[PDF] - Complexit y of nonrecursiv e logic programs with complex v PDF Document

SLIDE 1 Complexit y

f

nonrecursiv e logic programs with complex v alues Sergei V

rob

y

v

MPI f

ur

Informatik Saarbr

uc

k en German y Andrei V

ronk
v

Uppsala Univ ersit y Uppsala Sw eden S.V

rob

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v,

A.V

ronk
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Complexit y

f

nonrecursiv e . . . 1

SLIDE 2 Ov erview I Query languages and complexit y; I Complex v alues; I Nonrecursiv e logic programming; I T erm algebras; I Results; I T ec hniques. S.V

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Complexit y

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nonrecursiv e . . . 2

SLIDE 3 Query languages and complexit y Examples

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query languages: I Relational algebra; I First-order logic; I Logic programming. T yp es

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complexit y: 1. Complexit y (what resources do es it tak e to ev aluate a query); 2. Expressiv e p

w

er (what functions can w e express). S.V

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Complexit y

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nonrecursiv e . . . 3

SLIDE 4 Complex v alues An ything whic h is dieren t from tuples

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simple, atomic

b

jects, e.g., I Em b edded tuples; I Lists; I T rees; I Finite sets; I Finite m ultisets; I Images; I V

ices;

I HTML pages. S.V

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Complexit y

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nonrecursiv e . . . 4

SLIDE 5 Complex v alues in logic programming In logic programming v alues dep end

n

the signature used: I F unction-free signatures represen t tuples; I Signatures with unary function sym b

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represen t lists: the term f (g (h(a))) represen ts the list [f ; g ; a]. I Signatures with binary function sym b

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represen t trees. I Other complex v alues ma y b e represen ted b y using non-free constructors (e.g., set constructor), and c hanging equalit y in terpretation (unication),

r

b y using constrain ts. S.V

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Complexit y

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nonrecursiv e . . . 5

SLIDE 6 Logic programs A logic program is a nite set

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clauses, i.e., form ulas L 1 ^ : : : ^ L n

A;

where n

0,

A is an atom and L i are literals. Another notation for clauses: A L 1 ; : : : ; L n : Nonrecursiv e logic programs: the predicates are P 1 ; : : : ; P m . If the head con tains P i , then the b

dy

can

nly

use P 1 ; : : : ; P i1 . S.V

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Complexit y

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nonrecursiv e . . . 6

SLIDE 7 Sources

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expressiv e p

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er I Complex v alues, e.g., trees v ersus lists: P (f (x; x) ) Q(x): I Negation: P (x) Q(x); :R (x): I (Absence

f

) range restriction: P (f (x ); x) : S.V

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Complexit y

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nonrecursiv e . . . 7

SLIDE 8 Seman tics F

r

recursiv e programs there is no generally accepted seman tics (due to nonmonotone recursion). F

r

nonrecursiv e

nes

view a program as a set

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explicit denitions

f

predicates, using Clark's completion. F

r

example, P 3 (x; f (x)) P 1 (x; y ) P 3 (x; g (z )) P 2 (x; y ); P 2 (a; z ) denotes the explicit denition P 3 (x; u)

9y

(y = f (x) ^ P 1 (x; y ))_ 9y 9z (u = g (z ) ^ P 2 (x; y ) ^ P 2 (a; z )): Ev en tually , ev ery predicate ma y b e view ed as explicitly dened in terms

f

equalit y =. S.V

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Complexit y

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nonrecursiv e . . . 8

SLIDE 9 T erm algebra Explicit denitions

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er what? T erm algebra

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a signature

,

denoted T A() : 1. the domain is the set

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all ground terms

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; 2. ev ery term is in terpreted b y itself. The p erfect mo del

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a logic program L: the mo del induced b y the explicit denitions. Note: language-dep enden t in general, language-indep enden t for range-restricted programs. S.V

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Complexit y

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nonrecursiv e . . . 9

SLIDE 10 The SUCCESS problem SUCCESS(): sev eral equiv alen t reform ulations: 1. Giv en a logic program L and a predicate P , is P nonempt y in the p erfect mo del

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L? 2. Giv en a logic program and n ullary predicate suc c ess , is success true in the p erfect mo del

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L? 3. The com bined complexit y. 4. The program complexit y. S.V

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Complexit y

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SLIDE 11 Results function sym b

ls

no unary an y not range-restricted no negation PSP A CE NEXP NEXP with negation PSP A CE LA TIME(2 O (n) ) NONELEM (n) range-restricted no negation PSP A CE PSP A CE NEXP with negation PSP A CE PSP A CE LA TIME(2 O (n) ) S.V

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Complexit y

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nonrecursiv e . . . 11

SLIDE 12 Complexit y classes LA TIME (2 O (n) ) is the class

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problems solv able b y alternating T uring mac hines running in time 2 O (n) with a linear n um b er

f

alternations. NTIME(2 O (n) )

LA

TIME(2 O (n) )

