Effective Abstract State Machines

Size: px

Start display at page:

Download "Effective Abstract State Machines"

Thomasine Maxwell
6 years ago
Views:

1 Effective Abstract State Machines Nachum Dershowitz w/ Udi Boker & Yuri Gurevich February 2007

2 Church-Turing Thesis - All effective computational models are equivalent to, or weaker than, Turing machines. We need to formalize these notions

3 Abstract State Machines Assuming bounded exploration, runs are computable if the initial state is The initial state may contain uncomputable oracles

4 Martin Davis If non-computable inputs are permitted, then non-computable outputs are attainable.

5 Three Directions Initial state is arithmetic plus constants [Yuri] Initial state has decidable interpretation-induced equivalence on term-generated part [Wolfgang] Initial state contains only finite information [Udi]

6 G & D Church s Thesis is implied by the Sequential ASM Thesis

7 Mechanization

8 Ramon Lull (1274) Raymondus Lullus Ars Magna et Ultima

error at a glance, and when there are disputes among persons, we can

9 Leibniz (1666) The only way to rectify our reasonings is to make them as tangible as those of the Mathematician, so that we can find our error at a glance, and when there are disputes among persons, we can simply say: Let us calculate, without further ado, in order to see who is right.

10 Charles Babbage (1871) Arithmetic and printing units completed by son in Calculated first 25 multiples of π to 29 decimal places.

11 Ada Lovelace (1843) [The Analytical Engine is for] developing and tabulating any function whatever. The engine [is] the material expression of any indefinite function of any degree of generality and complexity.

12 Not Just For Numbers The bounds of arithmetic were however outstepped the moment the idea of applying the cards had occurred; and the Analytical Engine does not occupy common ground with mere "calculating machines." In enabling mechanism to combine together general symbols in successions of unlimited variety and extent, a uniting link is established between the operations of matter and the abstract mental processes of the most abstract branch of mathematical science. A.A.L.

13 f(n)={ 1 n=0 n f(n-1) otherwise Kurt Gödel (1930s)

14 Models of Computation

15 Program Input Model Output Memory

16 Axel Thue (1910) A. Markov ab bbaa

17 Turing Machines

18 a b a a b b B B B

19 a b a a a b B B B

20 R L 0? H!

21 My favorite computer

22 + +?? !

23 ? -? +! + - +

24 Equivalence of Models TM 2 TM [ 1 tape; 2 channels ] CM 2 TM 2 [ BBB... ] CM n CM 2 [ 2 i 3 j 5 k 7 l... ] RAM CM n [ 2 x (2y+1) ] Scheme RAM [ Abelson & Sussman ] TM Scheme [ Interpreter ]

25 Effectiveness

26 David Hilbert s Entscheidungsproblem #2. Provide an effective method to determine if a formula is valid.

27 Jules Henri Poincaré We might imagine a machine where we should put in axioms at one end and take out theorems at the other, like that legendary machine in Chicago where pigs go in alive and come out transformed into hams and sausages.

28 f(n)={ 1 n=0 n f(n-1) otherwise Kurt Gödel (1930s)

29 Alonzo Church (1936) [We] propose a definition of effective calculability which is thought to correspond satisfactorily to [a] somewhat vague intuitive notion...

30 Alonzo Church (1936) We now define... the notion of effective calculable function of the positive integers by identifying it with the notion of a recursive function of the positive integers.

31 Emil Leon Post (1936) To mask this identification under a definition... blinds us to the need of its continual verification.

32 Church (1941) Lambda Calculus λx. (x x)

33 Church The fact that two such widely different equally natural definitions of effective calculability turn out to be equivalent adds to the strength of the reasons for believing that they constitute as general a characterization of this notion as is consistent with the usual intuitive understanding of it.

34 Church s Thesis Recursive functions capture effective computability.

35 Alan Turing (1936) A man provided with paper, pencil, and rubber, and subject to strict discipline, is in effect a universal machine.

36 Alan Turing (1936) The theorem that all effectively calculable sequences are computable and its converse are proved...

37 Turing (1948) Logical Computing Machines can do anything that could be described as rule of thumb or purely mechanical. This is sufficiently well established that it is now agreed amongst logicians that calculable by means of an LCM is the correct rendering of such phrases.

38 Turing s Thesis Turing Machines capture mechanical human computation.

39 Kleene (1936) So Turing s and Church s theses are equivalent. We shall usually refer to them both as Church s thesis, or in connection with that one of its... versions which deals with Turing machines as the Church-Turing thesis.

40 Church-Turing Thesis - All effective computational models are equivalent to, or weaker than, Turing machines.

41 Laszlo Kalmár (1959) There are premathematical concepts which must remain [so]... Among these belong... such concepts as that of effective calculability... the extension of which cannot cease to change during the development of mathematics.

