Decision Methods for Concurrent Kleene Algebra with Tests : Based on Derivative

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1 Decision Methods for Concurrent Kleene Algebra with Tests : Based on Derivative Yoshiki Nakamura Tokyo Instutute of Technology, Oookayama, Meguroku, Japan, nakamura.y.ay@m.titech.ac.jp Abstract. Concurrent Kleene Algebra with Tests (CKAT) were introduced by Peter Jipsen[Jip14]. We give derivatives for CKAT to decide word problems, for example emptiness, equivalence, containment problems. These derivative methods are expanded from derivative methods for Kleene Algebra and Kleene Algebra with Tests[Brz64][Koz08][ABM12]. Additionally, we show that the equivalence problem of CKAT is in EXPSPACE. Keywords: concurrent kleene algebras with tests, series-parallel strings, Brzozowski derivative, computational complexity 1 Introduction In this paper, we assume [Jip14, theorem 1] and we use CKAT terms as expressions of guarded series-parallel language. Let Σ be a set of basic program symbols p 1, p 2,... and T a set of basic boolean test symbols t 1, t 2,, where we assume that Σ T =. Each α 1, α 2,... denotes a subset of T. Boolean term b and CKAT term p over T and Σ are defined by the following grammar, respectively. b := 0 1 t T b 1 + b 2 b 1 b 2 b 1 p := b p Σ p 1 + p 2 p 1 p 2 p 1 p 1 p 2 The guarded series-parallel strings set GS Σ,T over Σ and T is a smallest set such that follows α GS Σ,T for any α T α 1 pα 2 GS Σ,T for any α 1, α 2 T and any basic program p Σ if w 1 α, αw 2 GS Σ,T, then w 1 αw 2 GS Σ,T. if α 1 w 1 α 2, α 1 w 2 α 2 GS Σ,T, then α 1 { w 1, w 2 }α 2 GS Σ,T. Definition 1 (guarded series-parallel language). Let and be binary operators over GS Σ,T, respectably. { They are defined as follows. w w 1 w 2 = 1αw 2 (w 1 = w 1α and w 2 = αw 2) undef ined (o.w.) In particular, if w 1 = w 2 = α, then w 1 w 2 = α.

2 2 Y. Nakamura α 1 { w 1, w 2 }α 2 (w 1 = α 1 w 1α 2 and w 2 = α 1 w 2α 2 ) w 1 w 2 = α (w 1 = w 2 = α) undef ined (o.w.) L is a map from CKAT terms over Σ and T to this concrete model by L(0) =, L(1) = 2 T L(t) = {α T t α} for t T L(b) = 2 T \ L(b) L(p) = {α 1 pα 2 α 1, α 2 T } for p Σ L(p 1 + p 2 ) = L(p 1 ) L(p 2 ) L(p 1 p 2 ) = {w 1 w 2 w 1 G(p 1 ) and w 2 L(p 2 ) and w 1 w 2 is defined } L(p ) = {α 0 w 1 w n α 0 T and w 1,..., w n L(p) and α 0 w 1 w n is defined } n<ω L(p 1 p 2 ) = {w 1 w 2 w 1 L(p 1 ) and w 2 L(p 2 ) and w 1 w 2 is defined } We expand L to L(P ) = p P L(p), where P is a set of CKAT terms. Furthermore, let L α (p) = {αw αw L(p)}. In guarded series-parallel strings, α 1 { w 1, w 2 }α 2 has commutative(i.e. α 1 { w 1, w 2 }α 2 = α 1 { w 2, w 1 }α 2 ). We define p 1 = p 2 for two CKAT terms p 1 and p 2 as L(p 1 ) = L(p 2 ) (by means of [Jip14, Theorem 1]). 2 The Brzozowski derivative for CKAT Now, we give the naive derivative for CKAT. Derivative has applications to many language theoretic problems (e.g. membership problem, emptiness problem, equivalence problem, and so on). Definition 2 (Naive Derivative). We define E α and D w. They are maps from a CKAT term to a set of CKAT terms, respectively. E α is inductively defined as follows. We expand E α and D w to E α (P ) = p P E α(p) and D w (P ) = p P D w(p), where P is a set of CKAT terms, respectively. E α (0) = E α (p) = E α (1) = { E α (p 1) = {1} {1} (t α) E α (t) = (o.w.) E α (b) = {1} \ E α (b) E α (p 1 + p 2 ) = E α (p 1 ) E α (p 2 ) E α (p 1 p 2 ) = E α (p 1 p 2 ) = E α (p 1 )E α (p 2 ) D w is inductively defined as follows. For w = q { w 1, w 2 } and any series-parallel string w, D αwα w α (p) = D α w α (D αwα (p)) D αwα (p 1 + p 2 ) = D αwα (p 1 ) D αwα (p 2 ) D αwα (p 1 p 2 ) = D αwα (p 1 ){p 2 } E α (p 1 )D αwα (p 2 )

