89 research outputs found
A Correspondence between Maximal Abelian Sub-Algebras and Linear Logic Fragments
We show a correspondence between a classification of maximal abelian
sub-algebras (MASAs) proposed by Jacques Dixmier and fragments of linear logic.
We expose for this purpose a modified construction of Girard's hyperfinite
geometry of interaction which interprets proofs as operators in a von Neumann
algebra. The expressivity of the logic soundly interpreted in this model is
dependent on properties of a MASA which is a parameter of the interpretation.
We also unveil the essential role played by MASAs in previous geometry of
interaction constructions
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
Characterizing co-NL by a group action
International audienceIn a recent paper, Girard proposes to use his recent construction of a geometry of interaction in the hyperfinite factor in an innovative way to characterize complexity classes. We begin by giving a detailed explanation of both the choices and the motivations of Girard's definitions. We then provide a complete proof that the complexity class co-NL can be characterized using this new approach. We introduce as a technical tool the non-deterministic pointer machine, a concrete model to computes algorithms
Memoization for Unary Logic Programming: Characterizing PTIME
We give a characterization of deterministic polynomial time computation based
on an algebraic structure called the resolution semiring, whose elements can be
understood as logic programs or sets of rewriting rules over first-order terms.
More precisely, we study the restriction of this framework to terms (and logic
programs, rewriting rules) using only unary symbols. We prove it is complete
for polynomial time computation, using an encoding of pushdown automata. We
then introduce an algebraic counterpart of the memoization technique in order
to show its PTIME soundness. We finally relate our approach and complexity
results to complexity of logic programming. As an application of our
techniques, we show a PTIME-completeness result for a class of logic
programming queries which use only unary function symbols.Comment: Soumis {\`a} LICS 201
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