# kozen1 - Kleene Algebra with Tests Dexter Kozen Cornell...

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Kleene Algebra with Tests Dexter Kozen Cornell University Workshop on Logic & Computation Nelson, NZ, January 2004

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These Lectures 1. Tutorial on KA and KAT model theory complexity, deductive completeness relation to Hoare logic 2. Practical applications compiler optimization scheme equivalence static analysis 3. Theoretical applications automata on guarded strings & BDDs algebraic version of Parikh’s theorem representation dynamic model theory
pq + qp p*q {pq,qp} {q,pq,p 2 q,p 3 q, …} (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} Kleene Algebra (KA) is the algebra of regular expressions 0 0 0 1 1 1 p q p,q p q p q

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Standard Interpretation Regular sets over Σ A+B = A B AB = {xy | x A, y B} A* = U A = A A ...
Binary Relations R, S binary relations on a set X R+S = R S RS = R ° S = {(u,v) | 5 w (u,w) R, (w,v) S} R* = reflexive transitive closure of R

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Applications Automata and formal languages regular expressions Program logic and verification Dynamic Logic program analysis protocol verification compiler optimization Algorithms shortest paths connectivity computational geometry
Definition, relation to finite automata Kleene 56 No purely equational axiomatization Redko 64 Axiomatization of equational theory Salomaa 66 Algebraic theory Conway 71 Equational theory PSPACE complete (Stock+1)Meyer 74 Prehistory

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Axioms of KA [K91] 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 def x y x + y = y
Succinctly stated, A Kleene algebra is an idempotent semiring such that p*q is the least fixpoint of λ x.(q + px) qp* is the least fixpoint of λ x.(q + xp)

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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 a 11 x 1 + . .. + a n1 x n + b 1 x 1 . . . a n1 x 1 + . .. + a nn x n + b n x n
Alternative Characterizations of * Complete semirings (quantales) i I p i = supremum of {p i | i I} with respect to *-continuity pq*r = sup pq n r n 0

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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 qp = pq (p + q)* = (pq)*(p* + q*) Some Useful Properties
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## kozen1 - Kleene Algebra with Tests Dexter Kozen Cornell...

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