Skew confluence and the lambda calculus with letrec Original Research Article

We present an extension of the lambda calculus with the letrec construct. In contrast to current theories, which impose restrictions on where the rewriting can take place, our theory is very liberal, e.g., it allows rewriting under lambda abstractions and on cycles. As shown previously, the reduction theory is non-confluent. Thus, we searched for and found a new property that resembles confluence and that is equivalent to uniqueness of infinite normal forms: skew confluence. This notion is based on the intuition that some terms are better than others and that terms reduce to better terms. It states that if a term reduces to two other terms, the second of those terms can always be reduced to a term that is better than the first. Using skew confluence we define the infinite normal form of a term and show that the infinite normal form defines a congruence on the set of terms. We relate the lambda calculus plus letrec to the plain lambda calculus and to one of the infinitary lambda calculi. We also present a variant of our calculus, which follows the tradition of the explicit substitution calculi.