Complexity of Query Answering in Logic. Databases with Complex Values. Evgeny Dantsin. Steklov Institute of Mathematics at St.

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1 UPMAIL Technicl Report No. 149 November 27, 1997 Revised Februry 16, 1998 ISSN Complexity of Query Answering in Logic Dtbses with Complex Vlues Evgeny Dntsin Steklov Institute of Mthemtics t St.Petersburg Fontnk 27, St.Petersburg Russi Andrei Voronkov y Computing Science Deprtment Uppsl University Box 311, S Uppsl Sweden emil: fdntsin,voronkovg@csd.uu.se Supported by grnts from the Swedish Royl Acdemy of Sciences, the Swedish Institute nd RFBR/INTAS. Author's emil ddress t St.Petersburg: dntsin@pdmi.rs.ru y Supported by TFR grnt.

2 Abstrct This pper chrcterizes the computtionl complexity of nonrecursive queries in logic dtbses with complex vlues. Queries re represented by Horn cluse logic progrms. Complex vlues re represented by terms in equtionl theories (nite sets nd multisets re exmples of such complex vlues). We show tht the problem of whether query hs nonempty nswer is NEXP-hrd for nonrecursive rnge-restricted queries. We lso show tht this problem is in NEXP if complex vlues stisfy the following condition: the solvbility problem for equtions in the corresponding equtionl theory is in NP. Since trees, nite sets nd multisets stisfy this condition, the query nswering problem for logic dtbses with trees, nite sets nd multisets is shown to be NEXP-complete. 2 2 Copyright c 1997, 1998 Evgeni Dntsin nd Andrei Voronkov. This technicl report nd other technicl reports in this series cn be obtined t or t ftp.csd.uu.se in the directory pub/ppers/reports. Some reports cn be updted, check one of these ddresses for the ltest version.

3 Contents 1 Introduction 1 2 Preliminries Complex vlues Logic progrms with complex vlues NEXP-hrdness 4 4 Inclusion in NEXP 7 1 Introduction Reltionl query lnguges typiclly del with tuples of tomic vlues like strings or integers. Applictions require to hndle more complex vlues, for exmple sets or multisets. Vrious kinds of complex vlues in dtbses nd logic progrmming hve been considered in mny ppers, including [2, 3, 15, 6, 22, 20, 16, 5, 1, 14, 13, 10, 9, 23]. Our pper studies complexity spects of dtbses with complex vlues. There re two min kinds of complexity in dtbses: (1) query evlution complexity tht mesures time or spce needed to evlute query, nd (2) descriptive complexity tht chrcterizes the expressive power of query lnguges. In turn, query evlution complexity is considered in three forms: dt complexity, progrm complexity nd combined complexity. The dt complexity is expressed s function of the size of dt (represented by ground toms) for xed query (represented by logic progrm nd gol). The progrm complexity mens tht query is vrible, while dt re xed. The combined complexity considers both dt nd query to be vrible. By the query nswering problem we men the problem of whether given gol succeeds with respect to given logic progrm. In terms of logic dtbses, this problem is formulted s follows: given dtbse instnce nd query, does the outputted reltion contin t lest one tuple? Thus, the query nswering complexity, i.e. the complexity of the bove problem, is vrint of the combined complexity, nmely the combined complexity of Boolen queries. This kind of complexity seems to be especilly relevnt to dtbses with complex vlues since in this cse n nswer cn be too lrge or even innite. For Horn cluse logic progrms, the query nswering problem cn be nturlly formlized s the decision problem for corresponding frgments of rst-order logic. For non-horn cluse logic progrms, decidbility nd complexity hve been minly considered in the context of non-monotonic semntics ([7]). As for logic progrms with complex vlues, no previous complexity results re known to us. This pper chrcterizes the query nswering complexity for nonrecursive Horn cluse logic progrms with complex vlues. Complex vlues re represented by terms in equtionl theories (see detils in Section 2). Trees, nite sets nd multisets re exmples of such complex vlues. We show tht this problem for nonrecursive rnge-restricted logic progrms is NEXP-hrd. We lso show tht the query nswering problem for nonrecursive logic progrms is in NEXP if complex vlues stisfy the following condition: the solvbility problem for equtions in the corresponding equtionl theory is in NP. Since trees, nite sets nd multisets stisfy this condition, the query nswering problem for logic dtbses with trees, nite sets nd multisets is shown to be NEXPcomplete.

