A fully-abstract semantics of lambda-mu in the pi-calculus

This paper is part of a general programme of treating explicit substitutions as the primary λ-calculi from the point of view of foundations as well as applications. We work in a composition-free calculus of explicit substitutions and an augmented calculus obtained by adding explicit garbage-collection, and explore the relationship between intersection-types and reduction. We show that the terms that normalise by leftmost reduction and the terms that normalise by head reduction can each be characterised as the terms typable in a certain system. The relationship between typability and strong normalisation is subtly different from the classical case: we show that typable terms are strongly normalising but give a counterexample to the converse. Our notions of leftmost and head reduction are non-deterministic, and our normalisation theorems apply to any computations obeying these strategies. In this way we refine and strengthen the classical normalisation theorems. The proofs require some new techniques in the presence of reductions involving explicit substitutions. Indeed, our proofs do not rely on results from classical λ-calculus, which in our view is subordinate to the calculus of explicit substitution.

The teleparallel coframe gravity may be viewed as a generalization of the standard GR. A coframe (a field of four independent 1-forms) is considered, in this approach, to be a basic dynamical variable. The metric tensor is treated as a secondary structure. The general Lagrangian, quadratic in the first order derivatives of the coframe field is not unique. It involves three dimensionless free parameters. We consider a weak field approximation of the general coframe teleparallel model. In the linear approximation, the field variable, the coframe, is covariantly reduced to the superposition of the symmetric and antisymmetric field. We require this reduction to be preserved on the levels of the Lagrangian, of the field equations, and of the conserved currents. This occurs if and only if the pure Yang–Mills-type term is removed from the Lagrangian. The absence of this term is known to be necessary and sufficient for the existence of the viable (Schwarzschild) spherical-symmetric solution. Moreover, the same condition guarantees the absence of ghosts and tachyons in particle content of the theory. The condition above is shown recently to be necessary for a well-defined Hamiltonian formulation of the model. Here we derive the same condition in the Lagrangian formulation by means of the weak field reduction.

Standardization is a fundamental notion for connecting programming languages and rewriting calculi. Since both programming languages and calculi rely on substitution for defining their dynamics, explicit substitutions (ES) help further close the gap between theory and practice. This paper focuses on standardization for the linear substitution calculus, a calculus with ES capable of mimicking reduction in lambda-calculus and linear logic proof-nets. For the latter, proof-nets can be formalized by means of a simple equational theory over the linear substitution calculus. Contrary to other extant calculi with ES, our system can be equipped with a residual theory in the sense of Lévy, which is used to prove a left-to-right standardization theorem for the calculus with ES but without the equational theory. Such a theorem, however, does not lift from the calculus with ES to proof-nets, because the notion of left-to-right derivation is not preserved by the equational theory. We then relax the notion of left-to-right standard derivation, based on a total order on redexes, to a more liberal notion of standard derivation based on partial orders. Our proofs rely on Gonthier, Lévy, and Melliès' axiomatic theory for standardization. However, we go beyond merely applying their framework, revisiting some of its key concepts: we obtain uniqueness (modulo) of standard derivations in an abstract way and we provide a coinductive characterization of their key abstract notion of external redex. This last point is then used to give a simple proof that linear head reduction --a nondeterministic strategy having a central role in the theory of linear logic-- is standard.

In this paper, we put forward a weak conjunctive normal form for G3, which is similar to the conjunctive normal form for classical propositional logic. We also give two algorithms to reduce a formula to weak conjunctive normal form, where one is based on charactering countermodels in model-based-formulas, and the other is based on rewriting formulas according to transformation rules. In the end, we show that there is no polynomial time algorithm to reduce an arbitrary formula to the weak conjunctive normal form.

A completeness result for a realisability semantics for an intersection type system

We present a natural implementation of the Krivine machine in interaction nets: one rule for each transition and the usual rules for duplication and erasing. There is only one rule devoted to the so-called administration. This way, we obtain a graphical system encoding λ-calculus weak head reduction that can be extended to a λ-calculus normalizer by encoding left reduction.

