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On Bounded Time Turing Reducibility on the Recursive Sets
Logic Colloquium '88 Ferro, Bonotto, Valentini and Zanardo (Editors) 0 Elsevier Science Publishers B.V. (North-Honand), 1989
111
O n B o u n d e d Time T u r i n g Reducibility on the Recursive Sets
TIIEODORE -4.SLAMAN In this talk, we will discuss some recent progress in the understanding of the structure the recursive sets ordered by time bounded Turing reducibility. In particular, we will focus on (deterministic) polyiiomial time ( P T I M E )Turing reductions although our techniques are completely general. Let {O,l}* be the set of finite binary strings. Lower case Greek letters u and T denote strings; IuI is the length of u ;upper case Roman letters denote subsets of (0, I}*; upper case Greek letters ip and 0 denote Turing functionals. A Turing functional ip is in P T I M E if there is a polynoinial p such t1ia.t for every string Q and every X, the computation of @(o,S)is completed in less than p(lal)many steps. Note, the definition of P T I M E makes explicit reference to an unspecified model for computation. From our point of view, any of the standard choices, including Turing machines, is acceptable. DEFINITION.A < p B if there is a 9 in PTIAfE such that @ ( B )= A. A 3 p A
B if
We consider various structures associated with these reducibilities. Let REC be the class of recursive subsets of {0, l}*. (REC,< p ) gives this class the order induced from PTIAlE reducibility. (RECI e p ,S p ) orders its PTIAfE degrees. Early results of Ladner 111 show that these structures are quite rich. For example, he showed that the orderings are dense with incomparable elements. We will discuss some results addressing their more global properties. (Shinoda-Slaman) There is a n interpretation of the arithmetic in the first order theory of (RECI ~ p
TIIEOREhr 1.
first
order theory of
Theorem 1 has the immediate corollary that (RECI L p , < p ) is an undecidable structure. T I I E O R E h i 2.
(Shore-Slaman) E o c r y finite lattice c a n be embedded in ( R E C ,Sp).
Tlie methods used to prove Tlicorein 2 are conil>incdwith Ladner's techniques to show the following.
THEOREM 3. (Shore-Slaman) The 3 V - t h c o r y of (REC,< p ) is decidable. DEFINITION. D T I M E ( n ' ) is the class of recursive subsets of {O,l}* that have an algorithm such that for soiiie constant c and for all u,the algorithm returns the value at u in ~ steps. DTIAZE(n', A ) is the analogous class defined relative to A. less than c J Q Jmany Tlie author was partially supported by Presidential Yoouiig Iiivestigator Award DAIS-6451746 and N.S.F. Grant DAIS-SG0185G.
T.A. Slaman
112
THEOREM 4. (Haught-Slaman) There i s a recursive set A and an automorphism f of the lattice of sets {X I X < p -4) such that f preserves < p and f o r some B < p A, f ( B ) $ p B. Further, we m a y chose f to preserve the DTIME(n',A) complezity classes or not, as desired.
We may view Theorem 4 from two vantage points. As a theorem about ( R E C ,< p ) , we conclude that there is a principal ideal x-ith a nontrivial automorphism. It is open whether there is a nontrivial automorpliisin of the whole structure. As a theorem about the interaction between the various DTIAlE(n'), there is an A , not in P T I M E , such that the classes DTIAfE(n',A) are not definable in ( P T I M E ( A ) ,< p ) . Let < L denote the existence of a linear time reduction. It is open whether there is an automorphism of ( P T I A I E ,< L ) or cren whether DTIAIE(n) is defiiiable in this structure.
