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In a recent paper, Girard uses the geometry of interaction in the hyperfinite
factor in an innovative way to characterize complexity classes. The purpose of
this paper is two-fold: to give a detailed explanation of both the choices and
the motivations of Girard's definitions, and - since Girard's paper skips over
some non-trivial details and only sketches one half of the proof - to provide a
complete proof that co-NL can be characterized by an action of the group of
finite permutations. We introduce as a technical tool the non-deterministic
pointer machine, a concrete model that computes the algorithms represented in
this setting.

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you can request a copy directly from the authors.

... This approach was already used with complexity concerns [BP01,Gir12,AS16b,AS16a]. 1 It differs from usual ICC via proof theory because the restrictions on the expressivity of programs is not obtained through restrictions on type systems. Instead, limitations imposed on the objects representing proofs rule out computational principles on the semantics side and allow to capture complexity classes. ...

... This is an intuitive framework, yet expressive enough for our purposes. 1 For a more advanced discussion on the "sister approaches" relying on the theory of von Neumann algebras [Gir12,AS16a,AS16b] one should refer to the "related work" section, page 17. ...

... A first series of work [Sei12b, Aub13, AS16a, AS16b] deepened those intuitions by making formal the interpretation of proofs of linear logic in the hyperfinite factor, a type II 1 von Neuman Algebra. Thanks to the representation of infinite operators by matrices, it was proven [Sei12b,AS16a] that representations of programs in a specific sub-algebra were characterizing NLogspace. Later on, an additional restriction on the observations, phrased in terms of norm, was proven [Aub13, AS16b] to characterize Logspace. ...

We present an algebraic characterization of the complexity classes Logspace and NLogspace, using an algebra with a composition law based on unification. This new bridge between unification and complexity classes is rooted in proof theory and more specifically linear logic and geometry of interaction. We show how to build a model of computation in the unification algebra and then, by means of a syntactic representation of finite permutations in the algebra, we prove that whether an observation (the algebraic counterpart of a program) accepts a word can be decided within logarithmic space. Finally, we show that the construction naturally corresponds to pointer machines, a convenient way of understanding logarithmic space computation.

... This approach was initiated by Girard [12] and motivated by his work on Geometry of Interaction (goi) models, and more precisely the hyperfinit goi model [11]. Together with C. Aubert, the author showed how Girard's proposal lead to the characterisation of coNLogspace [20,3] and Logspace [4]. Unfortunately, technical reasons lead the authors to consider modifications of the initial hyperfinite goi framework, furthering characterisation results from the goi models construction. ...

... 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. ...

... Those graphings, however, are all obtained as representations of the same graph, corresponding to the set of axiom rules in the corresponding proof net. We refer the reader to an earlier paper for an illustrated discussion of how binary words can be represented as graphs [3]; we define in the next section the interpretation of binary words directly. ...

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.

... These new methods were initiated by Gi- rard [32] and have known a rapid development. They lead to a series of results in the form of new characterisations of the classes CONLOGSPACE [32, 2], LOGSPACE [3, 33] and PTIME [34]. Unfortunately, although the construction of Realisability models and the characterisations of classes are founded on similar techniques, they are two distinct, unrelated, constructions. ...

... In the models of linear logic we described, one can easily define the type Words (2) Σ of words over an arbitrary finite alphabet Σ. The definition of the representation of these binary words comes from the encoding of binary lists in lambda-calculus and is explained thoroughly in previous works [37, 2, 3]. We won't give the formal definition of what is a representation of a word w here, but let us sketch the main ideas. ...

... '0i', '0o', represent disjoint segments of unit length, e.g. [0, 1], [1, 2]. As mentioned in the caption, the plain edges are realised as translations. ...

We explain how recent developments in the fields of realizability 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-realizability theory to
be developed, and illustrates it with recent results.

... Still, they benefit from previous works in type theory: for instance the representation of integers used here comes from their representation in linear logic, translated in the GOI setting, whose interactive point of view on computation has proven crucial in characterizing logarithmic space computation [11]. The first results that used those innovative considerations were based on operator algebras [22], [23], [24]. Here we consider a more syntactic flavor of the GOI where untyped programs are represented in the so-called resolution semiring [21], a semiring based on the resolution rule [25] and a specific class of logic programs. ...

... Conversely, a loop in the automaton will refrain the wiring from being nilpotent. The point we need to be careful about is the encoding of loops: those should be represented as a re-initialization of the computation, as discussed in details in earlier work [23] . The reason for this is that another encoding may interfere with the representation of acceptation as termination: the existence of a loop in the observation O M representing the automaton M , even one that is not used in the computation with the input W , prevents the wiring O M ¯ W p from being nilpotent. ...

