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

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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 forCKATto decide word problems, for example emptiness, equivalence, containment problems. These derivative methods are expanded from derivative methods for Kleene Al- gebra and Kleene Algebra with Tests[Brz64][Koz08][ABM12]. Addition- ally, we show that the equivalence problem ofCKATis 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 useCKATterms as expressions of guarded series-parallel language.

LetΣbe a set ofbasic programsymbolsp1,p2, . . . andTa set ofbasic boolean test symbolst1,t2,· · ·, where we assume thatΣ∩T =∅. Eachα1, α2, . . . de- notes a subset ofT.Boolean termbandCKATtermpoverT andΣare defined by the following grammar, respectively.

b:=0|1|t∈T |b1+b2|b1b2|b1

p:=b|p∈Σ|p1+p2|p1p2|p^∗₁|p1∥p2

The guarded series-parallel strings setGS_Σ,T overΣandTis a smallest set such that follows

– α∈GS_Σ,T for anyα⊆T

– α₁pα₂∈GS_Σ,T for anyα₁, α₂⊆T and any basic programp∈Σ – ifw₁α, αw₂∈GS_Σ,T, thenw₁αw₂∈GS_Σ,T.

– ifα₁w₁α₂, α₁w₂α₂∈GS_Σ,T, thenα₁{|w₁, w₂|}α₂∈GS_Σ,T.

Definition 1 (guarded series-parallel language).Let⋄and∥be binary operators overGSΣ,T, respectably. They are defined as follows.

w1⋄w2= {

w₁^′αw₂^′ (w1=w₁^′αandw2=αw₂^′) undef ined (o.w.)

In particular, ifw1=w2=α, thenw1⋄w2=α.

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w₁∥w₂=







α₁{|w^′₁, w^′₂|}α₂ (w₁=α₁w^′₁α₂andw₂=α₁w^′₂α₂) α (w1=w2=α)

undef ined (o.w.)

Lis a map fromCKATterms overΣandTto this concrete model by – L(0) =∅,L(1) = 2^T

– L(t) ={α⊆T |t∈α}fort∈T – L(b) = 2^T \L(b)

– L(p) ={α₁pα₂|α₁, α₂⊆T}forp∈Σ – L(p1+p2) =L(p1)∪L(p2)

– L(p1p2) ={w1⋄w2| w1∈G(p1)andw2∈L(p2)andw1⋄w2is defined} – L(p^∗) = ∪

n<ω

{α0⋄w1⋄ · · · ⋄wn|α0⊆Tandw1, . . . , wn∈L(p)andα0⋄w1· · · ⋄wnis defined} – L(p1∥p2) ={w1∥w2|w1∈L(p1)andw2∈L(p2)and w1∥w2is defined}

We expandLtoL(P) =∪

p∈PL(p), whereP is a set ofCKATterms. Furthermore, letLα(p) ={αw|αw∈L(p)}.

In guarded series-parallel strings,α₁{|w₁, w₂|}α₂has commutative(i.e.α₁{|w₁, w₂|}α₂= α₁{|w₂, w₁|}α₂). We definep₁ = p₂for twoCKATtermsp₁andp₂asL(p₁) =

L(p₂)(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 CKATterm to a set of CKATterms, respectively.E_α is inductively defined as fol- lows. We expandE_αandD_wtoE_α(P) =∪

p∈PE_α(p)andD_w(P) =∪

p∈PD_w(p), whereP is a set ofCKATterms, respectively.

– Eα(0) =Eα(p) =∅ – Eα(1) =Eα(p^∗₁) ={1} – Eα(t) =

{{1} (t∈α)

∅ (o.w.) – E_α(b) ={1} \E_α(b)

– Eα(p1+p2) =Eα(p1)∪Eα(p2)

– Eα(p1p2) =Eα(p1∥p2) =Eα(p1)Eα(p2) Dwis inductively defined as follows.

Forw=q| {|w^′₁, w^′₂|}and any series-parallel stringw^′, – Dαwα^′w^′α^′′(p) =Dα^′w^′α^′′(Dαwα^′(p))

– D_αwα′(p₁+p₂) =D_αwα′(p₁)∪D_αwα′(p₂)

– Dαwα′(p1p2) =Dαwα′(p1){p2} ∪Eα(p1)Dαwα′(p2)

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– D_αwα′(p^∗₁) =D_αwα′(p₁){p^∗₁} – D_αwα′(b) =∅for any boolean termb – D_αqα′(p) =

{{1} (p=q)

∅ (o.w.) – Dαqα′(p1∥p2) =∅ – D_α_{|_w₁_,w₂_|}_α′(p) =∅

– D_α_{|_w₁_,w₂_|}_α′(p₁ ∥ p₂) = E_α′((D_αw₁_α′(p₁) ∥ D_αw₂_α′(p₂))∪(D_αw₁_α′(p₂) ∥ D_αw₂_α′(p₁)))

Theleft-quotientofL⊆GSΣ,T with regard tow∈GSΣ,T is the setw⁻¹L= {w^′|w⋄w^′∈L}.