4 This pper is structured s follows. In Section 2 we dene wht we men by complex vlues nd dene nottion for logic progrms with complex vlues. In Section 3 we prove NEXP-hrdness of the query nswering problem for nonrecursive rnge-restricted logic progrms. In Section 4 we prove the inclusion in NEXP for complex vlues stisfying the condition mentioned bove. 2 Preliminries 2.1 Complex vlues Equtionl theories. In order to dene wht we men by complex vlues, we rst dene the notion of n equtionl theory. Let F be set of function symbols. As usul, nonnegtive rity is ssigned to every function symbol. Constnts re function symbols of rity 0. We ssume tht F contins t lest one constnt. The notion of term is dened in the stndrd wy: constnt is term; vrible is term; if t 1 ; : : :; t n re terms nd f is function symbol of rity n then f(t 1 ; : : :; t n ) is term. (Some vritions re possible, for exmple vribles my hve sorts). A term without vribles is clled ground. By n eqution we men n expression s = t. An eqution is sid to be ground if s nd t re ground. An equtionl theory E over F is set of ground equtions closed under the logicl consequence reltion, i.e. such tht 1. E contins ll ground equtions of the form t = t. 2. If E contins s = t then E contins t = s. 3. If E contins r = s nd s = t then E contins r = t. 4. If E contins s 1 = t 1 ; : : :; s n = t n then E contins f(s 1 ; : : :; s n ) = f(t 1 ; : : :; t n ) for ech n-ry function symbol f 2 F. Bsiclly, ech equtionl theory E denes domin D E of complex vlues: the elements of D E re the equivlence clsses of ll ground terms over F with respect to the equlity reltion in E. Exmples of complex vlues. Trees cn be viewed s trivil exmple of complex vlues. Nmely, trees re vlues represented by ground terms, nd the equlity reltion on them is dened s identity. Thus, equtionl theories for trees consist of only equtions t = t. Lists re specil kind of trees. In logic progrmming, lists re constructed from the constnt [] (clled the empty list) by using the binry function symbol [s j t] (clled the list constructor). If term t represents list then term [s j t] represents the list obtined from the list t by dding the term s s the rst element. Equtionl theories for lists re the sme s for trees. Bgs (or nite multisets in nother terminology) re lists in which the order of elements is immteril. We denote bgs similr to lists, but using the empty bg hi insted of the empty list [], the bg constructor hs j ti insted of the list constructor [s j t] nd ht 1 ; : : :; t n i insted of [t 1 ; : : :; t n ]. Bgs cn be xiomtized by ground equtions hr j hs j tii = hs j hr j tii nd their logicl consequences. Sets cn be viewed s bgs in which the number of occurrences of n element is immteril. Similr to bgs, we use the empty set fg nd the set constructor fs j tg tht denotes the set obtined from the set t by dding s s n element. Sets cn be xiomtized s follows: (1) in the xioms for bgs, we replce the empty bg nd the bg constructor by the empty set nd the set

