Expansions with P = NP. Christine Gaßner Greifswald

Transcription

1 Expansions with P = NP Greifswald

2 Expansions with P = NP The complexity classes P Σ, NP Σ, DEC Σ 3. Why is it difficult to find an R with P ΣR = NP ΣR? 4. How to construct a relation R such that SAT ΣR P ΣR?

3 1. over structures Σ A structure: constants operations relations Σ = (U ; c 1,, c u ; f 1,, f v ; R 1,, R w, [=]) finite universe or infinite universe finite signature Examples: Σ bin = ({0, 1}; 0, 1; ; =) ( Turing machines), Σ R = ({a, b}*; ε; add a, add b, sub a, sub b ; R,, =) the empty string

4 The Σ-machines M Registers for elements of U : Z 1, Z 2, Z 3, (U the universe) Registers for indices: I 1, I 2,, I km Z 1 Z 2 Z 3 Z 4 Z 5 Z 6 Z 7 Z 8 Z 9 Z 10 Z 11 I 1 I 2 I km

5 The Σ-machines (input) The input: (Z 1,, Z n ) (x 1,, x n ); I 1 n; I 2 1; I k 1; M x 1 x 2 x 3 x 4 The size of the input Z 1 Z 2 Z 3 Z 4 Z 5 Z 6 Z 7 Z 8 Z 9 Z 10 Z 11 I 1 4 x 1 x 2 x 3 x 4 x 4 x 4 x 4 x 4 x 4 I 2 1 Z 1 Z 4 I km 1 Z 1

6 The Σ-machines (output) The output: : (Z( 1,, Z I1 ) The size of the output Z 1 Z 2 Z 3 Z 4 Z 5 Z 6 Z 7 Z 8 Z 9 Z 10 Z 11 I I I km 1

7 The Σ-machines (instructions) Structure: Σ = (U ; c 1,, c u ; f 1,, f v ; R 1,, R w, =) Computation: l: Z k f j (Z,, Z k ); 1 kf j l: Z k c j ; Branching: Copy: l: Z Z I k Ij ; l: if R j (Z,, Z k ) then goto l 1 kr 1 else goto l 2 ; j l: if Z k = Z j then goto l 1 else goto l 2 ; Index computation: I k 1; I k I k +1; if I k = I j then goto l 1 else goto l 2 ;

8 The non-deterministic machines The guessing: (Z n+1,, Z n+m ) (y 1,, y m ) U m non-deterministic deterministic! The size of the input I 1 I 2 I km Z 1 Z 2 Z 3 Z 10 x 1 x 2 x 3 x 4 y 1 y 2 y 3 y 4 y 5 Z 1 Z 4 Z 1 Z 4 Z 5 Z 6 Z 7 Z 8 Z 9 y 1 y 2 y 3 y 4 y 5... Z 11 Arbitrary elements can be guessed!

9 Representations of a Σ-machine A program 1: Input: (x 1,, x n ). Guess: (y 1,, y n ) {0, 1}. 2: I 2 I 1 + 1; 3: Z Z 1 * Z I 2 ; 4: if I 1 = I 3 then goto 9 else goto 5; 5: I 2 I 2 + 1; 6: I 3 I 3 + 1; 7: Z 1 Z 1 + Z I 3 * Z I2 ; 8: goto 4; 9: if Z 1 = 1 then goto 11 else goto 10; 10: Z 1 0; 11: I 1 1; 12: Output: Z 1. A flowchart Computation paths

10 2. The complexity classes DEC Σ, P Σ, NP Σ Decidable, (non-) deterministically recognizable in polynomial time Every element can be stored in one register! One operation can be executed in one time unit (one step)! Computation in polynomial time means output after p(n) ) steps for any (x( 1,, x n ) and some polynomial p. The usual problems (for structures with two constants): The Satisfiability Problem SAT Σ UNI Σ H Σ (Compare: the Blum-Shub Shub-SmaleSmale model introduced in 1989, the algebraic programming systems considered by Asveld Asveld and Tucker in 1982.) The NP-complete problem recognized by a usual universal machine The Halting Problem