DSP

A CE(2 O (n) ): e (n) = n; e k +1 (n) = 2 e k (n) ; e 1 (n) = e n (0): NONELEM(f (n)) is the class

f

problems with lo w er and upp er b

unds
f

the form e 1 (f (cn)) and e 1 (f (dn)). S.V

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Complexit y

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SLIDE 13 function sym b

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no unary an y not range-restricted no negation PSP A CE NEXP NEXP with negation PSP A CE LA TIME(2 O (n) ) NONELEM(n) range-restricted no negation PSP A CE PSP A CE NEXP with negation PSP A CE PSP A CE LA TIME(2 O (n) ) S | folklore; S | using complexit y

f

term algebras; S | upp er b

und

b y constrain t SLD-resolution, lo w er b y dierence lists; S | upp er b

und

b y constrain t SLD-resolution, lo w er b

und

b y tiling; S | upp er b

und

b y guessing terms

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restricted depth, lo w er trivial; S | upp er b

und

b y guessing terms

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restricted depth, lo w er using the theory

f

t w

sucessor

functions; S.V

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Complexit y

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SLIDE 14 T erm algebra and nonrecursiv e logic programs Theorem 1 T h(T A()) is p

lynomial-time

e quivalent to SUCCESS () . I SUCCESS() is PSP A CE

complete

for function-free

(Sto

c kmey er and Mey er, 1973); I SUCCESS() is LA TIME(2 O (n) )-complete for monadic

(F

erran te and Rac k

,

1979; V

lger

1983); I SUCCESS() is in NONELEM(n) for

with

binary sym b

ls

(upp er b

und

b y quan tier elimination Malcev, 1961, lo w er b

und

in V

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1996). S.V

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Complexit y

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SLIDE 15 Upp er b

unds

without negation Theorem 2 (Dantsin, V

r
nkov,

1997) If unic ation

ver

a domain D is solvable in nondeterministic p

lynomial

time, then the SUCCESS pr

blem

for lo gic pr

gr

ams without ne gation

ver

D is in NEXP. Pro

f

b y using constrain t SLD-resolution: clauses lik e P n (f (x; y )) P n1 (x); P n1 (y ) giv e exp

nen

tially long branc hes, but at the end w e solv e exp

nen

tially long sets

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equations b y using the nondeterministic unication algorithm. S.V

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Complexit y

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SLIDE 16 Upp er b

unds

with range restriction P n (f (x; y )) P n1 (x); P n1 (y ) F

r

P n to b e true

n

terms

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depth k + 1, P n1 m ust b e true

n

terms

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depth k : term depth gro ws slo wly. S.V

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Complexit y

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SLIDE 17 Lo w er b

unds

without negation The Tiling Problem: can w e co v er b y tiles the square

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size 2 n

2

n b y tiles from a giv en nite set, suc h that some conditions

n

adjacen t tiles are held? Represen t tilings b y terms: the term [t 1 ; t 2 ; t 3 ; t 4 ] represen ts the tiling t 1 t 2 t 3 t 4 S.V

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Complexit y

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SLIDE 18 Lo w er b

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without negation (con t.) 2 n+1 tiles z }| { 2 n+1 tiles z }| { X 1 X 2 Y 1 Y 2 X 3 X 4 Y 3 Y 4 Z 1 Z 2 U 1 U 2 Z 3 Z 4 U 3 U 4 X 1 X 2 X 3 X 4 X 2 Y 1 X 4 Y 3 Y 1 Y 2 Y 3 Y 4 X 3 X 4 Z 1 Z 2 X 4 Y 3 Z 2 U 1 Y 3 Y 4 U 1 U 2 Z 1 Z 2 Z 3 Z 4 Z 2 U 1 Z 4 U 3 U 1 U 2 U 3 U 4 S.V

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SLIDE 19 Enco ding tiling i+1 ([ [X 1 ; X 2 ; X 3 ; X 4 ] ; [ Y 1 ; Y 2 ; Y 3 ; Y 4 ]; [Z 1 ; Z 2 ; Z 3 ; Z 4 ]; [U 1 ; U 2 ; U 3 ; U 4 ]] ; T ) tiling i ([X 1 ; X 2 ; X 3 ; X 4 ]; T ); tiling i ([X 2 ; Y 1 ; X 4 ; Y 3 ]; ); tiling i ([Y 1 ; Y 2 ; Y 3 ; Y 4 ] ; ); tiling i ([X 3 ; X 4 ; Z 1 ; Z 2 ] ; ); tiling i ([X 4 ; Y 3 ; Z 2 ; U 1 ] ; ); tiling i ([Y 3 ; Y 4 ; U 1 ; U 2 ]; ); tiling i ([Z 1 ; Z 2 ; Z 3 ; Z 4 ]; ); tiling i ([Z 2 ; U 1 ; Z 4 ; U 3 ]; ); tiling i ([U 1 ; U 2 ; U 3 ; U 4 ]; ): S.V

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Complexit y

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SLIDE 20 Dierence lists Unary function sym b

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without range restriction: can create big dierences using dierence lists. dif (f (x); x) dif (g (x); x) : : : dif n+1 (x; y ) dif n (x; z ); dif n (z ; y ) dif n (s; t) holds if and

nly

if the \dierence" b et w een s and t is 2 n . Using large dierence, w e can still enco de the Tiling Problem . . . S.V

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Complexit y

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SLIDE 21 Bags and sets Some results ab

ut

bags and sets: 1. Without negation NEXP

complete

(Dan tsin and V

ronk
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1997). 2. Unrestricted: undecidable. S.V

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Complexit y

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nonrecursiv e . . . 21