Unary Resolution: Characterizing Ptime

Aubert, Clément; Bagnol, Marc; Seiller, Thomas

doi:10.1007/978-3-662-49630-5_22

Cited by 7 publications

(9 citation statements)

References 38 publications

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“…Firstly, complexity classes are here characterised as specific types in models of (fragments of) linear logic. It thus fills the gap between the above mentioned series of work goi-inspired results in computational complexity [3,4,1,2] and the actual semantics provided by goi models. Secondly, we obtain characterisations of several classes that were not available using previous techniques.…”

Section: Contributions and Outlinementioning

confidence: 83%

“…These semantic results were then rephrased in more syntactic terms, providing new characterisations related to logic programming results [1,2] but taking another step further from the initial framework of the hyperfinite goi model. After a first step which ended in the loss of an underlying logical framework, this second step ended in the loss of the rich mathematical theories the method was initially founded upon.…”

Section: Introductionmentioning

confidence: 99%

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Interaction Graphs: Nondeterministic Automata

Seiller¹

2016

Preprint

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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.

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Section: Contributions and Outlinementioning

confidence: 83%

Section: Introductionmentioning

confidence: 99%

Interaction Graphs: Nondeterministic Automata

Seiller¹

2016

Preprint

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“…Finally, the proof is concluded by a purely geometrical theorem (Thm. 61) 4 expressing a tension between the two geometrizations. Our work focuses here only on the construction of a set algebraic surfaces representing the computation of a pram; the remaining part of our proof follows Mulmuley's original technique closely.…”

Section: Mulmuley's Geometrizationmentioning

confidence: 99%

“…Luc Pellissier and Thomas Seiller reduction in λ-calculus [24]. More recently, a series of characterisations of complexity classes were obtained using goi techniques [3][4][5][6].…”

Section: Introductionmentioning

confidence: 99%

PRAMs over integers do not compute maxflow efficiently

Seiller¹,

Pellissier²,

Léchine³

2018

Preprint

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Finding lower bounds in complexity theory has proven to be an extremely difficult task. In this article, we analyze two proofs of complexity lower bound: Ben-Or's proof of minimal height of algebraic computational trees deciding certain problems and Mulmuley's proof that restricted Parallel Random Access Machines (prams) over integers can not decide P-complete problems efficiently. We present the aforementioned models of computation in a framework inspired by dynamical systems and models of linear logic : graphings.This interpretation allows to connect the classical proofs to topological entropy, an invariant of these systems; to devise an algebraic formulation of parallelism of computational models; and finally to strengthen Mulmuley's result by separating the geometrical insights of the proof from the ones related to the computation and blending these with Ben-Or's proof. Looking forward, the interpretation of algebraic complexity theory as dynamical system might shed a new light on research programs such as Geometric Complexity Theory.

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“…These notions are equivalent (Asperti et al, 1994), and their common core idea -describing computation by local and asynchronous conditions on routing of paths -inspired the design of efficient parallel abstract machines (Mackie, 1995;Danos et al, 1997;Laurent, 2001;Pinto, 2001;Pedicini and Quaglia, 2007;Dal Lago et al, 2014;Pedicini et al, 2014;Dal Lago et al, 2015, among others). More recently, the geometry of interaction (GoI) approach has been fruitfully employed for semantic investigations which characterised quantitative properties of programs, with respect to both time (Dal Lago, 2009;Perrinel, 2014;Aubert et al, 2016) and space complexity (Aubert and Seiller, 2014;Aubert and Seiller, 2015;Mazza, 2015b;Mazza and Terui, 2015).…”

Section: Introductionmentioning

confidence: 99%

Geometry of resource interaction and Taylor–Ehrhard–Regnier expansion: a minimalist approach

Solieri

2016

Math. Struct. Comp. Sci.

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International audienceThe resource λ-calculus is a variation of the λ-calculus where arguments are superpositions of terms and must be linearly used, hence it is a model for linear and non-deterministic programming languages. Moreover, it is the target language of the Taylor-Ehrhard-Regnier expansion of λ-terms, a linearisation of the λ-calculus which develops ordinary terms into infinite series of resource terms. In a strictly typed restriction of the resource λ-calculus, we study the notion of path persistence, and define a remarkably simple geometry of resource interaction (GoRI) that characterises it. In addition, GoRI is invariant under reduction and counts addends in normal forms. We also analyse expansion on paths in ordinary terms, showing that reduction commutes with expansion and, consequently, that persistence can be transferred back and forth between a path and its expansion. Lastly, we also provide an expanded counterpart of the execution formula, which computes paths as series of objects of GoRI, thus exchanging determinism and conciseness for linearity and simplicity

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Unary Resolution: Characterizing Ptime

Cited by 7 publications

References 38 publications

Interaction Graphs: Nondeterministic Automata

Interaction Graphs: Nondeterministic Automata

PRAMs over integers do not compute maxflow efficiently

Geometry of resource interaction and Taylor–Ehrhard–Regnier expansion: a minimalist approach

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