Yoshiharu Kojima 1,2

Transcription

1 Reachability Problems for Controlled Rewriting Systems Yoshiharu Kojima 1,2 joint work with Masahiko Sakai 1 and Florent Jacquemard 3 1 Nagoya University 2 Research Fellow of JSPS 3 INRIA Sacray, LSV AJSW2010, Obergurgl, 3 August, 2010 Y.Kojima, M.Sakai, F.Jacquemard () Reachability problems for CRS AJSW2010, 3 August 1 / 24

2 Motivation I have ever studied about reachability and regularity preservation under some strategies (e.g. innermost, context-sensitive, and etc.), but in some problems, these strategies are not sufficient to represent (e.g. XML update). Hence we should formalize another strategy. Related work F. Jacquremard and M. Rusinowitch, Rewrite based verification for XML update, PPDP This paper formalize controlled TRS by XPath expression and global constrained TRS to express XML update. Y.Kojima, M.Sakai, F.Jacquemard () Reachability problems for CRS AJSW2010, 3 August 2 / 24

3 Outline Introduction of controlled string rewriting systems (CSRS). Proofs of undecidability and decidability of reachability and regular model checking (R (L in ) L err where R is CSRS, L in and L err are regular language) for controlled string rewriting systems (CSRS). Extension of CSRS to the following three kinds of controlled term rewriting systems (CTRS) by means of tree automata: 1 full-control, 2 monadic-control, and 3 prefix-control. Proofs of undecidability and decidability of reachability and regular model checking for above CTRS. Y.Kojima, M.Sakai, F.Jacquemard () Reachability problems for CRS AJSW2010, 3 August 3 / 24

4 Controlled string rewriting systems (CSRS) [P. Butzbach 73] and [L. Chottin 79] formalized. CSRS can control prefix of redexes by regular languages. Definition (rewrite rule) Each rewrite rule is of the form: v 1 v 2 if Lv 1 where v 1,v 2 Σ and L is regular language over Σ for the set of alphabet Σ. Definition (rewrite relation) s can be rewritten to t by an above rewrite rule if s = uv 1 w,t = uv 2 w and u L. Y.Kojima, M.Sakai, F.Jacquemard () Reachability problems for CRS AJSW2010, 3 August 4 / 24

5 Controlled string rewriting systems Example a a if c a a R = c if c a b b if c a a d b b d if c a d b aaa bb a aa bb caa bb ca a bb a aa b b caa b b cca bb caa db R (a b ) {c,d} = {c i d j j i} (context-free language). Y.Kojima, M.Sakai, F.Jacquemard () Reachability problems for CRS AJSW2010, 3 August 5 / 24

6 Undecidable properties for CSRS Theorem Regular model checking is undecidable for length-one CSRS. Reachability is undecidable for flat CSRS. Flat CSRS CSRS where each rule is of the form a b, a bc, or ab c and latter two rules can be only applied at right-end of strings. Y.Kojima, M.Sakai, F.Jacquemard () Reachability problems for CRS AJSW2010, 3 August 6 / 24

7 Proof (Undecidability of reachability). Simulating emptiness problem of context-sensitive grammar. Context-sensitive grammar Grammar composed by quadruple N,T,P,S where each rule in P is of the form αaβ αγβ for α,β (N T),A N, and γ (N T) +. Simulated by the rules S SI,AB AC and A a [M. Penttonen 74] where I,A,B,C N and a T. Membership problems are PSPACE-complete, and emptiness problems are undecidable. Y.Kojima, M.Sakai, F.Jacquemard () Reachability problems for CRS AJSW2010, 3 August 7 / 24

8 Proof (Undecidable properties of reachability). Simulate a CSG G = N,T,P,S where each production rule in P is of the form S SI, AB AC, or A a. Let CSRS R as the following: R 1 = {S SI if S S SI P }, R 2 = {B C if (N T) AB AB AC P }, R 3 = {A a if (N T) A A a P }, and R 4 = {ab a if T ab,a if a a,b T }, R = R 1 R 2 R 3 R 4. Then, we have S R1 SI I R2 A 1 A n R3 a 1 a n R4 iff a 1 a n is produced by CSG G and hence we have S R iff L(G). Y.Kojima, M.Sakai, F.Jacquemard () Reachability problems for CRS AJSW2010, 3 August 8 / 24

9 Decidable property for CSRS Theorem Reachability for non-length-decreasing CSRS is PSPACE-complete. Lemma For CSRS R over Σ and CSG G s.t. L(G) = {a 1 a n }, we can construct a CSG G s.t. L(G ) = {t a 1 a n R t}. Y.Kojima, M.Sakai, F.Jacquemard () Reachability problems for CRS AJSW2010, 3 August 9 / 24

