arXiv subjects

Thomas Seiller

Publications and source records attributed to Thomas Seiller.

At least 37 records · Page 2Linked to original sources

arXiv2021

Lower bounds for algebraic machines, semantically

This paper presents a new semantic method for proving lower bounds in computational complexity. We use it to prove that maxflow, a PTIME complete problem, is not computable in polylogarithmic time on parallel random access machines (PRAMs) working with integers, showing that NCZ \neq PTIME, where NCZ is the complexity class defined by such machines, and PTIME is the standard class of polynomial time computable problems (on, say, a Turing machine). On top of showing this new separation result, we show our method captures previous lower bounds results from the literature: Steele and Yao's lower bounds for algebraic decision trees, Ben-Or's lower bounds for algebraic computation trees, Cucker's proof that NC is not equal to PTIME on the reals, and Mulmuley's lower bounds for "PRAMs without bit operations".

cs.CC↗

arXiv2020

Stellar Resolution: Multiplicatives

We present a new asynchronous model of computation named Stellar Resolution based on first-order unification. This model of computation is obtained as a formalisation of Girard's transcendental syntax programme, sketched in a series of three articles. As such, it is the first step towards a proper formal treatment of Girard's proposal to tackle first-order logic in a proofs-as-program approach. After establishing formal definitions and basic properties of stellar resolution, we explain how it generalises traditional models of computation, such as logic programming and combinatorial models such as Wang tilings. We then explain how it can represent multiplicative proof-structures, their cut-elimination and the correctness criterion of Danos and Regnier. Further use of realisability techniques lead to dynamic semantics for Multiplicative Linear Logic, following previous Geometry of Interaction models.

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arXiv2020

Probabilistic Complexity Classes through Semantics

In a recent paper, the author has shown how Interaction Graphs models for linear logic can be used to obtain implicit characterisations of non-deterministic complexity classes. In this paper, we show how this semantic approach to Implicit Complexity Theory (ICC) can be used to characterise deterministic and probabilistic models of computation. In doing so, we obtain correspondences between group actions and both deterministic and probabilistic hierarchies of complexity classes. As a particular case, we provide the first implicit characterisations of the classes PLogspace (un-bounded error probabilistic logarithmic space) and PPtime (unbounded error probabilistic polynomial time)

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arXiv2019

Interaction Graphs: Exponentials

This paper is the fourth of a series exposing a systematic combinatorial approach to Girard's Geometry of Interaction (GoI) program. The GoI program aims at obtaining particular realisability models for linear logic that accounts for the dynamics of cut-elimination. This fourth paper tackles the complex issue of defining exponential connectives in this framework. For that purpose, we use the notion of \emph{graphings}, a generalisation of graphs which was defined in earlier work. We explain how to define a GoI for Elementary Linear Logic (ELL) with second-order quantification, a sub-system of linear logic that captures the class of elementary time computable functions.

cs.LO↗

arXiv2019

Proceedings Third Joint Workshop on Developments in Implicit Computational complExity and Foundational & Practical Aspects of Resource Analysis

These proceedings present the accepted regular papers and some selected extended abstracts from the 3rd joint DICE-FOPARA workshop, which was held in Prague, Czech Republic on April 6-7, 2019, as a part of ETAPS. The joint workshop provides synergies by combining two complementary communities: The 10th DICE workshop explores the area of Implicit Computational Complexity (ICC), which grew out from several proposals to use logic and formal methods to provide languages for complexity-bounded computation (e.g. Ptime, Logspace computation). It aims at studying the computational complexity of programs without referring to external measuring conditions or a particular machine model, but only by considering language restrictions or logical/computational principles entailing complexity properties. Several approaches have been explored for that purpose, such as restrictions on primitive recursion and ramification, rewriting systems, linear logic, types and lambda calculus, interpretations of functional and imperative programs. The 6th FOPARA workshop serves as a forum for presenting original research results that are relevant to the analysis of resource (e.g. time, space, energy) consumption by computer programs. The workshop aims to bring together the researchers that work on foundational issues with the researchers that focus more on practical results. Therefore, both theoretical and practical contributions are encouraged. We also encourage papers that combine theory and practice. This third joint DICE-FOPARA workshop at ETAPS 2019 follows the successful experiences of co-location of DICE-FOPARA at ETAPS 2015 in London and ETAPS 2017 in Uppsala.

cs.CC↗

arXiv2019

Logarithmic Space and Permutations

In a recent work, Girard proposed a new and innovative approach to computational complexity based on the proofs-as-programs correspondence. In a previous paper, the authors showed how Girard proposal succeeds in obtaining a new characterization of co-NL languages as a set of operators acting on a Hilbert Space. In this paper, we extend this work by showing that it is also possible to define a set of operators characterizing the class L of logarithmic space languages.

cs.LO↗

arXiv2019

Coherent Interaction Graphs

We introduce the notion of coherent graphs, and show how those can be used to define dynamic semantics for Multiplicative Linear Logic (MLL) extended with non-determinism. Thanks to the use of a coherence relation rather than mere formal sums of non-deterministic possibilities, our model enjoys some finiteness and sparsity properties. We also show how studying the semantic types generated by a single "test" within our model of MLL naturally gives rise to a notion of proof net, which turns out to be exactly Retoré's R&B-cographs. This revisits the old idea that multplicative proof net correctness is interactive, with a twist: types are characterized not by a set of counter-proofs but by a single non-deterministic counter-proof.

