Loop Quasi-Invariant Chunk Motion by peeling with statement composition

Several techniques for analysis and transformations are used in compilers. Among them, the peeling of loops for hoisting quasi-invariants can be used to optimize generated code, or simply ease developers' lives. In this paper, we introduce a new concept of dependency analysis borrowed from the field of Implicit Computational Complexity (ICC), allowing to work with composed statements called Chunks to detect more quasi-invariants. Based on an optimization idea given on a WHILE language, we provide a transformation method - reusing ICC concepts and techniques - to compilers. This new analysis computes an invariance degree for each statement or chunks of statements by building a new kind of dependency graph, finds the maximum or worst dependency graph for loops, and recognizes if an entire block is Quasi-Invariant or not. This block could be an inner loop, and in that case the computational complexity of the overall program can be decreased. We already implemented a proof of concept on a toy C parser 1 analysing and transforming the AST representation. In this paper, we introduce the theory around this concept and present a prototype analysis pass implemented on LLVM. In a very near future, we will implement the corresponding transformation and provide benchmarks comparisons.Comment: In Proceedings DICE-FOPARA 2017, arXiv:1704.0516

SCT and the idempotence condition

5 pagesThese notes show how to get rid of the idempotence condition (G=G;G) in (delta)-Size Change Termination

On Quasi-Interpretations, Blind Abstractions and Implicit Complexity

Quasi-interpretations are a technique to guarantee complexity bounds on first-order functional programs: with termination orderings they give in particular a sufficient condition for a program to be executable in polynomial time, called here the P-criterion. We study properties of the programs satisfying the P-criterion, in order to better understand its intensional expressive power. Given a program on binary lists, its blind abstraction is the nondeterministic program obtained by replacing lists by their lengths (natural numbers). A program is blindly polynomial if its blind abstraction terminates in polynomial time. We show that all programs satisfying a variant of the P-criterion are in fact blindly polynomial. Then we give two extensions of the P-criterion: one by relaxing the termination ordering condition, and the other one (the bounded value property) giving a necessary and sufficient condition for a program to be polynomial time executable, with memoisation.Comment: 18 page

Study of the NP-completeness of the compact table problem

ISBN 978-5-94057-377-7International audienceThe problem of compact tables is to maximise the overlap when building a word that is to include permutations of every given words (all the words being the same length). This problem is shown to be NP-complete in the general case, and some specific restrictions are studied

Chains, Antichains, and Complements in Infinite Partition Lattices

We consider the partition lattice $\Pi_\kappa$ on any set of transfinite cardinality $\kappa$ and properties of $\Pi_\kappa$ whose analogues do not hold for finite cardinalities. Assuming the Axiom of Choice we prove: (I) the cardinality of any maximal well-ordered chain is always exactly $\kappa$; (II) there are maximal chains in $\Pi_\kappa$ of cardinality $> \kappa$; (III) if, for every cardinal $\lambda < \kappa$, we have $2^{\lambda} < 2^\kappa$, there exists a maximal chain of cardinality $< 2^{\kappa}$ (but $\ge \kappa$) in $\Pi_{2^\kappa}$; (IV) every non-trivial maximal antichain in $\Pi_\kappa$ has cardinality between $\kappa$ and $2^{\kappa}$, and these bounds are realized. Moreover we can construct maximal antichains of cardinality $\max(\kappa, 2^{\lambda})$ for any $\lambda \le \kappa$; (V) all cardinals of the form $\kappa^\lambda$ with $0 \le \lambda \le \kappa$ occur as the number of complements to some partition $\mathcal{P} \in \Pi_\kappa$, and only these cardinalities appear. Moreover, we give a direct formula for the number of complements to a given partition; (VI) Under the Generalized Continuum Hypothesis, the cardinalities of maximal chains, maximal antichains, and numbers of complements are fully determined, and we provide a complete characterization.Comment: 24 pages, 2 figures. Submitted to Algebra Universalis on 27/11/201

Quasi-interpretations a way to control resources

International audienceThis paper presents in a reasoned way our works on resource analysis by quasi- interpretations. The controlled resources are typically the runtime, the runspace or the size of a result in a program execution. Quasi-interpretations allow analyzing system complexity. A quasi-interpretation is a numerical assignment, which provides an upper bound on computed func- tions and which is compatible with the program operational semantics. Quasi- interpretation method offers several advantages: (i) It provides hints in order to optimize an execution, (ii) it gives resource certificates, and (iii) finding quasi- interpretations is decidable for a broad class which is relevant for feasible com- putations. By combining the quasi-interpretation method with termination tools (here term orderings), we obtained several characterizations of complexity classes starting from Ptime and Pspace

