Advanced Topics in Theoretical Computer Science Part 2: Register - PowerPoint PPT Presentation

Advanced Topics in Theoretical Computer Science Part 2: Register machines (3) 21.11.2013 Viorica Sofronie-Stokkermans Universit¨ at Koblenz-Landau e-mail: sofronie@uni-koblenz.de Wintersemester 2013-2014 1

Contents • Recapitulation: Turing machines and Turing computability • Register machines (LOOP, WHILE, GOTO) • Recursive functions • The Church-Turing Thesis • Computability and (Un-)decidability • Complexity • Other computation models: e.g. B¨ uchi Automata, λ -calculus 2

2. Register Machines • Register machines (Random access machines) • LOOP Programs • WHILE Programs • GOTO Programs • Relationships between LOOP, WHILE, GOTO • Relationships between register machines and Turing machines 3

Until now • Register machines (definition; state; input/output; semantics) Computed function Computable functions (LOOP, WHILE, GOTO, TM) • LOOP Programs (syntax, semantics) Every LOOP program terminates for every input All LOOP computable functions are total Additional instructions • WHILE Programs (syntax, semantics) WHILE programs do not always terminate WHILE computable functions can be undefined for some inputs • GOTO Programs (syntax, semantics) GOTO programs do not always terminate 4

LOOP Programs: Syntax Definition • Atomic programs: For each register x i : – x i := x i + 1 – x i := x i − 1 are LOOP instructions and also LOOP programs. • If P 1 , P 2 are LOOP programs then – P 1 ; P 2 is a LOOP program • If P is a LOOP program then – loop x i do P end is a LOOP program (and a LOOP instruction) 5

LOOP Programs: Semantics Definition (Semantics of LOOP programs) Let P be a LOOP program. ∆( P ) is inductively defined as follows: (1) On atomic programs: ∆( x i := x i ± 1)( s 1 , s 2 ) iff: • s 2 ( x i ) = s 1 ( x i ) ± 1 • s 2 ( x j ) = s 1 ( x j ) for all j � = i (2) Sequential composition: ∆( P 1 ; P 2 )( s 1 , s 2 ) iff there exists s ′ s.t.: • ∆( P 1 )( s 1 , s ′ ) • ∆( P 2 )( s ′ , s 2 ) (3) Loop programs: ∆(loop x i do P end)( s 1 , s 2 ) iff there exist states s ′ 0 , s ′ 1 , . . . , s ′ n with: • s 1 ( x i ) = n • s 1 = s ′ 0 • s 2 = s ′ n • ∆( P )( s ′ k , s ′ k +1 ) for 0 ≤ k < n 6

WHILE Programs: Syntax Definition • Atomic programs: For each register x i : – x i := x i + 1 – x i := x i − 1 are WHILE instructions and also WHILE programs. • If P 1 , P 2 are WHILE programs then – P 1 ; P 2 is a WHILE program • If P is a WHILE program then – while x i � = 0 do P end is a WHILE program (and a WHILE instruction) 7

WHILE Programs: Semantics Definition (Semantics of WHILE programs) Let P be a WHILE program. ∆( P ) is inductively defined as follows: (1) On atomic programs: ∆( x i := x i ± 1)( s 1 , s 2 ) iff: • s 2 ( x i ) = s 1 ( x i ) ± 1 • s 2 ( x j ) = s 1 ( x j ) for all j � = i (2) Sequential composition: ∆( P 1 ; P 2 )( s 1 , s 2 ) iff there exists s ′ s.t.: • ∆( P 1 )( s 1 , s ′ ) • ∆( P 2 )( s ′ , s 2 ) (3) While programs: ∆(while x i � = 0 do P end)( s 1 , s 2 ) iff there exists n ∈ N and there exist states s ′ 0 , s ′ 1 , . . . , s ′ n with: • s 1 = s ′ 0 • s 2 = s ′ n • ∆( P )( s ′ k , s ′ k +1 ) for 0 ≤ k < n • s ′ k ( x i ) � = 0 for 0 ≤ k < n • s ′ n ( x i ) = 0 8

GOTO Programs: Syntax Indices (numbers for the lines in the program) j ≥ 0 Definition • Atomic programs: – x i := x i + 1 – x i := x i − 1 are GOTO instructions for each register x i . • If x i is a register and j is an index then – if x i = 0 goto j is a GOTO instruction. • If I 1 , . . . , I k are GOTO instructions and j 1 , . . . , j k are indices then – j 1 : I 1 ; . . . ; j k : I k is a GOTO program 9

GOTO Programs: Semantics Let P be a GOTO program of the form: P = j 1 : I 1 ; j 2 : I 2 ; . . . ; j k : I k Let j k +1 be an index which does not occur in P (program end). Definition ∆( P )( s 1 , s 2 ) holds iff for every n ≥ 0 there exist states s ′ 0 , . . . , s ′ n and indices z 0 , . . . , z n s.t.: • s ′ 0 = s 1 , s ′ n = s 2 ; z 0 = j 1 , z n = j k +1 . • For 0 ≤ l ≤ n , if j s : I s is the line in P with j s = z l : s ′ i +1 ( x i ) = s ′ if I s = x i := x i ± 1 then: i ( x i ) ± 1 s ′ i +1 ( x j ) = s ′ i ( x j ) for j � = i z i +1 = j s +1 if I s = if x i = 0 goto j goto then: s ′ i +1 = s ′ i � j goto if x i = 0 z i +1 = j s +1 otherwise 10