5 constructor respectively, nd (2) we dd ll ground equtions hs j ti = hs j hs j tii. Note tht we dened hereditrily nite sets, i.e. nite sets whose elements cn be other (hereditrily nite) sets. There re other wys of integrting nite sets into logic progrmming. For exmple, in [10] sets re represented by terms fs 1 ; : : :; s n j tg where t cn be ny term not necessrily representing set (the uthors cll such objects colored sets). Thus, set is chrcterized not only by its elements but lso by some \type" (color). Our pproch is more trditionl: every til subterm t in fs 1 ; : : :; s n j tg is required to represent some set. 2.2 Logic progrms with complex vlues Herbrnd interprettions. Let be signture consisting of set F of function symbols (dened s bove) nd set of predicte symbols. We ssume tht contins t lest one constnt nd does not contin the equlity symbol =. Atoms, substitutions, instnces, vrints etc. re dened in the stndrd wy (see, for exmple, [4]). The Herbrnd universe U for is dened s the set of ll ground terms over. The set of ll ground toms over is clled the Herbrnd bse B for. By Herbrnd interprettion I for we men ny subset of B (ll toms in I re considered to be true, ll other ground toms re considered to be flse). Let E be n equtionl theory. Let I B be Herbrnd interprettion such tht if I contins n tom p(s 1 ; : : :; s n ) nd E contins equtions s 1 = t 1 ; : : :; s n = t n ; then I contins p(t 1 ; : : :; t n ). In this cse we cll I Herbrnd interprettion with respect to the equtionl theory E. Logic progrms. We x signture nd ssume tht ll terms nd toms below re terms nd toms over. A cluse is ny formul of the form A 1 ^ : : : ^ A n A, where A 1 ; : : :; A n ; A re toms nd n 0. Such cluse is usully written s A :- A 1 ; : : :; A n : nd simply s A when n = 0. The tom A is clled the hed, nd the sequence A 1 ; : : :; A n is clled the body of this cluse. By the length of the body we men the number n. A logic progrm is nite set of cluses. A gol is conjunction A 1 ^ : : :^ A n of toms, written s A 1 ; : : :; A n where n 0. When n = 0, it is clled the empty gol nd denoted by 2. By logic progrm with complex vlues we men pir (P; E) where P is logic progrm nd E is n equtionl theory. We lso cll such pir logic progrm over n equtionl theory. In prticulr, we sy tht logic progrm P over n equtionl theory E is logic progrm with sets if E is the equtionl theory of sets over F. Similrly, we use the terms logic progrms with bgs, trees, etc. Semntics. The semntics of logic progrms over equtionl theories cn be dened in mny wys by dpting the stndrd logic progrmming semntics [4, 17, 12] for the corresponding equtionl theory. For exmple, one cn use the lest xpoint semntics or generliztion of SLD-resolution. We dene the semntics by using the lest Herbrnd model. Let A :- A 1 ; : : :; A n be cluse without vribles. This cluse is sid to be true in Herbrnd interprettion I if A belongs to I whenever A 1 ; : : :; A n belong to I. We sy tht I is Herbrnd model for logic progrm P if ll ground instnces of ll cluses of P re true in I. If, in ddition,