42 Princeton Course The definition of a Turing machine is very complete. Multiple heads Multiple tapes Multiple states Multiple directions Multiple dimensions Multiple worlds

43 The Turing Tarpit Thue systems Post systems Lambda calculi Partial recursion Turing machines Markov normal algorithms Minsky counter machines Type 0 languages Kolmogorov-Uspenskii machines Neuring machines Wang machines Random access machines Quantum computers Billiard ball computers Least fixpoints Fortran, Algol, Lisp, C, Pascal, Logo, Ada, Java,...

44 Bernard Moret The remarkable result about these varied models is that all of them define exactly the same class of computable functions: whatever one model can compute, all the others can too!

45 Martin Davis How can we ever exclude the possibility of our presented, some day (perhaps by some extraterrestrial visitors), with a (perhaps extremely complex) device or oracle" that computes" an uncomputable function?"

46 Plan Prove that any model of computation over any (countable first-order) structure that satisfies effectiveness axioms can be simulated (up to isomorphism) by a Turing machine.

47 Extensional View Turing-machine model is TM = {{a,b} * ; f 1, f 2,...}. {a,b}* Input T M f 1 f 2 Output {a,b}*

48 TM ASM domain is alphabet plus states plus finitely written infinite tapes over alphabet if head(tape)=a and state=q then tape := left/right(write(b,tape)) state := r left, right, head, and write are given in initial state

49 Initial State left(ab^cbb...) = a^bcbb... right(ab^cbb...) = abc^bb... write(a,ab^cbb...) = ab^abb... head(ab^cbb...) = c

51 Wolfgang Reisig Follows that: ASM computation is effective iff initial state has decidable interpretation-induced equivalence on term-generated part

52 Axiomatics

53 Agenda If Ax = M, then TM > M That is, the Church-Turing Thesis One definition is worth three theorems. --Alfred Adler (New Yorker)

54 Church to Kleene (1935) [Gödel thought] that it might be possible... to state a set of axioms which would embody the generally accepted properties of [effective calculability], and to do something on that basis.

55 Joseph Shoenfield (1993) It may seem that it is impossible to give a proof of Church s Thesis. However, this is not necessarily the case. We can write down some axioms about computable functions which most people would agree are evidently true. It might be possible to prove Church s Thesis from such axioms.

56 Turing s Axioms Sequential symbol manipulation Deterministic Finite internal states Finite symbol space Finite observability and local action Linear external memory

57 Deterministic The behavior at any moment is determined by the symbols which he is observing and his state of mind at the moment

58 Finite Internal States If we admitted an infinity of states of mind, some of them will be arbitrarily close and will be confused

59 Finite Alphabet If we were to allow an infinity of symbols, then there would be symbols differing to an arbitrary small extent

60 Finiteness and Localness [Operations] so elementary that it is not easy to imagine them further divided Can only observe a finite number of symbols at each step

61 Linear External Memory The two-dimensional character of paper is no essential of computation

62 Robin Gandy (1980) States can be described by hereditarily finite sets The rank of these sets is bounded Machines can be assembled unambiguously from individual parts of bounded size Causation is local

63 Goal Representation-independent axioms Church uses numbers Turing uses string representation Gandy uses finite set representation ASMs use any structure

64 Abstract Effective Models An (abstract) effective computational model is any set of sequential (partial) state-transition algorithms with finite initial data over a fixed, finitelygenerated, domain.

65 Donald Knuth (1968) A computational method consists of: 1. States Q 2. Input I Q 3. Output O Q 4. Transitions τ : Q Q

66 Algorithm = Transition System I Q O State encapsulates all system data.

67 Church An algorithm consists of a method by which... a sequence of expressions (in some notation)... can be obtained; [each of which] is effectively calculable [given all prior expressions]; and where... the fact that the algorithm has terminated becomes effectively known and the [output] is effectively calculable.

68 Yuri Gurevich (2000) Abstract State Machines State-transition model Any first-order structure Output is the stable state

69 ASM Thesis Abstract state machines capture effective computations.

70 Abstract States 1. States are algebras (first-order structures). 2. Vocabulary does not change. Carrier fixed during computation. 3. States and transition functions are closed under isomorphism. 4. The glossary of control terms (vocabulary-terms that determine transitions) is of bounded size.

71 Procedures A procedure is a sequential algorithm, in which the initial states of all runs are the same, except for some constant In.

72 ASM Theorem Any (sequential) program satisfying these postulates is emulated by some ASM.

73 ASMT CT If initial state has only basic arithmetic (N,0,1,+,x,=,<), then any ASM computes a partial recursive function, and there s an ASM for each. If initial state has 0,succ,T, undefined pred and =, and 3 variables, plus input(s), then all ASM programs give all partial recursive functions.