Lemma 1. For any series-parallel stringαwα^′, 1. 1∈Eα(p) ⇐⇒ α∈Lα(p)

2. (αwα^′)⁻¹L_α(p) =L_α′(D_αwα′(p))

Proof (Sketch). 1. is proved by induction on the size ofp.

2. is proved by double induction on the size ofwand the size ofp.

We can decide whetherαwα^′∈L(p)to check1∈E_α′(D_αwα′(p))by Lemma 1. We now define efficient derivative. This derivative is another definition of derivative forCKAT. This derivative is useful for giving more efficient algorithm than naive derivative in computational complexity. (In naive derivative, we should memorizew1andw2to getD_α_{|_w₁_,w₂_|}_α′(p). In particular, the size of w1andw2can be double exponential size of input size in equivalence problem.) We expandCKATterms to express efficient derivative. We say these termsin- termediateCKATterms.IntermediateCKATtermis defined as following.

Definition 3 (intermediate CKAT term).IntermediateCKATtermis defined by the following grammar.

q:=b|p∈Σ|q₁+q₂|q₁q₂|q^∗₁|q₁∥q₂|D_x(q₁) We callxaderivative variableofDx(q1).

The efficient derivative d_pr(q) is defined in Definition 4, where q is an in- termediateCKATterm,pr is a sequence of assignments formedx += αpor x+=αT (The sequence of assignmentspris formedx1+=term1;. . .;xm+=

termm.) and T is formed by the following grammar. T := {|xlTl, xrTr|} | {|xlTl,prxr|} | {|plxl, xrTr|} | {|plxl,prxr|}. Intuitively,dx+=αw(. . . Dx(q). . .) means(. . . Dx( ˘Dαw(join_α(q))). . .).

Definition 4. Theefficient derivatived_pr(q)is inductively defined as follows, where we assume that any derivative variable occurred inT are different. To defined_pr(q), we also defineD˘_αwand join_α. We expandd_prtod_pr(Q) =∪

q∈Qd_pr(q), whereQis a set of intermediateCKATterms. We also expand join_αtojoin_α(Q) =∪

q∈Qjoin_α(q).

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– d_x+=αw;pr′(q) =d_pr′(d_x+=αw(q)) – dx+=αw(b) ={b}

– dx+=αw(p) ={p}

– dx+=αw(q1+q2) =dx+=αw(q1)∪dx+=αw(q2) – dx+=αw(q1q2) =dx+=αw(q1)dx+=αw(q2) – d_x+=αw(q₁^∗) =d_x+=αw(q₁)^∗

(={q^′∗|q^′∈d_x:=αw(q₁)})

– d_x+=αw(q₁∥q₂) =d_x+=αw(q₁)∥d_x+=αw(q₂) (={q₁^′ ∥q₂^′ |q₁^′ ∈dx+=αw(q1), q^′₂∈dx+=αw(q2)}) – dx+=αw(Dy(q1)) =Dy(dx+=αw(q1))

– dx+=αw(Dx(q1)) =Dx( ˘Dαw(join_α(q1))) – D˘_αp(q) =D_αp(q)

– D˘αT(b) = ˘DαT(p) =∅

– D˘αT(q1+q2) = ˘DαT(q1)∪D˘αT(q2)

– D˘αT(q1q2) = ˘DαT(q1){q2} ∪Eα(q1) ˘DαT(q2) – D˘_α_T(q₁^∗) = ˘D_α_T(q₁){q₁^∗}

– D˘_α_T(q₁∥q₂) =











(Dx_l( ˘Dαp_l(q1))∥Dx_r( ˘Dαp_r(q2)))

∪(D_x_r( ˘D_αp_r(q₁))∥D_x_l( ˘D_αp_l(q₂))) (T ={|p_lx_l,p_rx_r|}) (Dx_l( ˘DαTl(q1))∥Dx_r( ˘Dαp_r(q2)))