6 I is Herbrnd interprettion with respect to n equtionl theory E then we cll I Herbrnd model for P over E. Like the cse of ordinry logic progrms, Herbrnd models for logic progrms over equtionl theories hve the following property: the intersection of Herbrnd models for P over E is gin Herbrnd model for P over E. This property llows us to dene the lest Herbrnd model for logic progrm with complex vlues: it is the smllest Herbrnd model for P over E (with respect to the subset reltion ). Let P be logic progrm over n equtionl theory E nd G be gol. We sy tht G succeeds with respect to P over E if there exists ground instnce A 1 ; : : :; A n of G such tht ll A 1 ; : : :; A n belong to the lest Herbrnd model of P over E. In other words, the formul 9(A 1 ^ : : : ^ A n ) is true in the lest Herbrnd model of P over E. Nonrecursive progrms. A logic progrm P is sid to be nonrecursive if ll predicte symbols in P cn be numbered so tht the following condition is stised for every cluse in P: the number of the predicte in the hed is greter thn the number of ech predicte in the body. Rnge-restricted logic progrms. A logic progrm P is clled rnge-restricted if every cluse A :- A 1 ; : : :; A n in P hs the property: ll vribles of A occur in A 1 ; : : :; A n. Rnge-restricted queries ply n importnt role in dtbses becuse they re domin independent (see [21] for exmple). The query nswering problem. Consider set C of logic progrms over E. By the query nswering problem for C we men the following decision problem: given progrm P 2 C nd gol G, does G succeed with respect to P over E? Note tht the complexity with respect to the combined size of the progrm nd the gol is equivlent to the complexity with respect to the size of the progrm. Let P be progrm, A 1 ; : : :; A n be gol, nd yes be nullry predicte symbol foreign to. Consider the progrm P 0 = P [ fyes :- A 1 ; : : :; A n g. Evidently, G succeeds with respect to P if nd only if the gol consisting of yes succeeds with respect to P 0. This trnsformtion of P into P 0 preserves the clsses C of logic progrms considered in this pper (nonrecursive nd/or rnge-restricted progrms). 3 NEXP-hrdness Theorem 3.1 The query nswering problem for the clss of nonrecursive rnge-restricted logic progrms with lists is NEXP-hrd. Proof. We reduce the TILING problem, which is known to be NEXP-complete (see for exmple [19], pge 501), to the query nswering problem for nonrecursive rnge-restricted logic progrms with lists. Informlly, TILING is the problem of tiling the squre of size 2 n 2 n by tiles (squres of size 1 1). More precisely, there is nite set ft 0 ; : : :; t k g of tiles nd there re two binry reltions on nd to dened on the tiles. Tiles t i nd t j re sid to be verticlly comptible if on(t i ; t j ) holds nd, similrly, horizontlly comptible if to(t i ; t j ) holds. A tiling of the squre of size 2 n 2 n is function f from f1; : : :; 2 n g f1; : : :; 2 n g into ft 0 ; : : :; t k g such tht 1. f(i; j) nd f(i + 1; j) re verticlly comptible, for ll 1 i < 2 n nd 1 j 2 n ; 2. f(i; j) nd f(i; j + 1) re horizontlly comptible, for ll 1 i 2 n nd 1 j < 2 n.

7 We lso sy tht such f is tiling with t i t the top left corner if f(1; 1) is t i. The TILING problem is dened s follows. Suppose tht we re given set ft 0 ; : : :; t k g of tiles, comptibility reltions on nd to, nd number n (written in unry nottion). The question we sk is whether there is tiling of the squre of size 2 n 2 n with t 0 t the top left corner. The reduction we describe is polynomil-time lgorithm tht trnsforms every instnce I of the TILING problem into logic progrm P nd gol G such tht I hs tiling if nd only if G succeeds with respect to P. We think of tiles t 0 ; : : :; t k in I s constnts of the progrm P. The comptibility reltions on nd to re represented in P by the corresponding predictes: nmely, P contins cluses on(t i ; t j ): to(t l ; t m ): for ll pirs of comptible tiles. To proceed the construction of P nd G, we generlize tiles in nturl wy. A hypertile of rnk i is dened by induction s squre of size 2 i 2 i consisting of tiles. A hypertile of rnk 0 is ny tile of the set ft 0 ; : : :; t k g. Let H 1 ; H 2 ; H 3 nd H 4 be hypertiles of rnk i, then the qudruple (H 1 ; H 2 ; H 3 ; H 4 ) is hypertile of rnk i + 1. We think of this hypertile s the squre H 1 H 2 H 3 H 4 nd we represent it in P by the list [H 1 ; H 2 ; H 3 ; H 4 ] where H 1 ; H 2 ; H 3 nd H 4 re lists tht represent the corresponding hypertiles of rnk i. Here is n exmple of hypertile of rnk 2 (formed from tiles ; b nd c) nd its representtion by list: b c c c b b [[b,,c,],[c,,b,b],[c,,b,b],[,,,c]] b b c Clerly, we cn identify hypertiles of rnk i with functions from f1; : : :; 2 i g f1; : : :; 2 i g into ft 0 ; : : :; t k g. We cll hypertile tiling if the corresponding function is tiling. In ddition to the predictes on nd to, the progrm P denes n binry predictes tiling 1,..., tiling n. A sttement tiling i (H; T 0 ) is intended to hold if nd only if (i) H is hypertile of rnk i; (ii) H is tiling with the tile T 0 t the top left corner. Thus, the gol G consisting of one tom tiling n (H; t 0 ) will succeed with respect to P if nd only if there is tiling for given ft 0 ; : : :; t k g, on, to nd n. The predicte tiling 1 is dened in the following obvious wy:

