Adolfo Piperno

A note on optimal area algorithms for upward drawings of binary trees

The goal of this paper is to investigate the area requirements for upward grid drawings of binary trees. First, we show that there is a family of binary trees with n vertices that require ω(n log n) area; this bound is tight to within a constant factor, i.e. any binary tree with n vertices can be drawn in O(n log n) area. Then we present an algorithm for constructing an upward drawing of a complete binary tree with n vertices in O(n) area, and, finally, we extend this result to the drawings of Fibonacci trees.

A Filter Model for Concurrent

Type-free lazy

\lambda

\lambda

-Calculus

-calculus is enriched with angelic parallelism and demonic nondeterminism. Call-by-name and call-by-value abstractions are considered and the operational semantics is stated in terms of a must convergence predicate. We introduce a type assignment system with intersection and union types, and we prove that the induced logical semantics is fully abstract.

Nondeterministic Extensions of Untyped λ-Calculus

The main concern of this paper is the interplay between functionality and nondeterminism. We ask whether the analysis of parallelism in terms of sequentiality and nondeterminism, which is usual in the algebraic treatment of concurrency, remains correct in the presence of functional application and abstraction, We argue in favour of a distinction between nondeterminism and parallelism, due to the conjunctive nature of the former in contrast to the disjunctive character of the latter. This is the basis of our analysis of the operational and denotational semantics of the nondeterministic ?-calculus, which is the classical calculus plus a choice operator, and of our election of bounded indeterminacy as the semantic counterpart of conjunctive nondeterminism. This leads to operational semantics based on the idea of must preorder, coming from the classical theory of solvability and from the theory of process algebras. To characterize this relation, we build a model using the inverse limit construction over nondeterministic algebras, and we prove it fully abstract using a generalization of Bohm trees. We further prove conservativity theorems for the equational theory of the model and for other theories related to nondeterministic ?-calculus with respect to classical ?-theories.

Filter models for conjunctive-disjunctive l-calculi

Abstract The distinction between the conjunctive nature of non-determinism as opposed to the disjunctive character of parallelism constitutes the motivation and the starting point of the present work. λ-calculus is extended with both a non-deterministic choice and a parallel operator; a notion of reduction is introduced, extending β-reduction of the classical calculus. We study type assignment systems for this calculus, together with a denotational semantics which is initially defined constructing a set semimodel via simple types. We enrich the type system with intersection and union types, dually reflecting the disjunctive and conjunctive behaviour of the operators, and we build a filter model. The theory of this model is compared both with a Morris-style operational semantics and with a semantics based on a notion of capabilities.

Optimal-Area Upward Drawings of AVL Trees

We prove that any AVL tree admits a linear-area planar straight-line grid strictly-upward drawing, that is, a drawing in which (a) no two edges intersect, (b) each edge is mapped into a single straight-line segment, (c) each node is mapped into a point with integer coordinates, and (d) each node is placed below its parent.

Linear area upward drawings of AVL trees

Abstract We prove that any AVL tree admits a linear-area straight-line strictly-upward planar grid drawing, that is, a drawing in which (a) each edge is mapped into a single straight-line segment, (b) each node is placed below its parent, (c) no two edges intersect, and (d) each node is mapped into a point with integer coordinates.

Expanding Extensional Polymorphism

We prove the confluence and strong normalization properties for second order lambda calculus equipped with an expansive version of η-reduction. Our proof technique, based on a simple abstract lemma and a labelled λ-calculus, can also be successfully used to simplify the proofs of confluence and normalization for first order calculi, and can be applied to various extensions of the calculus presented here.

Intersection types and λ-definability

This paper presents a novel method for comparing computational properties of λ-terms that are typeable with intersection types, with respect to terms that are typeable with Curry types. We introduce a translation from intersection typing derivations to Curry typeable terms that is preserved by β-reduction: this allows the simulation of a computation starting from a term typeable in the intersection discipline by means of a computation starting from a simply typeable term. Our approach proves strong normalisation for the intersection system naturally by means of purely syntactical techniques. The paper extends the results presented in Bucciarelli et al. (1999) to the whole intersection type system of Barendregt, Coppo and Dezani, thus providing a complete proof of the conjecture, proposed in Leivant (1990), that all functions uniformly definable using intersection types are already definable using Curry types.