3 CKAT is in EXPSPACE 3 D αwα (p 1) = D αwα (p 1 ){p 1} D αwα (b) = for any boolean term b { {1} (p = q) D αqα (p) = (o.w.) D αqα (p 1 p 2 ) = D α{ w1,w 2 }α (p) = D α{ w1,w 2 }α (p 1 p 2 ) = E α ((D (p αw1α 1) D (p αw2α 2)) (D (p αw1α 2) D αw2 α (p 1))) The left-quotient of L GS Σ,T with regard to w GS Σ,T is the set w 1 L = {w w w L}. Lemma 1. For any series-parallel string αwα, 1. 1 E α (p) α L α (p) 2. (αwα ) 1 L α (p) = L α (D αwα (p)) Proof (Sketch). 1. is proved by induction on the size of p. 2. is proved by double induction on the size of w and the size of p. We can decide whether αwα L(p) to check 1 E α (D αwα (p)) by Lemma 1. We now define efficient derivative. This derivative is another definition of derivative for CKAT. This derivative is useful for giving more efficient algorithm than naive derivative in computational complexity. (In naive derivative, we should memorize w 1 and w 2 to get D α{ w1,w 2 }α (p). In particular, the size of w 1 and w 2 can be double exponential size of input size in equivalence problem.) We expand CKAT terms to express efficient derivative. We say these terms intermediate CKAT terms. Intermediate CKAT term is defined as following. Definition 3 (intermediate CKAT term). Intermediate CKAT term is defined by the following grammar. q := b p Σ q 1 + q 2 q 1 q 2 q 1 q 1 q 2 D x (q 1 ) We call x a derivative variable of D x (q 1 ). The efficient derivative d pr (q) is defined in Definition 4, where q is an intermediate CKAT term, pr is a sequence of assignments formed x += αp or x += αt (The sequence of assignments pr is formed x 1 += term 1 ;... ; x m += term m.) and T is formed by the following grammar. T := { x l T l, x r T r } { x l T l, p r x r } { p l x l, x r T r } { p l x l, p r x r }. Intuitively, d x+=αw (... D x (q)... ) means (... D x ( D αw (join α (q)))... ). Definition 4. The efficient derivative d pr (q) is inductively defined as follows, where we assume that any derivative variable occurred in T are different. To define d pr (q), we also define D αw and join α. We expand d pr to d pr (Q) = q Q d pr(q), where Q is a set of intermediate CKAT terms. We also expand join α to join α (Q) = q Q join α(q).