Register Machines Definition A register machine is a machine consisting of the following elements: • A finite (but unbounded) number of registers x 1 , x 2 , x 3 . . . , x n ; each register contains a natural number. • A LOOP-, WHILE- or GOTO-program. 11

Register Machines: Computable function Definition. A function f is • LOOP computable if there exists a register machine with a LOOP program, which computes f • WHILE computable if there exists a register machine with a WHILE program, which computes f • GOTO computable if there exists a register machine with a GOTO program, which computes f • TM computableif there exists a Turing machine which computes f 12

Computable functions LOOP = Set of all LOOP computable functions WHILE = Set of all total WHILE computable functions WHILE part = Set of all WHILE computable functions (including the partial ones) GOTO = Set of all total GOTO computable functions GOTO part = Set of all GOTO computable functions (including the partial ones) TM = Set of all total TM computable functions TM part = Set of all TM computable functions (including the partial ones) 13

Relationships between LOOP, WHILE, GOTO Theorem. LOOP ⊆ WHILE (every LOOP computable function is WHILE computable) 14

WHILE and GOTO Theorem. (1) WHILE = GOTO (2) WHILE part = GOTO part 15

WHILE and GOTO Theorem. (1) WHILE = GOTO (2) WHILE part = GOTO part Proof: To show: I. WHILE ⊆ GOTO and WHILE part ⊆ GOTO part II. GOTO ⊆ WHILE and GOTO part ⊆ WHILE part 16

WHILE and GOTO Theorem. (1) WHILE = GOTO (2) WHILE part = GOTO part Proof: I. WHILE ⊆ GOTO and WHILE part ⊆ GOTO part It is sufficient to prove that while x i � = 0 do P end can be simulated with GOTO instructions. We can assume without loss of generality that P does not contain any while (we can replace the occurrences of “while” from inside out). 17

WHILE and GOTO Proof (ctd.) while x i � = 0 do P end is replaced by: j 1 : if x i = 0 goto j 3 ; P ′ ; j 2 : if x n = 0 goto j 1 ; ** Since x n = 0 unconditional jump ** j 3 : x n := x n − 1 where: • x n is a new register, which was not used before. • P ′ is obtained from P by assigning to all instructions without an index an arbitrary new index. 18

WHILE and GOTO Proof (ctd.) while x i � = 0 do P end is replaced by: j 1 : if x i = 0 goto j 3 ; P ′ ; j 2 : if x n = 0 goto j 1 ; ** Since x n = 0 unconditional jump ** j 3 : x n := x n − 1 where: • x n is a new register, which was not used before. • P ′ is obtained from P by assigning to all instructions without an index an arbitrary new index. Remark: Totality is preserved by this transformation. Semantics is the same. 19

WHILE and GOTO Proof (ctd.) Using the fact that while x i � = 0 do P end can be simulated by a GOTO program we can show (by structural induction) that every WHILE program can be simulated by a GOTO program. 20

Relationships between LOOP, WHILE, GOTO Theorem. WHILE = GOTO; WHILE part = GOTO part Proof: I. WHILE ⊆ GOTO; WHILE part ⊆ GOTO part (WHILE programs expressible as GOTO programs). Proof by structural induction. Induction basis: We show that the property is true for all atomic WHILE programs, i.e. for programs of the form x i := x i ± 1 (expressible as j : x i := x i ± 1). Let P be a non-atomic WHILE program. Induction hypothesis: We assume that the property holds for all “subprograms” of P . Induction step: We show that then it also holds for P . Proof depends on form of P . Case 1: P = P 1 ; P 2 . By the induction hypothesis, there exist GOTO programs P ′ 1 , P ′ 2 with ∆( P i ) = ∆( P ′ i ). We can assume w.l.o.g. that the indices used for labelling the instructions are disjoint. Let P ′ = P ′ 1 ; P ′ 2 (a GOTO program). We can show that ∆( P ′ )( s 1 , s 2 ) iff ∆( P )( s 1 , s 2 ) as before. Case 2: P = while x i � = 0 do P 1 end . By the induction hypothesis, there exists a 1 ). Let P ′ be the following GOTO GOTO program P ′ 1 such that ∆( P 1 ) = ∆( P ′ program: j 1 : if xi = 0 goto j 3; P ′ ; j 2 : if xn = 0 goto j 1; j 3 : xn := xn − 1 It can be checked that ∆( P ′ )( s 1 , s 2 ) iff ∆( P )( s 1 , s 2 ). 21

WHILE and GOTO Theorem. (1) WHILE = GOTO (2) WHILE part = GOTO part Proof: II. GOTO ⊆ WHILE and GOTO part ⊆ WHILE part It is sufficient to prove that every GOTO program can be simulated with WHILE instructions. 22

WHILE and GOTO Proof (ctd.) j 1 : I 1 ; j 2 : I 2 ; . . . ; j k : I k is replaced by the following while program: x index := j 1 ; while x index � = 0 do if x index = j 1 then I ′ 1 end; if x index = j 2 then I ′ 2 end; . . . if x index = j k then I ′ k end; end 23