REFERENCES 1. Ladner, R. E., On lbe slruclure of po/ytioniinl lime reducibility, Jour. Assoc. Comp. hiachinery 22 l g i 5 ),.. 155- 171. 2. Haught, C . and Slaman, T. A . , .41ifoinorplitsn1sin the P T I A l E Turing degrees of flre recursive sets, (to appear). 3. Shinoda, J. and Slaman, T. A., OIL1hc theory of 16e P T I A I E degrees o/ flie recursive s e f s , (to appear). 4. Shore, R. A. and Slaman, T. A , , Lattice emleddings i a the P T I A I E degrees of the recursive s e f s , (to appear). %
Department of hlatliematics; The University of Cliicago; Chicago, IL GOG37; U . S. A
111
O n B o u n d e d Time T u r i n g Reducibility on the Recursive Sets
TIIEODORE -4.SLAMAN In this talk, we will discuss some recent progress in the understanding of the structure the recursive sets ordered by time bounded Turing reducibility. In particular, we will focus on (deterministic) polyiiomial time ( P T I M E )Turing reductions although our techniques are completely general. Let {O,l}* be the set of finite binary strings. Lower case Greek letters u and T denote strings; IuI is the length of u ;upper case Roman letters denote subsets of (0, I}*; upper case Greek letters ip and 0 denote Turing functionals. A Turing functional ip is in P T I M E if there is a polynoinial p such t1ia.t for every string Q and every X, the computation of @(o,S)is completed in less than p(lal)many steps. Note, the definition of P T I M E makes explicit reference to an unspecified model for computation. From our point of view, any of the standard choices, including Turing machines, is acceptable. DEFINITION.A < p B if there is a 9 in PTIAfE such that @ ( B )= A. A 3 p A
B if
We consider various structures associated with these reducibilities. Let REC be the class of recursive subsets of {0, l}*. (REC,< p ) gives this class the order induced from PTIAlE reducibility. (RECI e p ,S p ) orders its PTIAfE degrees. Early results of Ladner 111 show that these structures are quite rich. For example, he showed that the orderings are dense with incomparable elements. We will discuss some results addressing their more global properties. (Shinoda-Slaman) There is a n interpretation of the arithmetic in the first order theory of (RECI ~ p
TIIEOREhr 1.
first
order theory of
Theorem 1 has the immediate corollary that (RECI L p , < p ) is an undecidable structure. T I I E O R E h i 2.
(Shore-Slaman) E o c r y finite lattice c a n be embedded in ( R E C ,Sp).
Tlie methods used to prove Tlicorein 2 are conil>incdwith Ladner's techniques to show the following.
THEOREM 3. (Shore-Slaman) The 3 V - t h c o r y of (REC,< p ) is decidable. DEFINITION. D T I M E ( n ' ) is the class of recursive subsets of {O,l}* that have an algorithm such that for soiiie constant c and for all u,the algorithm returns the value at u in ~ steps. DTIAZE(n', A ) is the analogous class defined relative to A. less than c J Q Jmany Tlie author was partially supported by Presidential Yoouiig Iiivestigator Award DAIS-6451746 and N.S.F. Grant DAIS-SG0185G.
T.A. Slaman
112
THEOREM 4. (Haught-Slaman) There i s a recursive set A and an automorphism f of the lattice of sets {X I X < p -4) such that f preserves < p and f o r some B < p A, f ( B ) $ p B. Further, we m a y chose f to preserve the DTIME(n',A) complezity classes or not, as desired.
We may view Theorem 4 from two vantage points. As a theorem about ( R E C ,< p ) , we conclude that there is a principal ideal x-ith a nontrivial automorphism. It is open whether there is a nontrivial automorpliisin of the whole structure. As a theorem about the interaction between the various DTIAlE(n'), there is an A , not in P T I M E , such that the classes DTIAfE(n',A) are not definable in ( P T I M E ( A ) ,< p ) . Let < L denote the existence of a linear time reduction. It is open whether there is an automorphism of ( P T I A I E ,< L ) or cren whether DTIAIE(n) is defiiiable in this structure.
REFERENCES 1. Ladner, R. E., On lbe slruclure of po/ytioniinl lime reducibility, Jour. Assoc. Comp. hiachinery 22 l g i 5 ),.. 155- 171. 2. Haught, C . and Slaman, T. A . , .41ifoinorplitsn1sin the P T I A l E Turing degrees of flre recursive sets, (to appear). 3. Shinoda, J. and Slaman, T. A., OIL1hc theory of 16e P T I A I E degrees o/ flie recursive s e f s , (to appear). 4. Shore, R. A. and Slaman, T. A , , Lattice emleddings i a the P T I A I E degrees of the recursive s e f s , (to appear). %
Department of hlatliematics; The University of Cliicago; Chicago, IL GOG37; U . S. A