4 4 Y. Nakamura d x+=αw;pr (q) = d pr (d x+=αw (q)) d x+=αw (b) = {b} d x+=αw (p) = {p} d x+=αw (q 1 + q 2 ) = d x+=αw (q 1 ) d x+=αw (q 2 ) d x+=αw (q 1 q 2 ) = d x+=αw (q 1 )d x+=αw (q 2 ) d x+=αw (q1) = d x+=αw (q 1 ) (= {q q d x:=αw (q 1 )}) d x+=αw (q 1 q 2 ) = d x+=αw (q 1 ) d x+=αw (q 2 ) (= {q 1 q 2 q 1 d x+=αw (q 1 ), q 2 d x+=αw (q 2 )}) d x+=αw (D y (q 1 )) = D y (d x+=αw (q 1 )) d x+=αw (D x (q 1 )) = D x ( D αw (join α (q 1 ))) D αp (q) = D αp (q) D αt (b) = D αt (p) = D αt (q 1 + q 2 ) = D αt (q 1 ) D αt (q 2 ) D αt (q 1 q 2 ) = D αt (q 1 ){q 2 } E α (q 1 ) D αt (q 2 ) D αt (q1) = D αt (q 1 ){q1} (D xl ( D αpl (q 1 )) D xr ( D αpr (q 2 ))) (D xr ( D αpr (q 1 )) D xl ( D αpl (q 2 ))) (T = { p l x l, p r x r }) (D xl ( D αtl (q 1 )) D xr ( D αpr (q 2 ))) D (D xr ( αt (q 1 q 2 ) = D αpr (q 1 )) D xl ( D αtl (q 2 ))) (T = { T l x l, p r x r }) (D xl ( D αpl (q 1 )) D xr ( D αtr (q 2 ))) (D xr ( D αtr (q 1 )) D xl ( D αpl (q 2 ))) (T = { p l x l, T r x r }) (D xl ( D αtl (q 1 )) D xr ( D αtr (q 2 ))) (D xr ( D αtr (q 1 )) D xl ( D αtl (q 2 ))) (T = { T l x l, T r x r }) (b) = {b}, join α (p) = {p} (q 1 + q 2 ) = join α (q 1 ) join α (q 2 ), join α (q 1 q 2 ) = join α (q 1 )join α (q 2 ) (q 1 q 2 ) = join α (q 1 ) join α (q 2 ) (q1) = join α (q 1 ) (D y (q)) = E α (join α (q)) Efficient derivative is essentially equal to the derivative of Definition 1. Let sp x (pr) be the string corresponded to x of pr. (For example, sp x0 (x 0 += α{p 1 x 1, p 2 x 2 }; x 1 += α p 3 ; x 0 += α p 4 ) = α{p 1 α p 3, p 2 }α p 4. sp x1 (x 0 += α{p 1 x 1, p 2 x 2 }; x 1 += α p 3 ; x 0 += α p 4 ) = αp 1 α p 3 ) Lemma 2. join α (d pr (D x (p))) = E α (D spx (pr)α (p)) By Lemma 1 and Lemma 2, sp x (pr)α L(p) 1 join α (d pr (D x (p))). Therefore, we can use effective derivative instead of naive derivative. Next, we define the size of a intermediate CKAT term q, denoted by q as follows. 0 = 1 = t = p = 1 b = 1 + b q 1 = D x (q 1 ) = 1 + q 1

5 CKAT is in EXPSPACE 5 q 1 + q 2 = q 1 q 2 = q 1 q 2 = 1 + q 1 + q 2 Definition 5 (Closure). Cl X is a map from a intermediate CKAT term to a set of intermediate CKAT terms, where X is a set of intersection variables. Cl X is inductively defined as follows. Cl X (a) = {a} for a = 0 1 t Cl X (b) = {b} Cl X (b) for any boolean term b Cl X (p) = {p, 1} Cl X (q 1 + q 2 ) = {q 1 + q 2 } Cl X (q 1 ) Cl X (q 2 ) Cl X (q 1 q 2 ) = {q 1 q 2 } Cl X (q 1 ){q 2 } Cl X (q 2 ) Cl X (q 1) = {q 1} Cl X (q 1 ){q 1} Cl X (q 1 q 2 ) = {q 1 q 2 } {D x1 (q 1) D x2 (q 2) q 1 Cl X (q 1 ), q 2 Cl X (q 2 ), x 1, x 2 X} Cl X (D x (q 1 )) = {D x (q 1 )} D x (Cl X (q 1 )) We expand Cl X to Cl X (Q) = q Q Cl X(q), where Q is a set of intermediate CKAT terms. Cl X is a closed operator. In other words, Cl X satisfies (1) Q Cl X (Q), (2) Q 1 Q 2 Cl X (Q 1 ) Cl X (Q 2 ) and (3) Cl X (Cl X (Q)) = Cl X (Q). We also define the intersection width iw(q) over intermediate CKAT terms and iw(w) over GI Σ,T as follows. iw(b) = iw(p) = 1 for any boolean term b and any basic program p Σ iw(q 1 + q 2 ) = iw(q 1 q 2 ) = max(iw(q 1 ), iw(q 2 )) iw(q 1) = iw(d x (q 1 )) = iw(q 1 ) iw(q 1 q 2 ) = 1 + iw(q 1 ) + iw(q 2 ) iw(α) = 1 for any α T iw(α 1 pα 2 ) = 1 iw(w 1 αw 2 ) = max(iw(w 1 α), iw(αw 2 )) iw(α 1 { w 1, w 2 }α 2 ) = 1 + iw(w 1 ) + iw(w 2 ) Lemma 3 (closure is bounded). For any intermediate CKAT term q and any sequence of program pr and any set of derivative variables X, where X contains any derivative variables in pr, Cl X (q) 2 X 2 iw(q) q iw(q) Proof (Sketch). This is proved by induction on the structure of q. We only consider the case of q = q 1 q 2. Cl X (q 1 q 2 ) 1 + X Cl X (q 1 ) X Cl X (q 2 ) 1 + X 2 2 q 1 iw(q 1) X 2 iw(q 1) 2 q 2 iw(q 2) X 2 iw(q 2) = X 2 iw(q 1 q 2 ) q 1 iw(q 1) q 2 iw(q 2) 2 X 2 iw(q1 q2) ( q 1 + q 2 ) iw(q1)+iw(q2) 2 X 2 iw(q 1 q 2 ) q 1 q 2 iw(q 1 q 2 ) Lemma 4 (derivative is closed). For any intermediate CKAT term q and any sequence of program pr and any set of derivative variables X, where X contains any derivative variables in pr, d pr (q) Cl X (q) Proof (Sketch). This is proved by double induction on the size of pr and the size of q.

