1966) dBerLog - final lecture! • Summary – Universality – Duality – Self-reference – Program Verification • Life stories • About the exam dBerLog 2007 15 dBerLog 2007 16

Programs applied to programs Text/program manipulation - examples Happens ALL the time! Example: a Java interpreter • Encryption programs • Program optimizers Java program P Java interpreter: Output = • Indentation programs Java program I Input i “Output from P applied to i” dBerLog 2007 17 dBerLog 2007 18 Proof reading Self-reference All books in our library have a list of references Some books reference themselves (“see chapter..”) Let us call such a book self-referencing Task: write a book (for the library) containing a list of all the books (in the library), which are not self-referencing! dBerLog 2007 19 dBerLog 2007 20

Selfreference - Escher 1898-1972 Selfrerence - dBerLog dBerLog 2007 21 dBerLog 2007 22 dBerLog - final lecture! Program correctness Algoritme: Euklid (m, n) m, n ≥ 1 Inputbetingelse: • Summary Outputkrav: r = sfd(m, n) – Universality {m, n ≥ 1} Metode: p ← m; q ← n; – Duality I ={sfd(p, q) = sfd(m, n)} while p ≠ q do – Self-reference if p > q then p ← p - q – Program Verification else q ← q - p; r ← p • Life stories {r = sfd (m, n)} • About the exam dBerLog 2007 23 dBerLog 2007 24

✘ ✎ ✏ ✩✪ ✖ ✎ ✓★ ✏ ✧ ✦ ✥ ✏ ✒ ✌ ✗ ✗ ✎ ✗ ✥ ✚ ✙ ✘ ✗ ✔ ✎ ✓✖ ✔ ✔✕ ✫ ✬ ✏ ✔ ✗ ✔ ✎ ✖ ✑ ✥ ✗ ✕ ✘ ✳ ✲ ✘ ✲ ✏ ✲ ★ ✩ ✏ ✱ ✰ ✯ ✮ ✍ ✎ ✍ ✏ ✭ ✓ ✒ ✁ ☞ ✂ ✄☎ ✄ ✆ ✝ ✞ ✄ ✟ ✠ ✡ � ✟ ✟ ☛ ☞ ✌ ✍ ✎ ✏ ✑ Program incorrectness Program termination Algoritme: Euklid (m, n) m, n ≥ 1 Inputbetingelse: Outputkrav: r = sfd(m, n) {x > 1} {m, n ≥ 1} while x ≠ 1 do Metode: p ← m; q ← n; if even (x) then x := x div 2 else x:= 3 × x + 1 I ={sfd(p, q) = sfd(m, n)} while p ≠ q do {true} if p > q then p ← p - q else q ← q - p; r ← q {r = sfd (m, n)} dBerLog 2007 25 dBerLog 2007 26 Embedded Systems Automated Verification SyncMaster 17GLsi ✛✢✜✤✣ Mobile Phone Telephone Digital Watch Tamagotchi dBerLog 2007 27 dBerLog 2007 28

✼ ✳ ✭ ✮✯ ✰ ✮ ✪✱✲ ✲ ✴ ✧ ✯ ★✵ ✫✶ ✷ ✬ ✸ ✸ ✫✬ ✪ ✰✹ ✓ ✑ ✘ ✙ � ✚ ✁ ✔ ✛ ✩ ✜ ✢✣ ✤ ✥ ✦ ✧ ★ ✬ ✰ ✠ ✧ ✵ ✸ ✬ ✬ ✺ ✻ ✸ ✰ ✶ ✧ ★ ✺ ✦ ✷ ✬ ✺ ✬ ✬ ✸ ★ ✵ ✫ ✶ ✷ ✬ ✸ ✵ ✰ ✷ ✮ ✷ ✬ ✯ ✬ ✬ ✡ ☛ ☞ ✟ ✍ ✟ ✡ ✂✎ ✏ � ✞ ✑ ☞ ✒ ☞ ✡ ✂✓ ✓ ✔ ☎ ✒ ✞ ✞ ✑ ✠✡ � ✁ ✂ � ✄☎ ✆ ✝✞ ✟ � ✡ � � ☛ ✂ ☞ ✌ ☎ ✟ ☞ ✍ ✒ ☞ ✠ ☞ ✡ ✗ ✠ ✆ ✖ ☎ ✌ ✠ ✡ ☞ ☎ ✡ ✖ ☎ ✆ ✍ ✖ ✁ ☛ ✂ ✕ ✡ ☞ A REAL system Train Simulator dBerLog 2007 29 dBerLog 2007 30 dBerLog - final lecture! Life stories - A. Turing 1912-1956 • Summary – Universality – Duality – Self-reference – Program Verification • Life stories • About the exam dBerLog 2007 31 dBerLog 2007 32