Bisimulation in Untyped Lambda Calculus:: Böhm Trees and Bisimulation up to Context

We propose a lambda calculus λ^→_{\mathit{env}} where it is possible to handle first-class environments. This calculus is based on the idea of explicit substitution , that is λ σ -calculus. Syntax of λ^→_{\mathit{env}} is obtained by merging the class of terms and the one of substitutions. Reduction is made from the weak reduction of λ σ -calculus. Its type system also originates in the one of λ σ -calculus. Confluence of λ^→_{\mathit{env}} is proved by Hardin's interpretation method which is originally used for proving confluence of λ σ -calculus. We proved strong normalizability of λ^→_{\mathit{env}} by reducing it to strong normalizability of a simply typed record calculus. Finally, we propose a type inference algorithm which produced a principal typing for each typable term.

Context bisimulation [12, 1] has become an important notion of behavioural equivalence for higher-order processes. Weak forms of context bisimulation are particularly interesting, because of their high level of abstraction. We present a modal logic for this setting and provide a characterization of a variant of weak context bisimulation on second-order processes. We show how the logic permits compositional reasoning. In comparison to previous work by Amadio and Dam [2] on the strong case, our modal logic supports derived operators through a complete duality and thus constitutes an appealing extension of Hennessy-Milner logic.

In a paper entitled cocycles and spectra Helson and Par ry prove that for every T there exists a q~ such that V ~ has Lebesgue spectrum, moreover q~ can be chosen to be real, i.e., taking values + 1 and 1. The purpose of this paper is to extend this result to certain actions of countable groups which includes ergodic non-singular actions of countable abelian groups. We blend the method of Helson and Parry with the notions of weak equivalence and weak von Neumann transformations. In section 3 and 4 we discuss these results in connection with systems of imprimitivity. The problem of extending the result quoted above to countable groups was raised by H. Helson to one of us. It is a pleasure to acknowledge his interest and encouragement in this work.

First, we extend Leifer-Milner RPO theory, by giving general conditions to obtain IPO labelled transition systems (and bisimilarities) with a reduced set of transitions, and possibly finitely branching. Moreover, we study the weak variant of Leifer-Milner theory, by giving general conditions under which the weak bisimilarity is a congruence. Then, we apply such extended RPO technique to the lambda-calculus, endowed with lazy and call by value reduction strategies. We show that, contrary to process calculi, one can deal directly with the lambda-calculus syntax and apply Leifer-Milner technique to a category of contexts, provided that we work in the framework of weak bisimilarities. However, even in the case of the transition system with minimal contexts, the resulting bisimilarity is infinitely branching, due to the fact that, in standard context categories, parametric rules such as the beta-rule can be represented only by infinitely many ground rules. To overcome this problem, we introduce the general notion of second-order context category. We show that, by carrying out the RPO construction in this setting, the lazy observational equivalence can be captured as a weak bisimilarity equivalence on a finitely branching transition system. This result is achieved by considering an encoding of lambda-calculus in Combinatory Logic.

Team equivalences for finite-state machines with silent moves

This paper concerns the study of the global structure of measure-preserving actions of countable groups on standard probability spaces. Weak containment is a hierarchical notion of complexity of such actions, motivated by an analogous concept in the theory of unitary representations. This concept gives rise to an associated notion of equivalence of actions, called weak equivalence, which is much coarser than the notion of isomorphism (conjugacy). It is well understood now that, in general, isomorphism is a very complex notion, a fact which manifests itself, for example, in the lack of any reasonable structure in the space of actions modulo isomorphism. On the other hand, the space of weak equivalence classes is quite well behaved. Another interesting fact that relates to the study of weak containment is that many important parameters associated with actions, such as the type, cost, and combinatorial parameters, turn out to be invariants of weak equivalence and in fact exhibit desirable monotonicity properties with respect to the pre-order of weak containment, a fact that can be useful in certain applications. There has been quite a lot of activity in this area in the last few years, and our goal in this paper is to provide a survey of this work.