11 A question POIZAT (1995): Is there a structure of finite signature with P = NP?

12 3. Why is it difficult to find an R with P = NP? ΣR ΣR The idea: Some R with SAT ΣR P ΣR How can a tuple be encoded by one element? The undecidability of SAT ΣR for some R

13 The idea: A new relation R with SAT ΣR P ΣR tuple of elements (x 1,..., x n, code(ψ )) SAT ΣR? reduction? The first problem: Σ R R(CODE(x 1,..., x n, ψ ))? one element How can a tuple be encoded by one element?

14 How can a tuple be encoded by one element? Solution: We extend Σ = ( U; c 1,, c u ; f 1,, f v ; R 1,, R w, =). Σ R = (A; ε, a, b, c 1,, c u ;?,?,?, f 1 ',, f v '; R 1 ',, R w ', R, =) operations for concatenation, A = U* f j '(d,, d k 1 k fj ) = f j (d,, d k 1 k ); f fj j '(s k1,, s k fj ) = ε (x 1,..., x n, code(ψ )) SAT ΣR? d k1,, d kfj U, (s k1,, s kfj ) U k fj tuple of strings reduction Σ R R(string(x 1,..., x n,ψ ))? concat one string

15 Why is it difficult to find an R with SAT ΣR DEC? ΣR For many relations R : The assumption of the decidability of SAT ΣR (by means of R) implies the decidability of H ΣR. SAT ΣR DEC ΣR.

16 Some R h with SAT halt DEC Σ R h Σ R = ({a, b}*; ε; ; operations for concat.. ; R, =). TM set of all deterministic Turing machines. R h (s) = true if s=rcode TM (M), M TM, M halts on Code TM (M) after r steps false otherwise SAT halt = df {code(r h (Y 1 d 1 d k 1 )) M TM & Σ Rh Y 1 R h (Y 1 d 1 d k 1 )} SAT halt DEC ΣRh Code TM (M) Simulation by a TM H TM TM DEC TM H TM TM DEC TM SAT halt DEC ΣRh SAT ΣRh DEC ΣRh

17 4. How to construct a relation R such that SAT P? ΣR ΣR The idea: Σ R R(string(x, ψ )) (x, code(ψ )) SAT ΣR R is definable SAT ΣR is decidable! Can we find an R such that SAT ΣR is decidable? Can we give a definition of R such that SAT P ΣR Σ R?

18 Is R definable such that R(string(x,ψ )) Y ψ (x,y)? Problems: ψ can contain R. The guesses for Y 1,, Y m can be the codes of formulae. Can all possible guesses for any formula be finitely described by one quantifier-free free (,, )-formula over Σ R.? Solution: Definition of R such that R and SAT ΣR can be described by a tree which is similar to a tree of computation paths.

19 The description of SAT ΣR by a tree d k d k 1 d k 2 d 2 d 1 d 1 d k 1 d k A

20 The description of SAT ΣR by a tree d k d k 1 d k 2 d 2 d 1 d 1 d k 1 d k A The codes of the tuples in SAT ΣR A k in SAT ΣR A k+1 in SAT ΣR A k+2

21 Some R with SAT ΣR DEC ΣR R( x 1,..., x n, code(ψ) dbl ) = true iff (x 1,..., x n, code(ψ )) SAT ΣR a x, code(ψ)) x, code(ψ)) s A s dbl = df sa s A s 1,..., s k A code of (s 1,..., s k ) A k

22 Some R with SAT ΣR DEC ΣR a a x, code(ψ)) Padding of strings (by doubling the length) a a a R( x 1,..., x n, code(ψ) dbl ) = true iff (x 1,..., x n, code(ψ )) SAT ΣR a a x, code(ψ)) s A s dbl = df sa s A s 1,..., s k A code of (s 1,..., s k ) A k