Computability in the lattice of equivalence relations

We investigate computability in the lattice of equivalence relations on the natural numbers. We mostly investigate whether the subsets of appropriately defined subrecursive equivalence relations -for example the set of all polynomial-time decidable equivalence relations- form sublattices of the lattice.Comment: In Proceedings DICE-FOPARA 2017, arXiv:1704.0516

A toy model for approaching volcanic plumbing systems as complex systems

Magmas form at depth, move upwards and evolve chemically through a combination of processes. Magmatic processes are investigated by means of fieldwork combined with geophysics, geochemistry, analog and numerical models, and many other approaches. However, scientists in the field still struggle to understand how the variety of magmatic products arises, and there is no consensus yet on models of volcanic plumbing systems. This is because eruptions result from the integration of multiple processes, rooted in the magma source either in the mantle or lower crust that feeds a complex network of magma bodies linking magma source and volcano. In this work, we investigate the potential of the network approach through a prototype of magma pool interaction and magma transfer across the crust. In network terms, it describes a diffusion process on a dynamic spatial network, in which diffusion and network evolution are intertwined: the diffusion affects the network structure, and reciprocally. The diffusion process and network evolution mechanisms come from rules of behaviour derived from rock mechanics and melting processes. Nodes represent magma pools and edges physical connections between them, e.g., dykes or veinlets

The Murchison Greenstone Belt, South Africa: Accreted slivers with contrasting metamorphic conditions

International audienceThis paper presents new petrological and geochronological data for the ∼3.09-2.92 Ga Murchison Greenstone Belt (MGB), located in South Africa's Kaapvaal Craton, and discusses their geotectonic implications. The MGB is made of three tectono-metamorphic units: the Silwana Amphibolites, the Murchison Unit and the La France Formation. They underwent contrasting clockwise pressure-temperature-deformation (P-T-D) histories, and are separated from each other by relatively narrow, high-strain shear zones, with a sinistral, transpressive top-to-the-south movement, consistent with the deformation patterns observed throughout the belt. These patterns are explained by a N-S compressional stress field, affecting the Murchison Belt between 2.97 and 2.92 Ga. Results of new petrological investigations indicate that ultramafic to felsic volcano-sedimentary rocks of the Murchison Unit underwent a greenschist- to lower-amphibolite-facies metamorphism at maximum P-T conditions of 5.6 ± 0.6 kbar at 570 °C, along a relatively hot, minimum apparent geotherm of ∼30 °C/km. In contrast, the Silwana Amphibolites and the La France Formation were metamorphosed at much higher peak metamorphic conditions of 8.7-10 kbar, 630-670 °C, and 8-9 kbar, 600-650 °C, respectively, and require a colder apparent geotherm of ∼20 °C/km. A retrograde, nearly isothermal-decompression P-T path followed by isobaric cooling is also inferred for the La France Formation. The timing of the structural-metamorphic overprint is bracketed between 2.97 and 2.90 Ga, which is constrained by U-Pb zircon ages of a syn-deformation granite within the Murchison Unit and the post-deformation Maranda granite, respectively. Monazite and xenotime from La France metapelites yield much younger ages of ca. 2.75 Ga, with few inherited components at 2.92 Ga. They point to a later activation of the MGB, perhaps related with tectono-thermal events in the Rooiwater Complex and the Pietersburg Greenstone Belt. The relatively cold apparent geotherms recorded in the Silwana and La France rocks, the contrasted peak P-T conditions between the different units, and the near isothermal decompression of the La France Formation indicate that the Kaapvaal craton crust must have been cold enough to enable significant crustal thickening and strain localisation along narrow shear zones and, as a consequence, fast tectonic juxtaposition of rocks metamorphosed at different crustal depths. These features are similar to those observed along Palaeozoic or modern day, oblique subduction-collision zones, but different to those of hot Archaean provinces. We therefore interpret the MGB as representing part of an oblique collision-zone between two terrains of the Kaapvaal craton: the Witwatersrand and Pietersburg terrains