cs.LO↗

arXiv2017

An intensionally fully-abstract sheaf model for $π$ (expanded version)

Following previous work on CCS, we propose a compositional model for the $π$-calculus in which processes are interpreted as sheaves on certain simple sites. Such sheaves are a concurrent form of innocent strategies, in the sense of Hyland-Ong/Nickau game semantics. We define an analogue of fair testing equivalence in the model and show that our interpretation is intensionally fully abstract for it. That is, the interpretation preserves and reflects fair testing equivalence; and furthermore, any innocent strategy is fair testing equivalent to the interpretation of some process. The central part of our work is the construction of our sites, relying on a combinatorial presentation of $π$-calculus traces in the spirit of string diagrams.

cs.LO↗

arXiv2017

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.

cs.PL↗

arXiv2016

Interaction Graphs: Nondeterministic Automata

This paper exhibits a series of semantic characterisations of sublinear nondeterministic complexity classes. These results fall into the general domain of logic-based approaches to complexity theory and so-called implicit computational complexity (ICC), i.e. descriptions of complexity classes without reference to specific machine models. In particular, it relates strongly to ICC results based on linear logic since the semantic framework considered stems from work on the latter. Moreover, the obtained characterisations are of a geometric nature: each class is characterised by a specific action of a group by measure-preserving maps.

cs.LO↗

arXiv2016

Interaction Graphs: Full Linear Logic

Interaction graphs were introduced as a general, uniform, construction of dynamic models of linear logic, encompassing all "Geometry of Interaction" (GoI) constructions introduced so far. This series of work was inspired from Girard's hyperfinite GoI, and develops a quantitative approach that should be understood as a dynamic version of weighted relational models. Until now, the interaction graphs framework has been shown to deal with exponentials for the constrained system ELL (Elementary Linear Logic) while keeping its quantitative aspect. Adapting older constructions by Girard, one can clearly define "full" exponentials, but at the cost of these quantitative features. We show here that allowing interpretations of proofs to use continuous (yet finite in a measure-theoretic sense) sets of states, as opposed to earlier Interaction Graphs constructions were these sets of states were discrete (and finite), provides a model for full linear logic with second order quantification.

cs.LO↗

arXiv2016

From Dynamic to Static Semantics, Quantitatively

We exhibit a new relationship between dynamic and static semantics. We define the categorical outlay needed to define Interaction Graphs models, a generalisation of Girard's Geometry of Interaction models, which strongly relate to game semantics. We then show how this category is mapped to weighted relational models of linear logic. This brings into vision a new bridge between the dynamic and static approaches, and provides formal grounds for considering interaction graphs models as quantitative versions of GoI and game semantics models. We finally proceed to show how the interaction graphs models relate to a very general notion of quantitative coherence spaces.

cs.LO↗

arXiv2015

Interaction Graphs: Additives

Geometry of Interaction (GoI) is a kind of semantics of linear logic proofs that aims at accounting for the dynamical aspects of cut-elimination. We present here a parametrized construction of a Geometry of Interaction for Multiplicative Additive Linear Logic (MALL) in which proofs are represented by families of directed weighted graphs. Contrarily to former constructions dealing with additive connectives, we are able to solve the known issue of obtaining a denotational semantics for MALL by introducing a notion of observational equivalence. Moreover, our setting has the advantage of being the first construction dealing with additives where proofs of MALL are interpreted by finite objects. The fact that we obtain a denotational model of MALL relies on a single geometric property, which we call the trefoil property, from which we obtain, for each value of the parameter, adjunctions. We then proceed to show how this setting is related to Girard's various constructions: particular choices of the parameter respectively give a combinatorial version of his latest GoI, a refined version of older Geometries of Interaction based on nilpotency. This shows the importance of the trefoil property underlying our constructions since all known GoI construction to this day rely on particular cases of it.

cs.LO↗

arXiv2015

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.

math.LO↗

arXiv2015

Interaction Graphs: Graphings

In two previous papers, we exposed a combinatorial approach to the program of Geometry of Interaction, a program initiated by Jean-Yves Girard. The strength of our approach lies in the fact that we interpret proofs by simpler structures - graphs - than Girard's constructions, while generalizing the latter since they can be recovered as special cases of our setting. This third paper extends this approach by considering a generalization of graphs named graphings, which is in some way a geometric realization of a graph. This very general framework leads to a number of new models of multiplicative-additive linear logic which generalize Girard's geometry of interaction models and opens several new lines of research. As an example, we exhibit a family of such models which account for second-order quantification without suffering the same limitations as Girard's models.

cs.LO↗

arXiv2015

Towards a Complexity-through-Realisability Theory

We explain how recent developments in the fields of realisability models for linear logic -- or geometry of interaction -- and implicit computational complexity can lead to a new approach of implicit computational complexity. This semantic-based approach should apply uniformly to various computational paradigms, and enable the use of new mathematical methods and tools to attack problem in computational complexity. This paper provides the background, motivations and perspectives of this complexity-through-realisability theory to be developed, and illustrates it with recent results.

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arXiv2015

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.

cs.LO↗

arXiv2015

Characterizing co-NL by a group action

In 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.

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