8 X 1 X 2 X 3 X 4 tiling 1 ([X 1 ; X 2 ; X 3 ; X 4 ]; X 1 ) :- on(x 1 ; X 3 ); on(x 2 ; X 4 ); to(x 1 ; X 2 ); to(x 3 ; X 4 ): To dene tiling i+1, we use the following observtion. Consider hypertile H [[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 ] ] of rnk i + 1 nd 9 its subsqures, hypertiles of rnk i, shown below X 1 X 2 X 2 Y 1 Y 1 Y 2 X 3 X 4 X 4 Y 3 Y 3 Y 4 X 1 X 2 Y 1 Y 2 X 3 X 4 Y 3 Y 4 X 3 X 4 X 4 Y 3 Y 3 Y 4 Z 1 Z 2 U 1 U 2 Z 1 Z 2 Z 2 U 1 U 1 U 2 Z 3 Z 4 U 3 U 4 Z 1 Z 2 Z 2 U 1 U 1 U 2 Z 3 Z 4 Z 4 U 3 U 3 U 4 It is esy to see tht H is tiling if nd only if these 9 hypertiles re tilings too. Thus, tiling i+1 cn be dened s follows: 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 0 ) :- tiling i ([X 1 ; X 2 ; X 3 ; X 4 ]; T 0 ); 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 ]; ): The progrm P described bove is nonrecursive nd rnge-restricted. Clerly, P nd the gol G re constructed from the instnce I of TILING in polynomil time. 2 Theorem 3.2 The query nswering problem is NEXP-hrd for the following clsses of nonrecursive rnge-restricted logic progrms:

9 logic progrms with trees in ny signture with t lest one function symbol of rity 2; logic progrms with bgs; logic progrms with sets; logic progrms with ny combintion of the bove types of complex vlues (for exmple, logic progrms with trees nd bgs). Proof. The progrm P from the proof of Theorem 3.1 contins the list constructor, but insted we could hve used ny function symbol of rity 2 if this symbol is dierent from the bg nd set constructors. Thus, NEXP-hrdness holds for nonrecursive logic progrms with trees. How to construct P in the cse when contins only the bg constructor nd/or the set constructor? We cn simulte lists [s 1 ; s 2 ; s 3 ; s 4 ] by mens of bgs or sets, i.e. we cn use bgs nd sets to dene predictes equivlent to the predictes of P. For exmple, list [s 1 ; s 2 ; s 3 ; s 4 ] cn be represented by the set of pirs (s 1 ; 1); : : :; (s 4 ; 4) where 1; : : :; 4 re, in turn, represented by sets of crdinlities 1; : : :; Inclusion in NEXP Let E be n equtionl theory. We sy tht n eqution s = t hs solution in E if is substitution such tht the eqution s = t belongs to E (where t nd s denote the results of ppliction of to s nd t respectively). A system of equtions is nite set of equtions. A system of equtions is sid to be solvble in E if there exists substitution such tht is solution for ech eqution in the system. By the solvbility problem for E we men the following decision problem: given system of equtions, is the system solvble in E? Theorem 4.1 Let E be n equtionl theory such tht the solvbility problem for E is in NP. Then the query nswering problem for nonrecursive logic progrms over E is in NEXP. Proof. Let P be logic progrm over E. We dene binry reltion! on pirs (G; E), where G is gol nd E is system of equtions. Nmely, we hve if the following conditions re stised: 1. G contins n tom p(t 1 ; : : :; t n ). (G; E)! (G 0 ; E 0 ) 2. Some cluse in P hs vrint p(s 1 ; : : :; s n ) :- B 1 ; : : :; B m whose vribles re disjoint with the vribles of (G; E). 3. G 0 is (G n fp(t 1 ; : : :; t n )g) [ fb 1 ; : : :; B m g; 4. E 0 is E [ fs 1 = t 1 ; : : :; s n = t n g. By n constrint SLD-derivtion we men sequence of pirs (G 1 ; E 1 ); : : :; (G N ; E N ) (1)