23 The new operations for slow (!!!) computation Σ = ( U ; c 1,, c u ; f 1,, f v ; R 1,, R w, =), Σ R = (A; ε, a, b, c 1,, c u ; add, sub l, sub r, f 1 ',, f v '; R 1 ',, R w ', R, =) add(s, d) ) = sd sub l (sd)) = s sub r (sd)) = d s A, d U s d d k d k 1 d 2 d 1 s = d 1 d k 1 d k add(s, d) = d 1 d k d

24 Similar trees A tree of computation paths The paths corresponding to the strings satisfying R

25 Similar trees A tree of computation paths The paths corresponding to the strings satisfying R contains the inputs traversing the red path

26 Similar trees A tree of computation paths The paths corresponding to the strings satisfying R contains the inputs traversing the red path contains the codes of the k-tuples belonging to SAT Σ R Ak

27 Trees and polynomial time A tree of computation paths The paths corresponding to the strings satisfying R a cut a a cut

28 Trees and polynomial time A tree of computation paths The paths corresponding to the strings satisfying R cut cut We cannot distinguish between the inputs traversing these paths We cannot distinguish between the codes of the tuples which satisfy R and which begin here

29 The recursive definition is possible R and SAT ΣR can be described by a tree which is similar to a tree of computation paths. Reduction of the tuples (x 1,..., x n, code(ψ)) to small strings. Replacement of arbitrary guesses by small guesses. Restriction of the quantifier domains. Recursive definition of R.

30 Let us consider some details in the next talk.

31 Thank you for your attention! I thank Gerald van den Boogaart, Volkmar Liebscher, Rainer Schimming, Michael Schürmann of the Department of Mathematics and Computer Science, Ernst Moritz Arndt University of Greifswald for discussion on the presentation.

32 Appendix

33 DEC Σ, P Σ, NP Σ Decidable, (non-) ) deterministically recognizable in polynomial time The size of an input (x( 1,, x n ): n. Every u U can be stored in one register. The execution of any instruction: one time unit (one step). The execution of one operation = one time unit. Computation in polynomial time: output after p(n) ) steps for any input (x( 1,, x n ) and some polynomial p. Non-deterministic / deterministic acceptance: Deterministic rejection: P Σ NP Σ, P Σ DEC Σ output a (or halt). output b (or no halt). (a,( b U, a b) NP Σ DEC Σ P Σ NP Σ

34 NP-complete problems, Halting Problem The Satisfiability Problem SAT Σ = {(x, code(ψ )) U ψ L Σ & Σ Y ψ(x, Y )} The NP- complete problem recognized by a usual universal machine UNI Σ = {(x 1,, x n, Code(M), a,, a) U n+k+ t M NM Σ & M accepts (x 1,, x n ) within t steps} U The Halting Problem H Σ = {(x, Code(M )) U M M Σ & M halts on x} U the universe, L Σ formulae, NM Σ non-det det.. machines, M Σ det.. machines

35 SAT Σ Satisfiability Problem Σ = (U ; a, b, c 3,, c u ; f 1,, f v ; R 1,, R w, =) L Σ set of quantifier- free (,, )-formulae ψ L Σ (=,, R 1,, R w, R 1,, R w )-literals (a, b, c 3,, c u, f 1,, f v )-terms. code(ψ) {a, b} U = df U k 1 U k SAT Σ = {(x 1,, x n, code(ψ )) U ψ L Σ & Σ (Y 1,, Y m ) ψ(x 1,, x n, Y 1,, Y m )} SAT Σ NP Σ, SAT Σ is NP Σ -complete. complete. SAT Σ P P Σ Σ = NP Σ