∪(D_x_r( ˘D_αp_r(q₁))∥D_x_l( ˘D_α_T_l(q₂))) (T ={|Tlx_l,p_rx_r|}) (Dx_l( ˘Dαp_l(q1))∥Dx_r( ˘DαTr(q2)))

∪(D_x_r( ˘D_α_T_r(q₁))∥D_x_l( ˘D_αp_l(q₂))) (T ={|p_lx_l,Trx_r|}) (Dx_l( ˘DαTl(q1))∥Dx_r( ˘DαTr(q2)))

∪(D_x_r( ˘D_α_T_r(q₁))∥D_x_l( ˘D_α_T_l(q₂))) (T ={|Tlx_l,Trx_r|}) – join_α(b) ={b}, join_α(p) ={p}

– join_α(q1+q2) =join_α(q1)∪join_α(q2), join_α(q1q2) =join_α(q1)join_α(q2) – join_α(q1∥q2) =join_α(q1)∥join_α(q2)

– join_α(q^∗₁) =join_α(q1)^∗ – join_α(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 toxofpr. (For example,sp_x₀(x₀+=α{p₁x₁,p₂x₂};x₁+=

α^′p₃;x₀ += α^′′p₄) = α{p₁α^′p₃,p₂}α^′′p₄.sp_x₁(x₀ += α{p₁x₁,p₂x₂};x₁ +=

α^′p₃;x₀+=α^′′p₄) =αp₁α^′p₃)

Lemma 2. join_α′(d_pr(D_x(p))) =E_α′(D_sp_x_(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 thesizeof a intermediateCKATtermq, denoted by|q|as follows.

– |0|=|1|=|t|=|p|= 1 – |b|= 1 +|b|

– |q₁^∗|=|Dx(q1)|= 1 +|q1|

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– |q₁+q₂|=|q₁q₂|=|q₁∥q₂|= 1 +|q₁|+|q₂|

Definition 5 (Closure).Cl_Xis a map from a intermediateCKATterm to a set of in- termediateCKATterms, whereXis a set of intersection variables.Cl_Xis inductively defined as follows.

– ClX(a) ={a}fora=0|1|t

– ClX(b) ={b} ∪ClX(b)for any boolean termb – Cl_X(p) ={p,1}

– ClX(q1+q2) ={q1+q2} ∪ClX(q1)∪ClX(q2) – ClX(q1q2) ={q1q2} ∪ClX(q1){q2} ∪ClX(q2) – Cl_X(q^∗₁) ={q^∗₁} ∪Cl_X(q₁){q₁^∗}

– ClX(q1 ∥ q2) = {q1 ∥ q2} ∪ {Dx₁(q^′₁) ∥ Dx₂(q₂^′) | q^′₁ ∈ ClX(q1), q^′₂ ∈ ClX(q2), x1, x2∈X}

– Cl_X(D_x(q₁)) ={D_x(q₁)} ∪D_x(Cl_X(q₁)) We expandClX toClX(Q) = ∪

q∈QClX(q), whereQis a set of intermediate CKATterms.ClX is a closed operator. In other words,ClX satisfies (1) Q ⊆ ClX(Q), (2)Q1⊆Q2⇒ClX(Q1)⊆ClX(Q2)and (3)ClX(ClX(Q)) =ClX(Q).

We also define the intersection widthiw(q)over intermediateCKATterms and iw(w)overGIΣ,T as follows.

– iw(b) =iw(p) = 1for any boolean termband any basic programp∈Σ – iw(q1+q2) =iw(q1q2) = max(iw(q1), iw(q2))

– iw(q₁^∗) =iw(Dx(q1)) =iw(q1) – iw(q₁∥q₂) = 1 +iw(q₁) +iw(q₂) – iw(α) = 1for anyα⊆T

– iw(α1pα2) = 1

– iw(w₁αw₂) = max(iw(w₁α), iw(αw₂)) – iw(α1{|w1, w2|}α2) = 1 +iw(w1) +iw(w2)

Lemma 3 (closure is bounded).For any intermediateCKATtermqand any se- quence of program pr and any set of derivative variables X, whereX contains any derivative variables inpr,

|ClX(q)| ≤2∗ |X|²^∗^iw(q)∗ |q|^iw(q)

Proof (Sketch). This is proved by induction on the structure ofq. We only consider the case ofq=q1∥q2.