... This article extends modularly on our previous approaches [23], [24], [27], [28] to obtain a characterization of PTIME, by adding a sort of " stack plugin " to observations. This enhancement was guided by the intuition of a stack added to an automaton, allowing to move from LOGSPACE to PTIME and providing a decisive proof technique: memoization. ...

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.

... Since they can scan but not modify the input, they are usually called " pointer machines " . This model was already at the heart of previous works relating geometry of interaction and complexity theory [19,3,2]. ...

... Finally, Sect. 4 makes use of the tools introduced earlier to state and prove our complexity results. While the expressiveness part is quite similar to earlier presentations [3,2], the proof that acceptance can be decided within logarithmic space has been made more modular by reducing it to cycle search in a graph. ...

... This result implies that the notion of acceptance has the intended sense and is finitely verifyable: whether a word W is accepted by an observation O can be checked without considering all representations of W . This kind of situation where two semirings W and O are disjoint enough to obtain Corollary 32 can be formalized through the notion of normative pair considered in earlier works [19,3,2]. ...

We present an algebraic view on logic programming, related to proof theory
and more specifically linear logic and geometry of interaction. Within this
construction, a characterization of logspace (deterministic and
non-deterministic) computation is given via a synctactic restriction, using an
encoding of words that derives from proof theory.
We show that the acceptance of a word by an observation (the counterpart of a
program in the encoding) can be decided within logarithmic space, by reducing
this problem to the acyclicity of a graph. We show moreover that observations
are as expressive as two-ways multi-heads finite automata, a kind of pointer
machines that is a standard model of logarithmic space computation.

... We must also point out that geometry of interaction has been successful in providing tools for the study of computational complexity. The fact that it models the execution of programs explains that it is well suited for the study of complexity classes in time [BP01,Lag09], as well as in space [AS13,AS12]. It was also used to explain [GAL92] Lamping's optimal reduction of lambda-calculus [Lam90]. ...

... In the previously described models of multiplicative-additive linear logic, one can define the type of binary lists Nat 2 in a quite natural fashion (the representation of lists is thoroughly explained in previous papers on complexity [AS12,AS13]). Moreover, in a number of cases one can define exponential connectives in the model and therefore consider the type !Nat 2 ⊸ Bool. ...

... The set of languages characterized by the type !Nat 2 ⊸ Bool becomes larger and larger as we consider extensions of the microcosms. We can then work on this remark, and use intuitions gained from earlier work [AS12,AS13]. This leads to a perfect correspondance between a hierarchy of monoids on the measured space Z × [0,1] N and a hierarchy of classes of languages in between regular languages and logarithmic space predicates (both included) [Sei14b]. ...

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.

... In this aspect of reconstructing linear logic, several geometry of interaction models were defined using operators algebras [21,19,25], unification algebras [23], graphs [50,52] and graphings [54,56,53]. Although all these models did define rich models, that were in particular used to study computational complexity [5,2,3,55,57], they failed with respect to two different aspects. Firstly, the objects used to interpret even the most basic proofs were most of the time infinite objects and even when they were not, types were defined through an infinite number of tests. ...

... While this aspect is a failure somehow, it is also a feature as the models are very rich and open other paths of reflexion.2 The formulation is borrowed from Church's critic of von Mises notion of kollektiv[7]. ...

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.

... We now recall some definitions introduced in previous work by the author characterising complexity classes by use of graphings [30]; interested readers will find there (and in some references therein [3,4]) more detailed explanations ofand motivations for -the definitions. In particular the representation of binary words is related to ...

... This graph is the discrete representation of w. Detailed explanations on how these graphs relate to the proofs of the sequent ⊢ BList can be found in earlier work [23,3]. ...

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)

... It was already used with complexity concerns [12,13]. Recent works [13,14,15] studied the link between Geometry of Interaction and logarithmic space, relying on the theory of von Neumann algebras. Those three articles are indubitably sources of inspiration of this work, but the whole construction is made anew, in a simpler framework. ...

... Another pending question about this approach to complexity classes is to delimit the minimal prerequisites of the construction, its core. Earlier works [13,14,15] made use of von Neumann algebras to get a setting that is expressive enough, we ligthen the construction by using simpler objects. Yet, the possibility of representing the action of permutations on a unbounded tensor product is a common denominator that seems deeply related to logarithmic space and pointer machines. ...