10 Proof (sketch). S G, G A l 1A 2 A i A n G A l 1, q in A 2 A i A n G A l 1, q in A 2, q 2 A i, q f A n G G, G a 1 a i a n a i b 1 b m if La i a 1 b 1 b m a n G A l 1 A 2 B 1 B m A n G A l 1, q in A 2, q 2 B 1 B m A n Input: CSRS R = {a i b 1 b m if La i } and CSG G s.t. L(G) = {a 1 a i a n }. Output: CSG G s.t. L(G ) = {a 1 a i a n,a 1 b 1 b m a n } Y.Kojima, M.Sakai, F.Jacquemard () Reachability problems for CRS AJSW2010, 3 August 10 / 24

11 Proof (sketch). S A l 1 A 2 A i A n a 1 a i a n G G a i b 1 b m if La i a 1 b 1 b m a n a 1 a i a n is rewritten to a 1 b 1 b m a n by the rule a i b 1 b m if La i. CSG G produce a 1 a i a n as follows: 1 G has non-terminal A that corresponds to a Σ and its terminal is Σ 2 Firstly, G produces A l 1 A i A n. 3 Next, G produces a 1 a i a n by the rules of the form A k a k P. Terminals are only produced by such rules. Y.Kojima, M.Sakai, F.Jacquemard () Reachability problems for CRS AJSW2010, 3 August 11 / 24

12 Proof (sketch). S A l 1 G, G A 2 A i A n a 1 a i a n G, G G a i b 1 b m if La i A l 1, q in A 2 A i A n G a 1 b 1 b m a n A l 1, q in A 2, q 2 A i, q f A n G A l 1, q in A 2, q 2 B 1 B m A n G simulate DFA A that corresponds to L in the rewrite rule. 1 Produce non-terminal A l 1,q in from A l 1 (l is marker for left-end). 2 Simulate transition δ(a k,q k ) q k of A by the production rule A k,q k A k+1 A k,q k A k+1,q k. 3 If q f is final state of A, then it implies a 1 a i 1 L(A) and B 1 B m is produced by the rule A i,q f B 1 B m. Y.Kojima, M.Sakai, F.Jacquemard () Reachability problems for CRS AJSW2010, 3 August 12 / 24

13 Proof (sketch). S G, G A l 1A 2 A i A n G A l 1, q in A 2 A i A n G A l 1, q in A 2, q 2 A i, q f A n G G, G a 1 a i a n a i b 1 b m if La i a 1 b 1 b m a n G A l 1 A 2 B 1 B m A n G A l 1, q in A 2, q 2 B 1 B m A n Finally, a 1 b 1 b m a n is produced by the rules A k,q k A k and A k a k. Y.Kojima, M.Sakai, F.Jacquemard () Reachability problems for CRS AJSW2010, 3 August 13 / 24

14 Results for CSRS Undecidable properties Regular model checking for length-one CSRS. Reachability for flat CSRS. Decidable property Reachability for non-length decreasing CSRS is PSPACE-complete. Y.Kojima, M.Sakai, F.Jacquemard () Reachability problems for CRS AJSW2010, 3 August 14 / 24

15 Controlled term rewriting systems (CTRS) CTRS (R, A,S ) is a pair of rewrite rules R and a tree automaton A,S named selection automaton. Definition (rewrite rules) Rewrite rules are of the form: (s 1 t 1, A,S ) where s,t T (F,X), A(= Q,Q f, ) is a usual tree automaton and S Q. A,S is named selection automaton. Definition (rewrite relation) s can be rewritten to t by the above rule if: s = C[s 1 σ],t = C[t 1 σ], and q S.C[s 1 σ] A C[q] A q f Q f. Y.Kojima, M.Sakai, F.Jacquemard () Reachability problems for CRS AJSW2010, 3 August 15 / 24

16 Three kinds of CTRS Definition For every selection automaton A(= Q,Q f, ),S in CTRS R, if does not contain ε-transition and there exists a state q 0 Q\S s.t. every transition rule f(q 1,...,q n ) q meets the following condition: if q q 0, then there exists at most one i s.t. q i q 0, otherwise, q = q 1 = = q n = q 0, then the CTRS is monadic control. In addition to condition of monadic control, if every transition rule f(q 1,...,q n ) q meets following condition, then the CTRS is prefix control If q S, then q 1 = = q n = q 0. Otherwise, the CTRS is full-control. Y.Kojima, M.Sakai, F.Jacquemard () Reachability problems for CRS AJSW2010, 3 August 16 / 24

17 Three kinds of CTRS C s C C s s σ σ σ Full control Monadic control Prefix control Red part in the above figures represent controlled part. Full-control can control whole context and substitution. Monadic-control can control path. Prefix-control can control prefix. Generality (Context-sensitive ) Prefix-control Monadic-control Full-control. Y.Kojima, M.Sakai, F.Jacquemard () Reachability problems for CRS AJSW2010, 3 August 17 / 24

