A Completeness Theorem for Kleene Algebras and the Algebra of Regular Events Dexter Kozen Department of Computer Science Cornell University Ithaca New York kozencsco rn ell ed u Abstract Wegive a nitary axiomatization of the algebra of regular events involving only equations and equational implications Unlike Salo maas axiomatizations the axiomatization given here is sound for all interpretations over Kleene algebras Intro duction Kleene algebras are algebraic structures with op erators and satisfying certain axioms They arise in various guises in a number of set tings relational algebra semantics and logics of programs automata and formal language theory and the design and analysis of algorithms An imp ortant example of a Kleene algebra is Reg the family of regular sets over a nite alphab et The equational theory of this structure has b een called the algebraofregular events This theory was rst studied by Infor and Comput May A preliminary version of this pap er app eared as Kleene who p osed axiomatization as an op en problem Salomaa gavetwo complete axiomatizations of the algebra of regular events in but these axiomatizations dep ended on rules of inference that are not sound in general under nonstandard interpretations Redkoproved in that no nite set of equational axioms could characterize the algebra of regular events The algebra of regular events and its axiomatization is the sub ject of the extensive monograph of Conway The bulk of Conways treatment is innitary In wegave a complete innitary equational deductive system for the algebra of regular events that is sound over all continuous Kleene algebras A completeness theorem for relational algebras with a prop er sub class of Kleene algebras was given byNgandTarski but their axiomatization relies on the presence of a converse op erator Schematic equational axiomatizations for the algebra of regular events necessarily rep resenting innitely many equations havebeengiven by Krob and Blo om There is some disagreement regarding the denition of Kleene algebras The literature contains several inequivalent denitions In this pap er weintro duce yet another a Kleene algebra is any mo del of the equations and equational implications given in x By general considerations of equational logic the axioms of Kleene al gebra listed in x along with the usual axioms for equalityinstantiation and rules for the intro duction and elimination of implications constitute a complete deductive system for the universal Horn theory of Kleene algebras the set of univ ersally quantied equational implications n n true in all Kleene algebras The main result of this pap er is that this deductive system is complete for the algebra of regular events In other words two regular expressions and over denote the same regular set in Reg if and only if the equation is a logical consequence of the axioms Equivalently Reg is the free Kleene algebra on free generators This gives a more satisfactory solution to Kleenes question than Salo maas solution since the axiomatization is sound over an entire arrayof imp ortant nonstandard interpretations arising in computer science The re sult is proved by enco ding the classical combinatorial constructions of the theory of nite automata eg state minimization algebraically There is an extensive literature on the algebra of regular events and much of the development of this pap er is a recapitulation of previous work For example the construction of a transition matrix representing a nite automaton equivalent to a given regular expression is essentially implicit in the work of Kleene and app ears in Conways monograph the algebraic approach to the elimination of transitions app ears in the workofKuich and Salomaa and Sakarovitch and the results on the closure of Kleene algebras under the formation of matrices essentially go backtoConways monograph and the thesis of Backhouse We extend this program byshowing how to enco de algebraically two fundamental constructions in the theory of nite automata determinization of an automaton via the subset construction and state minimization via equivalence mo dulo a MyhillNero de equivalence relation We then use the uniqueness of the minimal deterministic nite automaton to obtain completeness Conway states a similar theorem without pro of in the latter part of his b o ok Theorem p Krob based on work of Boa and Archangelsky have recently indep endently obtained similar results based on dierenttechniques Examples of Kleene Algebras Kleene algebras ab ound in computer science Wehave already mentioned the regular sets Reg In the area of relational algebra the family of binary relations on a set with the op erations of for relational comp osition R S fx z jy x y R y z S g for the empty relation for the identity relation for and reexive tran sitive closure for constitute a Kleene algebra In semantics and logics of programs Kleene algebras are used to mo del programs in Dynamic Logic and Dynamic Algebra In the design and analysis of algorithms n n Bo olean matrices and ma trices over the socalled min algebra are used to derive ecient algorithms for reachability and shortest paths in directed graphs A Kleene alge bra in which gives the vector sum of two p olygons and gives the convex hull of the union of two p olygons has b een used to solve a cycle problem in graphs These Kleene algebras app ear in in the guise of closed semirings which are similar to the Salgebras of Conway also called Kleene semirings Closed semirings and Salgebras are dened in terms P of an innitary summation op erator whose sole purp ose it seems is to dene These structures are all closely related the precise relationship is drawn in Salomaas Axiomatizations Before one can fully appreciate the axiomatization of x it is imp ortantto understand Salomaas axiomatizations and their limitations Let R denote the interpretation of regular expressions over in the Kleene algebra Reg in which R a fag a This is called the standard interpretation Salomaa presented two axiomatizations F and F for the algebra of regular events and proved their completeness Aanderaa indep endently presented a system similar to Salomaas F Backhouse gave an algebraic version of F These systems are equational except for one rule of inference in each case that is sound under the standard interpretation R butnot sound in general over other interpretations To describ e the system F letussay a regular expression p ossesses the empty wordproperty EWP if the regular set it denotes under R contains the null string The EWP can b e characterized syntactically a regular expression has the EWP if either for some is a sum of regular expressions at least one of which has the EWP or is a pro duct of regular expressions b oth of whichhavetheEWP The system F contains the rule do es not havetheEWP The rule is sound under the standard interpretation R butthe proviso do es not have the EWP is not algebraic in the sense that it is not preserved under substititution Consequently is not valid under nonstandard interpretations For example if and are the single letters a b and c resp ectively then holds but it do es not hold after the substitution a b c Another waytosaythisisthatmust not b e interpreted as a universal Horn formula Salomaas system F is somewhat dierent from F but suers from a similar drawback In contrast the axioms for Kleene algebra given in x b elow are all equa tions or equational implications in which the symb ols are regarded as uni versally quantied so substitution is allowed Axioms for Kleene Algebra A Kleene algebra is an algebraic structure K K satisfying the following equations and equational implications a b c a bc a b b a a a a a a abc abc a a a a ab c ab ac a bc ac bc a a aa a a a a b ax x a b x b xa x ba x where refers to the natural partial order on K a b a b b Instead of and wemight take the equivalent axioms ax x a x x xa x xa x Axioms say that K is an idemp otent commutative monoid Axioms say that K is a monoid Axioms saythat K is an idemp otent semiring The remaining axioms deal with They say essentially that b ehaves like the Kleene star op erator of formal language theory or the reexiv e transitive closure op erator of relational algebra Using and the distributivity axiom we see that b aa b a b thus the left hand side of the implication is satised when a b is sub stituted for xmoreover says that a b is the least elementof K for which this is true In short a b is the least prexp oint of the monotone map x b ax Axioms are studied by Pratt who attributes and to Schroder and Dedekind The equivalence of and and by symmetry of and are proved in No attempt has b een made to reduce the axioms ab ove to a minimal set and no claim is made as to their indep endence All the structures mentioned in x in particular Reg are Kleene alge bras Elementary consequences In this section we derive some basic consequences of the Kleene algebra ax ioms These prop erties are quite elementary and have b een observed b efore by many dierent authors We do not claim any of them as original results but include them merely for the sake of completeness We refer the reader to for a comprehensiveintro duction It is straightforward to verify that the relation is a partial order and is monotone with resp ect