✼ ✥ ✸ ✶ ✧ ★ ✺ ✦ ✷ ✬ ✼ ✦ ✻ ✧ ★ ✩ ✪ ✧ ✫✬ ✭ ✮✯ ✰ ✮ ✧ ✺ ✲ ✷ ✫ ✶ ✷ ✬ ✸ ✸ ✵ ✷ ✮ ✬ ✬ ✯ ✬ ✬ ✰ ✬ ✰ ✺ ✵ ✸ ✬ ✪✱✲ ✳ ✬ ✬ ✬ ✬ ✰ ✬ ✰ ✺ ✵ ✸ ✬ ✺ ✬ ✻ ✧ ✸ ✶ ✧ ★ ✺ ✦ ✷ ✬ ✯ ✷ ✴ ✰ ✯ ★✵ ✫✶ ✷ ✬ ✸ ✸ ✬ ✰✹ ✬ ✮ ★ ✵ ✫ ✶ ✷ ✬ ✸ ✸ ✵ ✷ ✵ ★ ✰ ✸ ✰ ✬ ★ ✵ ✫ ✶ ✷ ✬ ✸ ✵ ✬ ✷ ✮ ✷ ✬ ✯ ✬ ✬ ✰ ✬ ✰ ✰✹ ✸ ✵ ✮✯ ✥ ✦ ✧ ★ ✩ ✪ ✧ ✫✬ ✭ ✰ ✸ ✮ ✪✱✲ ✲ ✳ ✴ ✯ ★✵ ✫✶ ✷ ✬ ✰✹ ✺ ✸ ✪✱✲ ✬ ✩ ✪ ✧ ✫✬ ✭ ✮✯ ✰ ✮ ✲ ✦ ✳ ✴ ✯ ★✵ ✫✶ ✷ ✬ ✸ ✸ ✬ ✧ ★ ✥ ✸ ✧ ★ ✧ ✻ ✺ ✦ ✷ ✬ ✼ ✺ ✬ ✶ 34 36 Ud og se med DSB - september 2007 Life stories - A. Church 1903 - 1995 dBerLog 2007 dBerLog 2007 33 35 Life stories - K. Gödel 1906-1978 Life stories - E. Post 1897 - 1958 dBerLog 2007 dBerLog 2007

✵ ✦ ✬ ✺ ✻ ✧ ✸ ✶ ✧ ★ ✺ ✷ ✸ ✬ ✼ ✧ ✶ ✸ ✥ ✦ ✧ ★ ✬ ✵ ✪ ✵ ✰ ✬ ★ ✫ ✶ ✷ ✬ ✸ ✸ ✷ ✺ ✮ ✷ ✬ ✯ ✬ ✬ ✰ ✬ ✰ ✩ ✧ ✬ ✬ ✫ ✶ ✷ ✬ ✸ ✮ ✷ ✬ ✯ ✬ ★ ✰ ✬ ✰ ✺ ✵ ✸ ✬ ✬ ✺ ✵ ✬ ✫ ✯ ✬ ✭ ✮✯ ✰ ✮ ✪✱✲ ✲ ✳ ✴ ★ ✰ ✵ ✫ ✶ ✷ ✬ ✸ ✸ ✬ ✰✹ ✰✹ ✸ ✧ ✶ ✸ ✸ ✬ ✰✹ ✰ ✬ ★ ✵ ✫ ✷ ✷ ✬ ✸ ✸ ✵ ✷ ✮ ✷ ✬ ✯ ✬ ✫✶ ✬ ✧ ✬ ✷ ✦ ✥ ✦ ✧ ★ ✩ ✪ ✫✬ ★✵ ✭ ✮✯ ✰ ✮ ✪✱✲ ✲ ✳ ✴ ✯ ✬ ✰ ✸ ✪✱✲ ★ ✩ ✪ ✧ ✫✬ ✭ ✮✯ ✰ ✮ ✲ ✬ ✳ ✴ ✯ ★ ✵ ✫ ✶ ✷ ✬ ✧ ✦ ✥ ★ ✰ ✺ ✵ ✸ ✬ ✬ ✺ ✻ ✧ ✸ ✶ ✧ ★ ✺ ✦ ✷ ✬ ✼ ✺ ✻ 38 40 Life stories - J. Kelly ?? - 1995 Life stories - T. Hoare 1934 - dBerLog 2007 dBerLog 2007 37 39 Life stories - N. Chomsky 1928 - Life stories - J. C. Martin ?? - dBerLog 2007 dBerLog 2007