74 Effectiveness Axiom 4. An effective procedure is one in which the shared part comprises a base structure (Herbrand universe) an almost finite structure (all but finitely many locations are equal)

75 Base Structure Every element is the value of a unique term. It is isomorphic to the free term algebra. If one wants more than one name, then one must program them

76 Examples of Base Structures A = {N; 0, S} 0 S(0) S(S(0)) B = {binary trees; nil, cons} cons(nil,nil) nil cons(cons(nil,nil),cons(nil,nil)) cons(nil,cons(nil,nil))

77 Almost-Finite Structure A structure is almost-finite if all but a finite number of locations have the same value. Example, f (0)=2, f (1)=5, f (n >1)=

78 We get rid of ASM s Booleans equality undefined If you want them, then program them.

79 Effective Procedure An effective procedure consists of Σ finite signature; In, Out Σ D countable domain B base structure F almost-finite structure S the states, a set of structures S 0 S initial states. S 0 = B + F + {In} τ : S S transition function, closed under isomorphism T glossary, finite set of Σ-terms that determine transitions

80 Effective Models An effective model is a set of effective procedures sharing the same base structure

81 Main Claim Any effective model can be simulated by a Turing machine.

82 Proving the Thesis

83 Church-Turing Theorem 1. Turing machines are effective 2. Turing Machines simulate (up to isomorphism) all effective models

84 Effectiveness Turing machines are effective Counter machines are effective Etc.

85 TMs are Effective domain = {a,b} * B base structure = B, a:, b: a-c structure In, Out,??? glossary {state,head(tape), }

86 Transition Function (One lousy version) if q = q0 if Head(Right) = a Left := a:left Right := Tail(Right) q := q3 Tail(b:Left) := Left Head(b:Left) := b if Head(Right) = b Left := Tail(Left) Right := a:right q := q1 Tail(a:Right) := Right Head(a:Left) := a if q = q1... ** initialization code ** if q = qf % the halting state Out := Right

87 Counter Machines

88 Church-Turing Theorem 1. Turing machines are effective 2. Turing Machines simulate (up to isomorphism) all effective models That is, TM E for every model E ~ > satisfying the effectiveness axioms.

89 Simulation D f D ρ ρ D' f ' D' ρ is a bijection ρ y = iff y =

90 Example TM Rec π : N 1* [Tally numbers] π : Σ* N [Radix numbers]

91 Proof Steps 1. ASM Sequential algorithm 2. ASM with effective domain computer program 3. computer program TM

92 ASMs ASM program: conditional equations if a 1 =b 1 then c 1 :=d 1... if a k =b k then c k :=d k Gurevich: Every sequential algorithm has an equivalent ASM program

93 Domain is Programmable Let π : F N be a recursive enumeration of the free Σ-term algebra F Program a Σ-algebra over N inductively: [f (n 1,,n k )] := π (f (π -1 (n 1 ),, π -1 (n k )))

94 Example Σ = {a,b,f,g} π = a b f(a) f(b) g(a) g(b) f(f(a)) f(f(b)) f(g(a)) f(g(b)) then f (5)=9

95 Necessity Sequential time is necessary, since transfinite computations can be hypercomputational. First-order structures are necessary to preclude infinitary functions, like limits. With unbounded exploration, programs could perform infinite lookup. Isomorphism invariance is needed for bounded exploration. Without limiting the initial data, an oracle could be included in initial states.

96 Conclusion Super-recursive ability requires one of: Uncountable domain and/or continuous transition Infinite initial information Infinite program or unbounded work

97 Problems

98 Yuri s Problem Good Bad r s Bad Good

99 Angelic Automata Good Bad Lucifer Bad Gabriel Good

100 Isomorphism Isomorphic structures have the same computational power x x f f (x ) Mapping A ~ A f f (x ) 101 f f 6

101 My Problem Who says that TMs can t simulate super-turing devices?!

102 Hartley Rogers, Jr. The use of codings raises an immediate question of invariance. Once a coding is chosen, will the formal concept partial recursive function on code numbers correspond to the informal notion algorithmic mapping on the uncoded expressions?... Church s thesis provides an affirmative answer.

103 Potential Problems 1. Information can be hidden in a mapping 2. Different mappings can have opposite effects mapping #1 A B mapping #2

104 Mark Burgin To show that [Inductive Turing machines] are more powerful [than ordinary TMs], we need to find a problem solvable by an ITM and insolvable by a TM.

Gottfried Wilhelm Leibniz (1666)

Gottfried Wilhelm Leibniz (1666) Euclid (c. -300) Euclid s GCD algorithm appeared in his Elements. Formulated geometrically: Find common measure for 2 lines. Used repeated subtraction of the shorter segment from the longer. Gottfried