36 UNI Σ and H Σ Recognition by universal machine, Halting Problem NM Σ set of all non-deterministic Σ-machines. UNI Σ = {(x 1,, x n, Code(M), a,, a) U n+k+ t M NM Σ & M accepts (x 1,, x n ) within t steps} UNI Σ NP Σ, UNI Σ is NP Σ -complete. UNI Σ P Σ P Σ = NP Σ M Σ set of all deterministic Σ-machines. H = Σ {(x 1,, x n, Code(M)) (x 1,, x n ) U n & M M Σ & M halts on (x 1,..., x n )} H Σ DEC Σ

37 Some historical remarks P versus NP for special structures GÖDEL, TURING, CHURCH, KLEENE, COOK, KARP (1931, 1936, 1971, 1972): The classical theory of computation ENGELER, FRIEDMAN, MANSFIELD (1967, 1971): Computation over structures for functions of fixed arities ASVELD, TUCKER (1980, 1982): Deterministic and binary non-deterministic Satisfiability Problem BLUM, SHUB, SMALE (1989): CUCKER (1990): deterministic algebraic programming systems and an abstract over the real numbers Investigation of computation paths

38 Some historical remarks P versus NP for special structures MEER (1992, 1993): P NP for (R ; 0, 1; +, ; =) and for (R ; 0, 1; sin, cos,, +, ; =) KOIRAN (1994): DNP = NP for (R ; 0, 1; +, ; ) DN digitally (binary) non-deterministic: y 1,, y m {0, 1} POIZAT (1995): The question: Is there a structure of finite signature with P = NP? PRUNESCU (2001, Greifswald): Talk on Poizat s idea to define an additional relation for a structure over {0,1}* which implies P = NP MAINHARDT (2001, Greifswald): A structure of infinite signature with P = NP

39 Some R h with SAT halt DEC SAT halt DEC ΣRh (The idea of the proof) R h (s) iff s = r Code TM (M ), M TM, M halts on Code TM (M ) after r steps Assume: P 0 decides SAT P 0 SAT halt code(r h (Y 1 d 1 d k 1 )) Code TM (M ) = d 1 d k 1 M 0 Acceptance iff M halts on Code TM (M ) Contradiction since H TM DEC TM Simulation of P 0 by a TM M 0 Simulation of any test R h (rcode TM (M s )) by simulation of r steps of M s H TM TM DEC TM SAT halt DEC ΣRh SAT ΣRh DEC ΣRh P ΣRh NP ΣRh

40 The description of SAT Σ by a tree R for Σ R = ({a, b}*; ε; add a, add b, sub a, sub b ; R, =) d k d k 1 d k 2 d 2 d 1 d 1 d k 1 d k A

41 The description of SAT Σ by a tree R for Σ R = ({a, b}*; ε; add a, add b, sub a, sub b ; R, =) d k d k 1 d k 2 d 2 d 1 d 1 d k 1 d k A SAT ΣR A k SAT ΣR A k+1 SAT ΣR A k+2

42 The description of SAT Σ by a tree R for Σ R = ({a, b}*; ε; add a, add b, sub a, sub b ; R, =) d k d d R( x k 1 k 2 1,..., d x n, d 1 code(ψ) 1 dbl d k 1 ) = dtrue k A iff (x 1,..., x n, code(ψ )) SAT ΣR a x, code(ψ)) Padding of strings (by doubling the length) s A s dbl = sa s A x, code(ψ)) s 1,..., s k A code of (s 1,..., s k ) A k

43 Σ R = ({a, b}*; ε; add Trees for add a, add b, sub a, sub b ; R, =) A tree of computation paths The paths corresponding to the strings satisfying R contains the inputs traversing the red path contains the codes of the k- tuples belonging to SAT Σ R Ak

44 Trees and polynomial time for Σ R = ({a, b}*; ε; add a, add b, sub a, sub b ; R, =) A tree of computation paths The paths corresponding to the strings satisfying R cut cut

45 Trees and polynomial time for Σ R = ({a, b}*; ε; add a, add b, sub a, sub b ; R, =) A tree of computation paths The paths corresponding to the strings satisfying R cut cut We cannot distinguish between the inputs traversing these paths We cannot distinguish between the codes of the tuples beginning here