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Representation of Kleene Algebra with Tests

Representation of Kleene Algebra with Tests. Dexter Kozen Cornell University. Kleene Algebra (KA) is the algebra of regular expressions. pq + qp p*q {pq,qp} {q,pq,p 2 q,p 3 q, …}

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Representation of Kleene Algebra with Tests

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  1. Representation ofKleene Algebra with Tests Dexter Kozen Cornell University

  2. Kleene Algebra (KA)is the algebra of regular expressions pq + qp p*q {pq,qp} {q,pq,p2q,p3q, …} (p + q)* = (p*q)*p* (pq)*p = p(qp)* {all strings over p,q}{p,pqp,pqpqp,} (0 + 1(01*0)*1)* {multiples of 3 in binary} p p q q q p p,q q p 0 1 0 1 0 1

  3. Standard Interpretation Regular sets over  A+B = A  B AB = {xy | x  A, y  B} A* = Un0 An = A0 A1 A2 ... 1 = {} 0 =  p   interpreted as {p}

  4. Binary Relations • R, S binary relations on a set X • R+S = R  S • RS = R ° S = {(u,v) | w (u,w)  R, (w,v)  S} • R* = reflexive transitive closure of R • = Un0 Rn = 1  R  R2 R3 • 1 = identity relation = {(u,u) | u  X} • 0 = 

  5. Applications • Automata and formal languages regular expressions • Relational algebra • Program logic and verification Dynamic Logic program analysis optimization • Design and analysis of algorithms shortest paths connectivity

  6. Conway’s Definition A Kleene algebra is any structure (K, +, ·, *, 0, 1) satisfying the same equations as the regular sets established the equational theory as the primary theory of interest—not necessarily the best choice

  7. Fundamental Questions • Can we axiomatize the valid equations? YES[Salomaa 66] —but not equationally[Redko 64] • Can we decide if expressions are equivalent? YES, PSPACE complete[(Stock+1)Meyer 74]

  8. def x  y  x + y = y Axioms of KA • K is an idempotent semiring under +, ·, 0, 1 • (p + q) + r = p + (q + r) (pq)r = p(qr) • p + q = q + p p1 = 1p = p • p + p = p p0 = 0p = 0 • p + 0 = p • p(q + r) = pq + pr • (p + q)r = pr + qr • p*q = least x such that q + px  x • qp* = least x such that q + xp  x

  9. This is a universal Horn axiomatization • p*q = least x such that q + px  x • q + p(p*q) p*q • q + px  x  p*q x • qp* = least x such that q + xp  x • q + p(qp*) qp* • q + px  x  qp* x • Every system of linear inequalities • a11x1 + ... + an1xn + b1 x1 • . • . • . • an1x1 + ... + annxn + bn xn • has a unique least solution

  10. Complete semirings (quantales) • i  I pi = supremum of {pi | i  I} • with respect to  • *-continuity • pq*r = sup pqnr • infinitary • same equational theory Eq(KA) = Eq(KA*) n  0 Alternative Characterizations of *

  11. Some Useful Properties 1 + pp* = 1 + p*p = p* p*p* = p** = p* (pq)*p = p(qp)* sliding (p*q)*p* = (p + q)*denesting px = xq  p*x = xq* bisimulation qp = 0  (p + q)* = p*q* loop distribution

  12. Proof of the Sliding Rule (ab)*a  a(ba)* a + aba(ba)* = a(1 + ba(ba)*) distributivity = a(ba)* 1 + pp* = p*. a + aba(ba)*  a(ba)* (ab)*a  a(ba)*q + px x  p*q x The reverse inequality  is symmetric.

  13. Equational Completeness [K91] • Reg, the KA regular sets over , is the • free KAon generators  • p  q as regular sets •  • p = q is a consequence of the KA axioms • KA iscompleteover relational models • Eq(REL) = Eq(KA) = Eq(Reg)

  14. Matrices over a KA a b c d e f g h a+e b+f c+g d+h def + = ae+bg af+bh ce+dg cf+dh a b c d e f g h · def = 1 0 0 1 0 0 0 0 def def = 0 = 1 a b c d * (a+bd*c)* (a+bd*c)*bd* (d+ca*b)*ca* (d+ca*b)* def =

  15. Matrices over a KA * a b c d (a+bd*c)* (a+bd*c)*bd* (d+ca*b)*ca* (d+ca*b)* def = b d a c

  16. Matrices over a KA • Representation of finite automata • Construction of regular expressions • Solution of linear equations over a KA • Connectivity and shortest path algorithms