6 6 Y. Nakamura 3 CKAT equational theory is in EXPSPACE By Lemma 1 and Lemma 2, L(p 1 ) = L(p 2 ) iff join α (d pr (D x (p 1 ))) = join α (d pr (D x (p 2 ))) for any pr and any α. Thus we find some pr such that join α (d pr (D x (p 1 ))) join α (d pr (D x (p 2 ))) to decide p 1 p 2. We must consider all the patterns of pr at first glance. But, we need not to check if pr is too long. We are enough to check the cases of iw(sp(pr)) max(iw(p 1 ), iw(p 2 ))( l) by the following Lemma 5. Lemma 5. If iw(sp(pr)) > iw(q), d pr (q) =. By Lemma 5, we are enough to check the case of iw(sp(pr)) max(iw(p 1 ), iw(p 2 )) l. By iw(sp(pr)) l, We are enough to prepare (l 1) derivative variables. By Lemma 3, Cl X (q) 2 q iw(q) X 2 iw(q) 2 l l (1 + 3 (l 1)) 2 l. Therefore, Cl X (D x (p 1 )) = O(2 p(l) ) and Cl X (D x (p 2 )) = O(2 p(l) ), where p(l) is a polynomial function of l. We can give a nondeterministic algorithm. We nondeterministically select the syntax of pr. (pr is x += αp or x += αt.) If there exists a sequent of assignments pr and α such that join α (d pr (D x (p 1 ))) join α (d pr (D x (p 2 ))), p 1 p 2. Otherwise, p 1 = p 2. (See Algorithm 1 if you know more details.) It holds the Theorem 1 by this algorithm. Theorem 1. CKAT equivalence problem is in EXPSPACE. Corollary 1. if iw(p) is a fixed parameter, then CKAT equivalence problem is PSPACEcomplete. Note that PSPACE-hardness is derived by [Hun73]. 4 Concluding Remarks We have given the derivative for CKAT and shown that CKAT equational theory is in EXPSPACE. We finish with the following some of our future works. Is this equivalence problem EXPSPACE-complete? (We expect that this claim is T rue.) If we allow ϵ (for example, α{ p, ϵ }α), can we give efficient derivative? (It become a little difficult because we have to memorize α in the case of x += α{ p 1 x 1, ϵ }. We should give another derivative to show the result like Corollary 1.) A Pseudo Code

7 REFERENCES 7 Algorithm 1 Decide p 1 = p 2, given two CKAT terms p 1 and p 2 Ensure: Whether p 1 p 2 or not?(true or False) step 0, P 1 {D x0 (p 1)}, P 2 {D x0 (p 2)} while step 2 Cl X (D x0 (p 1 )) 2 Cl X (D x0 (p 2 )) do Let α be a subset of T, which is picked up nondeterministically. if join α (P 1 ) join α (P 2 ) then return T rue end if Let pr be x += αp or x += αt, which is picked up nondeterministically, where iw(pr) max(iw(p 1 ), iw(p 2 )). step step + 1, P 1 d pr(p 1), P 2 d pr(p 2) end while return F alse References [ABM12] [Brz64] [Hun73] [Jip14] [Koz08] Ricardo Almeida, Sabine Broda, and Nelma Moreira. Deciding KAT and Hoare Logic with Derivatives. In: Proceedings Third International Symposium on Games, Automata, Logics and Formal Verification, GandALF 2012, Napoli, Italy, September 6-8, , pp Janusz A Brzozowski. Derivatives of regular expressions. In: Journal of the ACM (JACM) 11.4 (1964), pp Harry B Hunt III. On the time and tape complexity of languages I. In: Proceedings of the fifth annual ACM symposium on Theory of computing. ACM. 1973, pp Peter Jipsen. Concurrent Kleene algebra with tests. In: Relational and Algebraic Methods in Computer Science. Springer, 2014, pp Dexter Kozen. On the Coalgebraic Theory of Kleene Algebra with Tests. Tech. rep. Computing and Information Science, Cornell University, Mar