Left-Handed Completeness

Abstract

Abstract. We give a new, significantly shorter proof of the completeness of the left-handed star rule of Kleene algebra. The proof exposes the rich interaction of algebra and coalgebra in the theory of Kleene algebra.

On the Fine-Structure of Regular Algebra

“…Boffa proved completeness of a simplified left-handed variant of Kleene algebra [7] relative to Krob's result. An alternative proof has recently been published by Kozen and Silva [23]. Our analysis sheds some new light on Kleene algebra, showing that they are, in fact, equipollent to Conway's variants.…”

Left-Handed Completeness for Kleene algebra, via Cyclic Proofs

“…15.15]. Kozen and Silva recently gave a shorter but still highly technical proof [23], by relying on the coalgebraic semantics of regular expressions: Brzozowski's derivatives [11]. We give a new proof in this work.Removing the implication (R) above disqualifies Kozen's technique for proving completeness [20]: both (L) and (R) are needed to replay automaton algorithms algebraically.…”

“…15.15]. Kozen and Silva recently gave a shorter but still highly technical proof [23], by relying on the coalgebraic semantics of regular expressions: Brzozowski's derivatives [11]. We give a new proof in this work.…”

“…In fact, the key difficulty for left-handed completeness consists in finding invariants to exploit the remaining implication (L). A key feature of our proof is that we compute these invariants explicitly: we do not need to compute decompositions and solve systems of linear equations, as in [23].Our completeness proof builds on a sequent-style calculus, HKA, which was proposed recently in [14]. This system admits 'cyclic' proofs, where proofs are finite graphs with cycles rather than trees or dags, and soundness and completeness for rational language inclusions holds under a simple correctness criterion.…”

“…In fact, the key difficulty for left-handed completeness consists in finding invariants to exploit the remaining implication (L). A key feature of our proof is that we compute these invariants explicitly: we do not need to compute decompositions and solve systems of linear equations, as in [23].…”