  17. Solution of Linear Inequalities • a11x1 + ... + an1xn + b1 x1 • . • . • . • an1x1 + ... + annxn + bn xn • a11 ... an1 • . • . +  • . • an1 ... ann x1 . . . xn b1 . . . bn x1 . . . xn

  18. Shortest PathsThe min,+ algebra R+ {} r + s = min r,s rs = r + s r* = 0 0 =  1 = 0  =  .9 1.4 3.2 * 0 1.4 2.3 0 .9   0 0 1.4 3.2 0 .9   0 =

  19. Other Models Convex polyhedra [Iwano & Steiglitz 90] AB = {ax + by | x  A, y  B} A* = convex hull of A A B

  20. Kleene Algebra with Tests (KAT) • (K, B, +, ·, *, ¯, 0, 1) • (K, +, ·, *, 0, 1) is a Kleene algebra • (B, +, ·, ¯, 0, 1) is a Boolean algebra • B K • p,q,r, range over K • a,b,c, range over B

  21. Kleene Algebra with Tests (KAT) • +, ·, 0, 1 serve double duty • applied to programs, denote choice, composition, fail, and skip, resp. • applied to tests, denote disjunction, conjunction, falsity, and truth, resp. • these usages do not conflict! • bc = b  c b + c = b  c

  22. Models • Relational models • K = binary relations on a set X • B = subsets of the identity relation • Trace models • K = sets of traces u0p0u1p1u2 … un-1pn-1un • B = sets of traces of length 0 • Language-theoretic models • K = regular sets of guarded strings over  • B = atoms of a finite free Boolean algebra

  23. Guarded Strings [Kaplan 69] P atomic programs B atomic tests , , atoms (minimal nonzero elements) of the free Boolean algebra on generators B e.g. if B = {b1,...,b6}, then b1b2b3b4b5b6 is an atom guarded strings 0p01p12p23 n-1pn-1n A+B = A  B AB = {xy | x  A, y  B} A* = Un0 An 1 = {atoms} 0 = 

  24. Theorem [Kozen & Smith 96] • The family of regular sets of guarded strings over P,B is the free KAT on generators P,B. • Corollary • KAT is complete over relational models. • Eq(GS) = Eq(KAT) = Eq(KAT*) = Eq(REL)

  25. Matrices over a KAT The n x n matrices over a KAT (K,B) forms a KAT (K’,B’) B’ = diagonal matrices over B

  26. Modeling Programs[Fischer & Ladner 79] p;q pq if b then p else q bp + bq while b do p (bp)*b

  27. Hoare Logic [C. A. R. Hoare 69] {b}p{c} partial correctness assertion ‘‘If b holds in the current state, and if p is executed starting in the current state, then if p halts, c will be true of the halting state.’’

  28. {b}p{c},{c}q{d} {b}pq{d} b’  b, {b}p{c}, c  c’ {b’}p{c’} Rules of Hoare Logic assignment rule {b[x/e]} x:=e {b} composition rule {bc}p{d},{bc}q{d} {c}if b then p else q{d} conditional rule {bc}p{c} {c}while b do p{bc} while rule weakening rule

  29. KAT subsumes PHL {b}p{c} modeled by bp=bpcor bpc= 0

  30. {b}p{c},{c}q{d} {b}pq{d} composition rule {bc}p{d},{bc}q{d} {c}if b then p else q{d} conditional rule {bc}p{c} {c}while b do p{bc} while rule

  31. {b}p{c},{c}q{d} {b}pq{d} composition rule bpc = 0  cqd = 0bpqd = 0 {bc}p{d},{bc}q{d} {c}if b then p else q{d} conditional rule bcpd = 0  bcqd = 0 c(bp+bq)d = 0 {bc}p{c} {c}while b do p{bc} while rule bcpc = 0c(bp)*b bc = 0

  32. {b1}p1{c1}, ..., {bn}pn{cn} {b}p{c} • Theorem • These are all theorems of KAT • Completeness Theorem [K 99] • All relationally valid rules of the form • are derivable in KAT (not so for PHL)

  33. Counterexample {c}if b then p else p{c} {c}p{c} is trivially unprovable in Hoare Logic, but c(bp + bp)c = 0  cpc = 0 is easily provable in KAT

  34. Hoare formulas • p1 = 0  p2 = 0  ... pn = 0q = r • Theorem • KAT is complete for the Hoare theory of relational algebras • ... not for the Horn theory! • Counterexample: p  1  p2 = p

  35. Complexity of KAT and PHL • Theorem [Cohen 94] • The Hoare theory of KA (Horn formulas with • premises p = 0) is PSPACE-complete • Theorem [Cohen, Kozen & Smith 96] • The Hoare theory of KAT is PSPACE-complete • Theorem • PHL is PSPACE-complete

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