J. Barkley Rosser, 81, a professor emeritus of mathematics and computer science at the University of Wisconsin who had served in government, died

Transcription

1 Church-Rosser

2 J. Barkley Rosser, 81, a professor emeritus of mathematics and computer science at the University of Wisconsin who had served in government, died Sept. 5, Along with Alan Turing and John von Neumann, his 1930s work in mathematical logic laid the foundation for modern computer science theory. Dr. Rosser did mathematics work for the space program and served on the National Security Agency's scientific advisory board

3 442

4

5 ?

6 Chamelion Problem

7 Grecian Urn

8 Urn Problem

9 Urn System

10 Rewriting Lambda calculi Recursive functions Functional languages Logic languages Symbolic computation

11 Functional Programming Languages Lisp, Scheme APL, FP, M ML, OCaml SASL, Miranda Haskell, Curry F#

12 ML Robin Milner ( ) Logic for Computable Functions Stanford & Edinburgh Meta-Language Theorem proving Type system Higher-order functions

13 ML Examples fun length nil = 0 length (x::s) = 1 + length(s); fun append(nil, ys) = ys append (x::xs, ys) = x :: append (xs, ys); fun reverse nil = nil reverse (x::xs) = append ((reverse xs), [x]);

14 Abstract Rewriting An abstract rewriting system is composed of elements T binary relation T T there may be several relations,,... labelled transition system

15 Properties Termination (always halts) Normalization (sometimes halts) Has at least one normal form Confluence (can always recover) Unique normalization Has at most one normal form Order of evaluation may matter!

16 Normal Form Descendent reachable Normal form cannot be rewritten An object s nf is any descendent nf Ambiguous has more than one nf Weakly normalizing at least one nf Immortal has a non-halting path Strongly normalizing mortal (not immortal)

17 Why Nontermination? Full computability, including interpreters A nontrivial PL with only halting programs cannot interpret itself Streams c(x) := I(x,x)+1 ones := 1 :: ones map f (x::s) := f(x) :: map f s int := 1 :: map +1 int

18 Interpreter

19 Notation Binary relation Inverse Symmetric closure Reflexive-Transitive closure (*)

20 Notation Given Exists

21 Notation

22 Church-Rosser

23 Confluence

24 Confluence for all M,P,Q * * * * there exists Z s.t.

25 Confluence = Church-Rosser

26 Semi-Confluence

27 Semi-Confluence = Confluence

28 1C CR

29

30

31 Very Strong Confluence

32 Strong Confluence 0,1

33 Strong Confluence = Semi-Confluence

34 Strong Confluence

35 Strong Confluence

36 Strong Confluence

37 Hindley-Rosen Lemma is SC is SC U is CR

38 Local Confluence

39 Question LC CR

40

41 Counterexample [Curry]

42 Counterexample

43 Newman s Lemma If terminating (SN), then local confluence (LC) implies (global) confluence (CR)

44 Newmann s Lemma LC + SN CR

45 Complete Induction x. {[ y. x > y P(y)] P(x)} x. P(x)

46 Emmy Noether

47 Well-founded Induction x. {[ y. x > y P(y)] P(x)} x. P(x)

48 Noetherian Induction [ y. x y P(y)] P(x) x. P(x)

49 Proof

50 Diamond Proof

51 Orthogonality A system is orthogonal if it s left sides are linear and there is no overlap between left sides. A term is linear if no variable occurs more than once. A term overlaps another if it unifies with a non-variable subterm.

52 Example

53

54 Combinatory Logic Ix x (Kx)y x ((Sx)y)z (xz)(yz)

55 Combinatory Logic Ix x (Kx)y x ((Sx)y)z (xz)(yz) (Ex)x 1 (Ex)(Sx) 0

56 Currying Treat a function of many parameters as many functions of one parameter cf. message-passing in object-oriented languages

57 Orthogonality Theory of functional programming Confluence Normalization

58 Main Theorems Orthogonal systems are Church- Rosser. The lambda calculus is CR. Outermost rewriting is normalizing.

59 Notions Redex Descendant of redex A redex that is a descendant of a redex is a residual If not, it is created

60 Counterexample f(x,x) a f(x,g(x)) b c g(c)

61 Counterexample f(x,x) a g(x) f(x,g(x)) c g(c)

62 Disjoint Redexes

63 Nested Redexes

64 Cont d

65 Overlapping Redexes

66 Intermediate Relation Suppose A B A* Then In particular B CR A CR B SC A CR

67 Multistep Reduction x x s1 t1,..., sn tn f(s1... sn) f(t1... tn) l r R, σ τ lσ rτ

68 Proof 1. * 2. is SC 3. So CR

69 is SC If you do some, then do rest to resolve.

70 Normalization Ambiguous has more than one nf (Weakly) normalizing (WN) at least one nf Strongly normalizing (SN) not immortal Perpetual if immortal before rewrite, then also after

71 Normalization Strategies Normalizing (n) computes at least one nf Perpetual (p) if immortal before step, then also after

72 Example

73 Strategy: Parallel Innermost

74 Strategy: Leftmost Innermost

75 Strategy: Leftmost Outermost

76 Strategy: Parallel Outermost

77 Strategy: Full Substitution

78 Outermost Rewriting Outermost is normalizing. Parallel outermost is normalizing. Fair outermost is normalizing.

79 Counter/example F(x,B) D A B C C F(C,A)

80 Fixpoints Outermost and full substitution compute the least fixpoint of recursive definitions.

81 Critical Pair Lemma A rewrite system is locally confluent iff all its critical pairs are joinable

82 Summary