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We have formalized the semantics of lazy evaluation for the lambda calculus using the two-level grammar formalism. The resulting semantics enjoys several properties, e.g., there is a sharing in the recursive computation, there is no α conversion, the heap is automatically reclaimed, an attempt to evaluate an argument is done at most once and there is a sharing in the evaluation of partial application to functions

The notion of hypergraph cyclicity is important in numerous fields of application of hypergraph theory in computer science and relational database theory. The database scheme and query can be represented as a hypergraph. The database scheme (or query) has a cycle if the corresponding hypergraph has a cycle. An Acyclic database has several desired computational properties such as making query optimization easier and can be recognized in linear time. In this paper, we introduce a new type of cyclicity in hypergraphs via the notions of Quasi α-cycle(s) and the set of α-nodes in hypergraphs, which are based on the existence of an αcycle(s). Then, it is proved that a hypergraph is acyclic if and only if it does not contain any α-nodes. Moreover, a polynomial-time algorithm is proposed to detect the set of α-nodes based on the existence of Quasi α-cycle(s), or otherwise claims the acyclicity of the hypergraph. Finally, a systematic discussion is given to show how to use the detected set of α-nodes to convert the cyclic hypergraph into acyclic one if the conversion is possible. The acyclic database and acyclic query enjoy time and/or space-efficient access paths for answering a query.

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ML-style multi-abstraction calculus, ߣ letmx , is introduced as an extension of the Hindley-Milner type system. The calculus has a new multiabstraction, a simultaneous application and a simultaneous explicit substitution. In comparison with some other multi-abstraction formalization, the calculus has the advantage of saving the usual α equivalence between all terms. The calculus can be used to represent contexts, where contexts in lambda calculus are lambda terms with holes. The calculus has a subject reduction property, is confluent and has a sound and complete type inference algorithm.