The (In)Efficiency of interaction

Evaluating higher-order functional programs through abstract machines inspired by the geometry of the interaction is known to induce space efficiencies, the price being time performances often poorer than those obtainable with traditional, environment-based, abstract machines. Although families of lambda-terms for which the former is exponentially less efficient than the latter do exist, it is currently unknown how general this phenomenon is, and how far the inefficiencies can go, in the worst case. We answer these questions formulating four different well-known abstract machines inside a common definitional framework, this way being able to give sharp results about the relative time efficiencies. We also prove that non-idempotent intersection type theories are able to precisely reflect the time performances of the interactive abstract machine, this way showing that its time-inefficiency ultimately descends from the presence of higher-order types

On Irrelevance and Algorithmic Equality in Predicative Type Theory

Dependently typed programs contain an excessive amount of static terms which are necessary to please the type checker but irrelevant for computation. To separate static and dynamic code, several static analyses and type systems have been put forward. We consider Pfenning's type theory with irrelevant quantification which is compatible with a type-based notion of equality that respects eta-laws. We extend Pfenning's theory to universes and large eliminations and develop its meta-theory. Subject reduction, normalization and consistency are obtained by a Kripke model over the typed equality judgement. Finally, a type-directed equality algorithm is described whose completeness is proven by a second Kripke model.Comment: 36 pages, superseds the FoSSaCS 2011 paper of the first author, titled "Irrelevance in Type Theory with a Heterogeneous Equality Judgement

Local consistency as a reduction between constraint satisfaction problems

We study the use of local consistency methods as reductions between constraint satisfaction problems (CSPs), and promise version thereof, with the aim to classify these reductions in a similar way as the algebraic approach classifies gadget reductions between CSPs. This research is motivated by the requirement of more expressive reductions in the scope of promise CSPs. While gadget reductions are enough to provide all necessary hardness in the scope of (finite domain) non-promise CSP, in promise CSPs a wider class of reductions needs to be used.We provide a general framework of reductions, which we call consistency reductions, that covers most (if not all) reductions recently used for proving NP-hardness of promise CSPs. We prove some basic properties of these reductions, and provide the first steps towards understanding the power of consistency reductions by characterizing a fragment associated to arc-consistency in terms of polymorphisms of the template. In addition to showing hardness, consistency reductions can also be used to provide feasible algorithms by reducing to a fixed tractable (promise) CSP, for example, to solving systems of affine equations. In this direction, among other results, we describe the well-known Sherali-Adams hierarchy for CSP in terms of a consistency reduction to linear programming

On the logical complexity of cyclic arithmetic

We study the logical complexity of proofs in cyclic arithmetic ($\mathsf{CA}$), as introduced in Simpson '17, in terms of quantifier alternations of formulae occurring. Writing $C\Sigma_n$ for (the logical consequences of) cyclic proofs containing only $\Sigma_n$ formulae, our main result is that $I\Sigma_{n+1}$ and $C\Sigma_n$ prove the same $\Pi_{n+1}$ theorems, for all $n\geq 0$. Furthermore, due to the 'uniformity' of our method, we also show that $\mathsf{CA}$ and Peano Arithmetic ($\mathsf{PA}$) proofs of the same theorem differ only exponentially in size. The inclusion $I\Sigma_{n+1} \subseteq C\Sigma_n$ is obtained by proof theoretic techniques, relying on normal forms and structural manipulations of $\mathsf{PA}$ proofs. It improves upon the natural result that $I\Sigma_n$ is contained in $C\Sigma_n$. The converse inclusion, $C\Sigma_n \subseteq I\Sigma_{n+1}$, is obtained by calibrating the approach of Simpson '17 with recent results on the reverse mathematics of B\"uchi's theorem in Ko{\l}odziejczyk, Michalewski, Pradic & Skrzypczak '16 (KMPS'16), and specialising to the case of cyclic proofs. These results improve upon the bounds on proof complexity and logical complexity implicit in Simpson '17 and also an alternative approach due to Berardi & Tatsuta '17. The uniformity of our method also allows us to recover a metamathematical account of fragments of $\mathsf{CA}$; in particular we show that, for $n\geq 0$, the consistency of $C\Sigma_n$ is provable in $I\Sigma_{n+2}$ but not $I\Sigma_{n+1}$. As a result, we show that certain versions of McNaughton's theorem (the determinisation of $\omega$-word automata) are not provable in $\mathsf{RCA}_0$, partially resolving an open problem from KMPS '16

Proving Soundness of Extensional Normal-Form Bisimilarities

International audienceNormal-form bisimilarity is a simple, easy-to-use behavioral equivalence that relates terms in λ-calculi by decomposing their normal forms into bisimilar subterms. Besides, they allow for powerful up-to techniques, such as bisimulation up to context, which simplify bisimulation proofs even further. However, proving soundness of these relations becomes complicated in the presence of η-expansion and usually relies on ad-hoc proof methods which depend on the language. In this paper, we propose a more systematic proof method to show that an extensional normal-form bisimilarity along with its corresponding bisimulation up to context are sound. We illustrate our technique with the call-by-value λ-calculus, before applying it to a call-by-value λ-calculus with the delimited-control operators shift and reset. In both cases, there was previously no sound bisimulation up to context validating the η-law. Our results have been formalized in the Coq proof assistant