We present an algebraic characterization of the complexity classes Logspace
and NLogspace, using an algebra with a composition law based on unification.
This new bridge between unification and complexity classes is inspired from
proof theory and more specifically linear logic and Geometry of Interaction.
We show how unification can be used to build a model of computation by means
of specific subalgebras associated to finite permutations groups. We then prove
that the complexity of deciding whenever an observation (the algebraic
counterpart of a program) accepts a word can be decided within logarithmic
space.
We also show that the construction can naturally be related to pointer
machines, an intuitive way of understanding logarithmic space computing.

... Several GoI models were defined using operator algebras [44,42,49,50], term unification algebras [46], graphs [98,99] and graphings [101,103,100]. Although all these models did define rich models, that were in particular used to study computational complexity [15,10,11,102,104], they had two main drawbacks. ...

We present the stellar resolution, a "flexible" tile system based on Robinson's first-order resolution. After establishing formal definitions and basic properties of the stellar resolution, we show its Turing-completeness and to illustrate the model, we exhibit how it naturally represents computation with Horn clauses and automata as well as nondeterministic tiling constructions used in DNA computing. In the second and main part, by using the stellar resolution, we formalise and extend ideas of a new alternative to proof-net theory sketched by Girard in his transcendental syntax programme. In particular, we encode both cut-elimination and logical correctness for the multiplicative fragment of linear logic (MLL). We finally obtain completeness results for both MLL and MLL extended with the so-called MIX rule. By extending the ideas of Girard's geometry of interaction, this suggests a first step towards a new understanding of the interplay between logic and computation where linear logic is seen as a (constructed) way to format computation.

... 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]. ...

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.

... 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 [e.g. Dal Lago, 2009, Perrinel, 2014, Aubert et al., 2016 and space complexity [Aubert and Seiller, 2014, 2015, Mazza, 2015b, Mazza and Terui, 2015]. ...

Elegant semantics and eﬃcient implementations of functional programming languages can both be described by the very same mathematical structures, most prominently with in the Curry-Howard correspondence, where programs, types and execution respectively coincide with proofs, formulæ and normalisation. Such a ﬂexibility is sharpened by the deconstructive and geometrical approach pioneered by linear logic (LL) and proof-nets, and by Lévy-optimal reduction and sharing graphs (SG).Adapting Girard’s geometry of interaction, this thesis introduces the geometry of resource interaction (GoRI), a dynamic and denotational semantics, which describes, algebra-ically by their paths, terms of the resource calculus (RC), a linear and non-deterministic variation of the ordinary lambda calculus. Inﬁnite series of RC-terms are also the domain of the Taylor-Ehrhard-Regnier expansion, a linearisation of LC. The thesis explains the relation between the former and the reduction by proving that they commute, and provides an expanded version of the execution formula to compute paths for the typed LC. SG are an abstract implementation of LC and proof-nets whose steps are local and asynchronous, and sharing involves both terms and contexts. Whilst experimental tests on SG show outstanding speedups, up to exponential, with respect to traditional implementations, sharing comes at price. The thesis proves that, in the restricted case of elementary proof-nets, where only the core of SG is needed, such a price is at most quadratic, hence harmless.

... 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 [e.g. Dal Lago, 2009, Perrinel, 2014, Aubert et al., 2016 and space complexity [Aubert and Seiller, 2014, 2015, Mazza, 2015b, Mazza and Terui, 2015]. ...

Elegant semantics and efficient implementations of functional programming languages can both be described by the very same mathematical structures, most prominently within the Curry-Howard correspondence, where programs, types and execution respectively coincide with proofs, formulæ and normalisation. Such a flexibility is sharpened by the deconstructive and geometrical approach pioneered by linear logic (LL) and proof-nets, and by Lévy-optimal reduction and sharing graphs (SG).Adapting Girard’s geometry of interaction, this thesis introduces the geometry of resource interaction (GoRI), a dynamic and denotational semantics, which describes, algebraically by their paths, terms of the resource calculus (RC), a linear and non-deterministic variation of the ordinary lambda calculus. Infinite series of RC-terms are also the domain of the Taylor-Ehrhard-Regnier expansion, a linearisation of LC. The thesis explains the relation between the former and the reduction by proving that they commute, and provides an expanded version of the execution formula to compute paths for the typed LC.SG are an abstract implementation of LC and proof-nets whose steps are local and asynchronous, and sharing involves both terms and contexts. Whilst experimental tests on SG show outstanding speedups, up to exponential, with respect to traditional implementations, sharing comes at price. The thesis proves that, in the restricted case of elementary proof-nets, where only the core of SG is needed, such a price is at most quadratic, hence harmless.