|ClX(q1∥q2)| ≤1 +|X| ∗ |ClX(q1)| ∗ |X| ∗ |ClX(q2)|

≤1 +|X|²∗2∗ |q1|îw(q¹⁾∗ |X|²^∗îw(q¹⁾∗2∗ |q2|îw(q²⁾∗ |X|²^∗îw(q²⁾

= 1 + 4∗ |X|²^∗îw(q¹^∥^q²⁾∗ |q₁|îw(q¹⁾∗ |q₂|îw(q²⁾

≤2∗ |X|²^∗^iw(q¹^∥^q²⁾∗(|q1|+|q2|)^iw(q¹^)+iw(q²⁾

≤2∗ |X|²^∗^iw(q¹^∥^q²⁾∗ |q₁∥q₂|^iw(q¹^∥^q²⁾

Lemma 4 (derivative is closed). For any intermediateCKATterm qand any se- quence of program pr and any set of derivative variables X, whereX contains any derivative variables inpr,

dpr(q)⊆ClX(q)

Proof (Sketch). This is proved by double induction on the size ofprand the size ofq.

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3 CKAT equational theory is in EXPSPACE

By Lemma 1 and Lemma 2,L(p1) =L(p2)iff join_α′(dpr(Dx(p1))) =join_α′(dpr(Dx(p2))) for anypr and anyα^′. Thus we find someprsuch that join_α′(dpr(Dx(p1))) ̸= join_α′(dpr(Dx(p2)))to decidep1̸=p2. We must consider all the patterns ofprat first glance. But, we need not to check ifpris too long. We are enough to check the cases ofiw(sp(pr))≤max(iw(p₁), iw(p₂))(≤l)by the following Lemma 5.

Lemma 5. Ifiw(sp(pr))> iw(q),dpr(q) =∅.

We can give a nondeterministic algorithm. We nondeterministically select the syntax ofpr. (prisx+=αporx+=αT.) If there exists a sequent of assign- mentsprandα^′ such that join_α′(dpr(Dx(p1))) ̸=join_α′(dpr(Dx(p2))),p1 ̸=p2. Otherwise,p1=p2. (See Algorithm 1 if you know more details.)

It holds the Theorem 1 by this algorithm.

Theorem 1. CKATequivalence problem is in EXPSPACE.

Corollary 1. ifiw(p)is a fixed parameter, thenCKATequivalence problem is PSPACE- complete.

Note that PSPACE-hardness is derived by [Hun73].

4 Concluding Remarks

We have given the derivative forCKATand shown thatCKATequational 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 isT rue.)

– If we allowϵ (for example, α{|p, ϵ|}α), can we give efficient derivative?

(It become a little difficult because we have to memorizeαin the case of x += α{|p1x1, ϵ|}. We should give another derivative to show the result like Corollary 1.)

A Pseudo Code

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Algorithm 1Decidep₁=p₂, given two CKAT termsp₁andp₂ Ensure: Whetherp1̸=p2or not?(True or False)

step⇐0,P1⇐ {Dx₀(p1)},P2⇐ {Dx₀(p2)}

whilestep≤2^|^Cl^X^(D^x⁰^(p¹⁾⁾^|∗2^|^Cl^X^(D^x⁰^(p²⁾⁾^|do

Letαbe a subset ofT, which is picked up nondeterministically.

ifjoin_α(P1)̸=join_α(P2)then return T rue

end if

Letprbex +=αporx+=αT, which is picked up nondeterministically, where iw(pr)≤max(iw(p1), iw(p2)).

step⇐step+ 1,P1⇐dpr(P1),P2⇐dpr(P2) end while

return F alse

References

[ABM12] Ricardo Almeida, Sabine Broda, and Nelma Moreira. “Deciding KAT and Hoare Logic with Derivatives”. In: Proceedings Third Interna- tional Symposium on Games, Automata, Logics and Formal Verification, GandALF 2012, Napoli, Italy, September 6-8, 2012.2012, pp. 127–140.

[Brz64] Janusz A Brzozowski. “Derivatives of regular expressions”. In:Jour- nal of the ACM (JACM)11.4 (1964), pp. 481–494.

[Hun73] Harry B Hunt III. “On the time and tape complexity of languages I”. In:Proceedings of the fifth annual ACM symposium on Theory of com- puting. ACM. 1973, pp. 10–19.

[Jip14] Peter Jipsen. “Concurrent Kleene algebra with tests”. In:Relational and Algebraic Methods in Computer Science. Springer, 2014, pp. 37–48.

[Koz08] Dexter Kozen.On the Coalgebraic Theory of Kleene Algebra with Tests.

Tech. rep.http://hdl.handle.net/1813/10173. Computing and Information Science, Cornell University, Mar. 2008.