18 Example Full-CTRS can represent a rule like: a c if a is brother of b and descendant of f by the TA A = {q, q a,q b,q qb,q f }, {q f }, where is the union of the followings and S = { q a }: {a q a,b q b }, {g(q 1,...,q n ) q g f, i,j.(q i = q a q j = q b )}, {g(q 1,...,q n ) q ab g f, i,j.(q i = q a q j = q b )}, {g(q 1,...,q n ) q ab g f, i.q i q f i.q i = q ab }, {g(q 1,...,q n ) q f g f, i.q i = q f }, {f(q 1,...,q n ) q i.q i = q ab }, {f(q 1,...,q n ) q f i.(q i = q ab q i = q f )}. We have the transition A C 1 [f(c 2 [g(a,b,t 1,...,t n )])] C 1 [f(c 2 [g( q a,q b,q,...,q)])] C 1 [f(q ab )] C A 1 [q f ] q f Q f A A Y.Kojima, M.Sakai, F.Jacquemard () Reachability problems for CRS AJSW2010, 3 August 18 / 24

19 Undecidability and decidability for CTRS Undecidability Reachability is undecidable for flat prefix-ctrs. Regular model checking is undecidable for depth-one prefix-ctrs. Reachability is undecidable for ground full-ctrs. Decidability Reachability is decidable for non-size-decreasing full-ctrs. Y.Kojima, M.Sakai, F.Jacquemard () Reachability problems for CRS AJSW2010, 3 August 19 / 24

20 Theorem (Results of undecidability) The following problems are undecidable: 1 Reachability is undecidable for flat prefix-ctrs. 2 Regular model checking is undecidable for depth-one prefix-ctrs. 3 Reachability is undecidable for ground full-ctrs. Proof. By Simulating the problems for string case. 1,2 By representing every string a 1 a n as the tree a 1 ( (a n )), depth-one CTRS can simulate length-one CSRS and flat prefix-ctrs can simulate flat CSRS. 3 By introducing binary symbol f and representing every string a 1 a n as the tree f(a 1,f( f(a n 1,a n ))), ground full-ctrs can simulate flat-csrs. Y.Kojima, M.Sakai, F.Jacquemard () Reachability problems for CRS AJSW2010, 3 August 20 / 24

21 Proof. (ground full-ctrs). Ground full-ctrs R can simulate CSRS R that simulates CSG G where each production rule is of the form: S SI, AB AC, A a S R SI is simulated by S f(s,i) R and I f(i,i) R. A R a is simulated by A a R. ab R a and a R is simulated by f(a,b) a R and a R. AB R AC is simulated by B C if parents brother is A R. This reduction is like f(a,f(b, )) f(a,f(c, )). f f f S f f a 1 f S R I A 1 A 2 a 2 R R f R f f I I A n 1 A n a n 1 a n Y.Kojima, M.Sakai, F.Jacquemard () Reachability problems for CRS AJSW2010, 3 August 21 / 24

22 Theorem (Results of decidability) Reachability is decidable for non-size-decreasing full-ctrs. Proof. For every term t, the set of terms reachable to t is finite. However, complexity of this problem is still open. In string cases, we have a corresponding grammar (context-sensitive grammar), but we don t have a corresponding grammar for tree (context-sensitive tree grammar?). Y.Kojima, M.Sakai, F.Jacquemard () Reachability problems for CRS AJSW2010, 3 August 22 / 24

23 Conclusion Proofs of decidability and undecidability for some problems for CSRS. Reachability is undecidable for flat CSRS. Regular model checking is undecidable for length-one CSRS. Reachability for non-length-decreasing CSRS is PSPACE-complete. Formalizing CTRS by selection automata. Proofs of decidability and undecidability for some problems for CTRS. Reachability is undecidable for flat prefix-ctrs. Regular model checking is undecidable for depth-one CTRS. Reachability is decidable for non-size-decreasing CTRS. Y.Kojima, M.Sakai, F.Jacquemard () Reachability problems for CRS AJSW2010, 3 August 23 / 24

24 Future works The following problems are still open: Reachability for ground monadic- or ground prefix-ctrs. Complexity of reachability for non-size-decreasing CTRS. We can prove this by formalizing context-sensitive tree grammar? Subclasses of CTRS s.t. regular model-checking is decidable. There exists decidable results of confluence, equivalence problem, and or so for basic strict CSRS [G. Senizergues 90]. Y.Kojima, M.Sakai, F.Jacquemard () Reachability problems for CRS AJSW2010, 3 August 24 / 24