Is Constructive Logic relevant for Computer Science?
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1 Is Constructive Logic relevant for Computer Science? Thorsten Altenkirch University of Nottingham BCTCS 05 p.116
2 Birth of Modern Mathematics BCTCS 05 p.216
3 Birth of Modern Mathematics Isaac Newton ( ) BCTCS 05 p.216
4 Birth of Modern Mathematics Isaac Newton ( ) 1687: Philosophiae Naturalis Principia Mathematica BCTCS 05 p.216
5 1920th century: Foundations? BCTCS 05 p.316
6 1920th century: Foundations? Frege ( ) Russell ( ) BCTCS 05 p.316
7 1925: ZF set theory Zermelo ( ) Fraenkel ( ) BCTCS 05 p.416
8 1925: ZF set theory Zermelo ( ) Fraenkel ( ) End of story? BCTCS 05 p.416
9 Mathematics is universal The foundations which are good for mathematical reasoning within natural sciences are equally useful in Computer Science. BCTCS 05 p.516
10 Constructivism? BCTCS 05 p.616
11 Constructivism? Computer Science focusses on constructive solutions to problems. BCTCS 05 p.616
12 Constructivism? Computer Science focusses on constructive solutions to problems. Classical Mathematics is based on the platonic idea of truth. BCTCS 05 p.616
13 Constructivism? Computer Science focusses on constructive solutions to problems. Classical Mathematics is based on the platonic idea of truth. Constructive Mathematics is based on the notion of evidence or proof. BCTCS 05 p.616
14 BHK: Programs are evidence BCTCS 05 p.716
15 BHK: Programs are evidence Brouwer ( ) Heyting ( ) Kolmogorov ( ) BCTCS 05 p.716
16 BCTCS 05 p.816, classically
17 , classically BCTCS 05 p.816
18 , classically The same truth table shows that BCTCS 05 p.816
19 BHK semantics BCTCS 05 p.916
20 BHK semantics Evidence for is given by pairs:! #"! BCTCS 05 p.916
21 ! BCTCS 05 p.916 BHK semantics is given by pairs: Evidence for! #"!. or is tagged evidence for Evidence for ')(, + * ')(! % $&%
22 !! -! BHK semantics Evidence for is given by pairs:! #"! Evidence for is tagged evidence for or. ')(, + * ')(! % $&% Evidence for from evidence for. is a program constructing evidence for BCTCS 05 p.916
23 BCTCS 05 p.1016, constructively
24 -! * 1 1 BCTCS 05 p.1016, constructively 1! 1!.0! * #" ')( ')(. " #" ' (, ')(,. "
25 -! * 1 1 BCTCS 05 p.1016, constructively 1! 1!.0! * #" ')( ')(. " #" ' (, ')(,. " The program is invertible, because the right hand sides are patterns.
26 -! * 1 1 BCTCS 05 p.1016, constructively 1! 1!.0! * #" ')( ')(. " #" ' (, ')(,. " The program is invertible, because the right hand sides are patterns. This shows that the types are isomorphic.
27 Predicate logic BCTCS 05 p.1116
28 5 9 : 8 : Predicate logic Evidence for evidence for is a function which assigns to each BCTCS 05 p.1116
29 5 9 : 8 : 5 8 < : : < : " Predicate logic Evidence for evidence for Evidence for ; 3 is a function is a pair which assigns to each where and. BCTCS 05 p.1116
30 5 9 : 8 : 5 8 < : : < : " Predicate logic Evidence for evidence for Evidence for ; 3 is a function is a pair which assigns to each where and. We need dependent types! BCTCS 05 p.1116
31 Propositions = Types BCTCS 05 p.1216
32 Propositions = Types Per Martin-Löf BCTCS 05 p.1216
33 Propositions = Types Martin-Löf Type Theory Per Martin-Löf BCTCS 05 p.1216
34 Propositions = Types Martin-Löf Type Theory Per Martin-Löf Implementations: NuPRL, LEGO, ALF, COQ, AGDA, Epigram... BCTCS 05 p.1216
35 = BCTCS 05 p.1316
36 ? = We cannot prove undecided proposition, where., for an BCTCS 05 p.1316
37 ?, for an BCTCS 05 p.1316 =, where. We cannot prove undecided proposition F G D)E I@ FHG D)E C 6 A&B 24@
38 ? = We cannot prove undecided proposition, where., for an A&B C 6 FHG D)E I@ F G D)E is provable, i.e. Prime is decidable. BCTCS 05 p.1316
39 ? = We cannot prove undecided proposition, where., for an A&B C 6 FHG D)E I@ F G D)E is provable, i.e. Prime is decidable. Indeed, the proof is the program which decides Prime. BCTCS 05 p.1316
40 ? = We cannot prove undecided proposition, where., for an A&B C 6 FHG D)E I@ F G D)E is provable, i.e. Prime is decidable. Indeed, the proof is the program which decides KC J&B C 6 A&B 24@ J&B BCTCS 05 p.1316
41 ? = We cannot prove undecided proposition, where., for an A&B C 6 FHG D)E I@ F G D)E is provable, i.e. Prime is decidable. Indeed, the proof is the program which decides KC J&B C 6 A&B 24@ J&B is not provable, because Halt is undecidable. BCTCS 05 p.1316
42 The classical Babelfish BCTCS 05 p.1416
43 The classical Babelfish Classical reasoner says: Babelfish translates to: BCTCS 05 p.1416
44 The classical Babelfish Classical reasoner says: Babelfish translates to: BCTCS 05 p.1416
45 The classical Babelfish Classical reasoner says: Babelfish translates to: BCTCS 05 p.1416
46 The classical Babelfish Classical reasoner says: Babelfish translates to: ; 3 BCTCS 05 p.1416
47 The classical Babelfish Classical reasoner says: Babelfish translates to: ; 3 BCTCS 05 p.1416
48 The classical Babelfish Classical reasoner says: Babelfish translates to: ; 3 Negative translation BCTCS 05 p.1416
49 The classical Babelfish Classical reasoner says: Babelfish translates to: ; 3 Negative translation is traslated to BCTCS 05 p.1416
50 The classical Babelfish Classical reasoner says: Babelfish translates to: ; 3 Negative translation is traslated to which is constructively provable. BCTCS 05 p.1416
51 The classical Babelfish Classical reasoner says: Babelfish translates to: ; 3 Negative translation is traslated to which is constructively provable. A classical reasoner is somebody who is unable to say anything positive. BCTCS 05 p.1416
52 The Axiom of Choice? BCTCS 05 p.1516
53 OP BCTCS 05 p.1516 The Axiom of Choice? N 3 L M 6 ;4L N M ;
54 OP BCTCS 05 p.1516 The Axiom of Choice? N 3 L M 6 ;4L N M ; is provable constructively.
55 OP 2 BCTCS 05 p.1516 The Axiom of Choice? N 3 L M 6 ;4L N M ; is provable constructively. However, its negative translation: N 3 L M 6 2 L OP P 9 3 N M is not.
56 OP 2 BCTCS 05 p.1516 The Axiom of Choice? N 3 L M 6 ;4L N M ; is provable constructively. However, its negative translation: N 3 L M 6 2 L OP P 9 3 N M is not. There is empirical evidence that CAC is consistent.
57 Summary BCTCS 05 p.1616
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