A type system for Continuation Calculus

Continuation Calculus (CC), introduced by Geron and Geuvers, is a simple foundational model for functional computation. It is closely related to lambda calculus and term rewriting, but it has no variable binding and no pattern matching. It is Turing complete and evaluation is deterministic. Notions like "call-by-value" and "call-by-name" computation are available by choosing appropriate function definitions: e.g. there is a call-by-value and a call-by-name addition function. In the present paper we extend CC with types, to be able to define data types in a canonical way, and functions over these data types, defined by iteration. Data type definitions follow the so-called "Scott encoding" of data, as opposed to the more familiar "Church encoding". The iteration scheme comes in two flavors: a call-by-value and a call-by-name iteration scheme. The call-by-value variant is a double negation variant of call-by-name iteration. The double negation translation allows to move between call-by-name and call-by-value.Comment: In Proceedings CL&C 2014, arXiv:1409.259

Very Simple Chaitin Machines for Concrete AIT

In 1975, Chaitin introduced his celebrated Omega number, the halting probability of a universal Chaitin machine, a universal Turing machine with a prefix-free domain. The Omega number's bits are {\em algorithmically random}--there is no reason the bits should be the way they are, if we define ``reason'' to be a computable explanation smaller than the data itself. Since that time, only {\em two} explicit universal Chaitin machines have been proposed, both by Chaitin himself. Concrete algorithmic information theory involves the study of particular universal Turing machines, about which one can state theorems with specific numerical bounds, rather than include terms like O(1). We present several new tiny Chaitin machines (those with a prefix-free domain) suitable for the study of concrete algorithmic information theory. One of the machines, which we call Keraia, is a binary encoding of lambda calculus based on a curried lambda operator. Source code is included in the appendices. We also give an algorithm for restricting the domain of blank-endmarker machines to a prefix-free domain over an alphabet that does not include the endmarker; this allows one to take many universal Turing machines and construct universal Chaitin machines from them

Graph Lambda Theories

to appear in MSCSInternational audienceA longstanding open problem in lambda calculus is whether there exist continuous models of the untyped lambda calculus whose theory is exactly lambda-beta or the the least sensible lambda-theory H (generated by equating all the unsolvable terms). A related question, raised recently by C. Berline, is whether, given a class of lambda models, there are a minimal lambda-theory and a minimal sensible lambda-theory represented by it. In this paper, we give a positive answer to this question for the class of graph models à la Plotkin-Scott-Engeler. In particular, we build two graph models whose theories are respectively the set of equations satisfied in any graph model and in any sensible graph model. We conjecture that the least sensible graph theory, where ''graph theory" means ''lambda-theory of a graph model", is equal to H, while in one of the main results of the paper we show the non-existence of a graph model whose equational theory is exactly the beta-theory. Another related question is whether, given a class of lambda models, there is a maximal sensible lambdatheory represented by it. In the main result of the paper we characterize the greatest sensible graph theory as the lambda-theory B generated by equating lambda-terms with the same Boehm tree. This result is a consequence of the main technical theorem of the paper: all the equations between solvable lambda-terms, which have different Boehm trees, fail in every sensible graph model. A further result of the paper is the existence of a continuum of different sensible graph theories strictly included in B

Coinductive big-step operational semantics

Using a call-by-value functional language as an example, this article illustrates the use of coinductive definitions and proofs in big-step operational semantics, enabling it to describe diverging evaluations in addition to terminating evaluations. We formalize the connections between the coinductive big-step semantics and the standard small-step semantics, proving that both semantics are equivalent. We then study the use of coinductive big-step semantics in proofs of type soundness and proofs of semantic preservation for compilers. A methodological originality of this paper is that all results have been proved using the Coq proof assistant. We explain the proof-theoretic presentation of coinductive definitions and proofs offered by Coq, and show that it facilitates the discovery and the presentation of the results

On the Termination Problem for Probabilistic Higher-Order Recursive Programs

In the last two decades, there has been much progress on model checking of both probabilistic systems and higher-order programs. In spite of the emergence of higher-order probabilistic programming languages, not much has been done to combine those two approaches. In this paper, we initiate a study on the probabilistic higher-order model checking problem, by giving some first theoretical and experimental results. As a first step towards our goal, we introduce PHORS, a probabilistic extension of higher-order recursion schemes (HORS), as a model of probabilistic higher-order programs. The model of PHORS may alternatively be viewed as a higher-order extension of recursive Markov chains. We then investigate the probabilistic termination problem -- or, equivalently, the probabilistic reachability problem. We prove that almost sure termination of order-2 PHORS is undecidable. We also provide a fixpoint characterization of the termination probability of PHORS, and develop a sound (but possibly incomplete) procedure for approximately computing the termination probability. We have implemented the procedure for order-2 PHORSs, and confirmed that the procedure works well through preliminary experiments that are reported at the end of the article