... 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). ...

The 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.

... Moreover, the great generality and flexibility of the definition of exponentials also seem promising when it comes to the study of complexity. Some results in this direction were already obtained: using a new technique proposed by Girard [19], the author obtained, in a joint work with Clément Aubert, new characterizations of the computational complexity classes co-NL [3] and L [4] as sets of operators in the hyperfinite type II 1 factor. ...

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.

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. This construction stems from an interactive interpretation of the cut-elimination procedure of linear logic known as the geometry of interaction.
This framework is restricted to terms (logic programs, rewriting rules) using only unary symbols, and this restriction is shown to be complete for polynomial time computation by encoding pushdown automata. Soundness w.r.t. Ptime is proven thanks to a saturation method similar to the one used for pushdown systems and inspired by the memoization technique.
A Ptime-completeness result for a class of logic programming queries that uses only unary function symbols comes as a direct consequence.

This research in Theoretical Computer Science extends the gateways between Linear Logic and Complexity Theory by introducing two innovative models of computation. It focuses on sub-polynomial classes of complexity: AC and NC --the classes of efficiently parallelizable problems-- and L and NL --the deterministic and non-deterministic classes of problems efficiently solvable with low resources on space. Linear Logic is used through its Proof Net presentation to mimic with efficiency the parallel computation of Boolean Circuits, including but not restricted to their constant-depth variants. In a second moment, we investigate how operators on a von Neumann algebra can be used to model computation, thanks to the method provided by the Geometry of Interaction, a subtle reconstruction of Linear Logic. This characterization of computation in logarithmic space with matrices can naturally be understood as a wander on simple structures using pointers, parsing them without modifying themWe make this intuition formal by introducing Non Deterministic Pointer Machines and relating them to other well-known pointer-like-machines. We obtain by doing so new implicit characterizations of sub-polynomial classes of complexity.

The aim of Implicit Computational Complexity is to study algorithmic complexity only in terms of restrictions of languages and computational principles. Based on previous work about realizability models for linear logic, we propose a new approach where we consider semantical restrictions instead of syntactical ones. This leads to a hierarchy of models mirroring subtle distinctions about computational principles. As an illustration of the method, we explain how to obtain characterizations of the set of regular languages and the set of logarithmic space predicates.

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.

In two previous papers, we exposed a combinatorial approach to the program of Geometry of Interaction (GoI), a program initiated by Jean-Yves Girard which aims at giving a semantics of proofs that accounts for the dynamics of cut-elimination. 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 tackles the complex issue of defining exponential connectives in this framework. In order to succeed in this, we consider a generalization of graphs named graphings, which is in some way a geometric realization of a graph. We explain how we can then define a GoI for Elementary Linear Logic (ELL), a sub-system of linear logic where representable functions
are exactly the functions computable in elementary time. This construction is moreover parametrized by the choice of a map, giving rise to a whole family of models.

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.

Girard’s Geometry of Interaction (GoI) develops a mathematical framework for modelling the dynamics of cut-elimination. We
introduce a typed version of GoI, called Multiobject GoI (MGoI) for multiplicative linear logic without units in categories
which include previous (untyped) GoI models, as well as models not possible in the original untyped version. The development
of MGoI depends on a new theory of partial traces and trace classes, as well as an abstract notion of orthogonality (related
to work of Hyland and Schalk) We develop Girard’s original theory of types, data and algorithms in our setting, and show his
execution formula to be an invariant of Cut Elimination. We prove Soundness and Completeness Theorems for the MGoI interpretation
in partially traced categories with an orthogonality.

We introduce a geometry of interaction model given by an algebra of clauses equipped with resolution (following (Gir95)) into which proofs of Elementary Linear Logic can be interpreted. In order to extend geometry of interaction computation (the so called execution formula) to a wider class of programs in the algebra than just those coming from proofs, we define a variant of execution (called weak execution). Its application to any program of clauses is shown to terminate with a bound on the number of steps which is elementary in the size of the program. We establish that weak execution coincides with standard execution on programs coming from proofs.

Using a proofs-as-programs correspondence, Terui was able to compare two
models of parallel computation: Boolean circuits and proof nets for
multiplicative linear logic. Mogbil et. al. gave a logspace translation
allowing us to compare their computational power as uniform complexity classes.
This paper presents a novel translation in AC0 and focuses on a simpler
restricted notion of uniform Boolean proof nets. We can then encode
constant-depth circuits and compare complexity classes below logspace, which
were out of reach with the previous translations.