10 such tht (G i ; E i )! (G i+1 ; E i+1 ) holds for ll 1 i N? 1. By the stndrd technique (see for exmple [18, 11]) one cn prove tht gol G succeeds with respect P if nd only if there is constrint SLD-derivtion (1) such tht (1) G 1 = G; (2) E 1 = ;; (3) G N = 2; nd (4) E N is solvble in E. It is esy to dene nondeterministic Turing mchine M tht decides the query nswering problem by generting constrint SLD-derivtions for every input P nd G. Nmely, this mchine nondeterministiclly mkes trnsitions from (G i ; E i ) to (G i+1 ; E i+1 ) (connected by the reltion!). When G i is not empty nd it is impossible to pply this step, M rejects the input. When G i is empty, M should check whether E i is solvble in E. By the theorem condition, this cn be done in nondeterministic polynomil time (in the size of E i ). If E i is solvble then M ccepts the input, otherwise the input is rejected. We show tht M works within time 2 no(1) where n is the combined size of P nd G. Since P is nonrecursive, by denition there is numbering p 1 ; : : :; p r of ll predictes in P such tht for every cluse, if p i occurs in the hed nd p j occurs in the body then i > j. Let m be the mximum length of cluse bodies in P. We dene the weight of n tom p i (t 1 ; : : :; t k ) s the number (m + 1) i+1. The weight of gol is the sum of weights of its toms. Consider ny constrint SLD-derivtion generted by M. Evidently, the weight of the initil gol G is exponentil in n. For ech trnsition (G i ; E i )! (G i+1 ; E i+1 ), the weight of G i+1 is less thn the weight of G i. Therefore, the totl size of ll gols nd the totl size of ll systems of equtions is exponentil in n. The running time of M consists of the time of generting constrint SLD-derivtion nd the time of checking solvbility of the nl system of eqution. Since the size of the constrint SLD-derivtion is exponentil in n, the time of generting is exponentil too. The time of checking solvbility is 2 no(1) becuse the nl system of equtions hs the size of the sme order s the constrint SLD-derivtion, i.e. exponentil size, nd the solvbility problem for E is in NP by the condition of the theorem. 2 Theorem 4.2 The query nswering problem is NEXP-complete for the following clsses of nonrecursive logic progrms: logic progrms with trees in ny signture with t lest one function symbol of rity 2; logic progrms with bgs; logic progrms with sets; logic progrms with ny combintion of the bove types of complex vlues. Proof. NEXP-hrdness ws proved in Theorem 3.2. The pper [8] describes nondeterministic polynomil-time uniction lgorithm for terms representing ny combintion of trees, bgs nd sets. Thereby, the solvbility problem for corresponding equtionl theories is in NP. Therefore, the inclusion in NEXP immeditely follows from Theorem 4.1 bove nd [8]. 2 References [1] S. Abiteboul nd C. Beeri. The power of lnguges for the mnipultion of complex vlues. VLDB Journl, 4:727{794, 1995.

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