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A representation for lambda terms is described based on the scheme of de Bruijn for eliminating variable names. The new notation provides for a class of terms that can encode other terms together with substitutions to be performed on them. The notion of an environment is used to realize this "delaying" of substitutions.The precise mechanism that is used is, however, more complex than the usual so as to support the ability to examine subterms embedded under abstractions. A virtue of our representation is that it permits substitution to be realized as an atomic operation and thereby provides for efficient implementations of P-reduction.Operations on X-terms are described based on our representation and the relationship of these to the conventional definitions are exhibited.

We present a novel definition of an algorithm and its corresponding algorithm language called CoLweb. The merit of CoLweb [1] is that it makes algorithm design so versatile. That is, it forces us to a high-level, proof-carrying, distributed-style approach to algorithm design for both non-distributed computing and distributed one. We argue that this approach simplifies algorithm design. In addition, it unifies other approaches including recursive logical/functional algorithms, imperative algorithms, object-oriented imperative algorithms, neural-nets, interaction nets, proof-carrying code, etc.As an application, we refine Horn clause definitions into two kinds: blinduniverally-quantified (BUQ) ones and parallel-universally-quantified (PUQ) ones. BUQ definitions corresponds to the traditional ones such as those in Prolog where knowledgebase is not expanding and its proof procedure is based on the backward chaining. On the other hand, in PUQ definitions, knowledgebase is expanding and its proof procedure leads to forward chaining and automatic memoization.