We present QBAL, an extension of Girard, Scedrov and Scott's bounded linear
logic. The main novelty of the system is the possibility of quantifying over
resource variables. This generalization makes bounded linear logic considerably
more flexible, while preserving soundness and completeness for polynomial time.
In particular, we provide compositional embeddings of Leivant's RRW and
Hofmann's LFPL into QBAL.

We introduce a geometry of interaction model given by an algebra of clauses equipped with resolution (following [Gir95]) into which proofs of Elementary Linear Logic can be interpreted. In order to extend geometry of interaction computation (the so called execution formula) to a wider class of programs in the algebra than just those coming from proofs, we define a variant of execution (called weak execution). Its application to any program of clauses is shown to terminate with a bound on the number of steps which is elementary in the size of the program. We establish that weak execution coincides with standard execution on programs coming from proofs. Keywords: Elementary Linear Logic, Geometry of interaction, Complexity, Semantics.

Incompleteness—the absence of alternative natural numbers—can be ascribed to a ready-made normativity, inducing a rigid departure syntax/semantics. Geometry of Interaction, set in the non-commutative universe of von Neumann algebras, makes normative assumptions explicit, thus rending possible their internalisation, a possible way out from the semantic aporia. As an illustration, we define an alternative “model”: logspace integers.

Incompleteness—the absence of alternative natural numbers—can be ascribed to a ready-made normativity, inducing a rigid departure syntax/semantics. Geometry of Interaction, set in the non-commutative universe of von Neumann algebras, makes normative assumptions explicit, thus rending possible their internalisation, a possible way out from the semantic aporia. As an illustration, we define an alternative “model”: logspace integers.

This chapter describes the development of a semantics of computation free from the twin drawbacks of reductionism (that leads to static modification) and subjectivism (that leads to syntactical abuses, in other terms, bureaucracy). The new approach initiated in this chapter rests on the use of a specific C*-algebra Λ* that has the distinguished property of bearing a (non associative) inner tensor product. The chapter describes that a representative class of algorithms can be modelized by means of standard mathematics.

In a previous work, Hofmann and Schöpp have introduced the programming language purple to formalise the common intuition of logspace-algorithms as pure pointer programs that take as input some structured data (e.g. a graph) and store in memory only a constant number of pointers to the input (e.g. to the graph nodes). It was shown that purple is strictly contained in logspace, being unable to decide st-connectivity in undirected graphs.
In this paper we study the options of strengthening purple as a manageable idealisation of computation with logarithmic space that may be used to give some evidence that ptime-problems such as Horn satisfiability cannot be solved in logarithmic space.
We show that with counting, purple captures all of logspace on locally ordered graphs. Our main result is that without a local ordering, even with counting and nondeterminism, purple cannot solve tree isomorphism. This generalises the same result for Transitive Closure Logic with counting, to a formalism that can iterate over the input structure, furnishing a new proof as a by-product.

Let Mn, be the class of languages defined by n-head finite automata. The Boolean and Kleene closure properties of Mn, are investigated, and a relationship between Mn, and the class of sets of n-tuples of tapes defined by n-tape finite automata is established. It is shown that the classes Mi form a hierarchy; and that, moreover, for all n, there is a context-free language (CFL) in Mn+1, - Mn. It is further shown that there is a CFL which is in no Mn for any integer n. Finally, several decision properties of the multi-head languages are established.

We introduce a graph-theoretical representation of proofs of multiplicative
linear logic which yields both a denotational semantics and a notion of truth.
For this, we use a locative approach (in the sense of ludics) related to game
semantics and the Danos-Regnier interpretation of GoI operators as paths in
proof nets. We show how we can retrieve from this locative framework both a
categorical semantics for MLL with distinct units and a notion of truth.
Moreover, we show how a restricted version of our model can be reformulated in
the exact same terms as Girard's latest geometry of interaction. This shows
that this restriction of our framework gives a combinatorial approach to J.-Y.
Girard's geometry of interaction in the hyperfinite factor, while using only
graph-theoretical notions.

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.

We present a subsystem of second-order linear logic with restricted rules for exponentials so that proofs correspond to polynomial time algorithms, and vice versa.

We study pointer programs as a model of structured computation within logspace. Pointer programs capture the common description of logspace algorithms as programs that take as input some structured data (e.g. a graph) and that store in memory only a constant number of point- ers to the input (e.g. to the graph nodes). Starting from pure pointer programs, in which only abstract pointers without any internal structure are allowed, we consider pointer programs with constructs for iterating over the input structure and for counting. We classify with which of these constructs it is possible to write a program for solving s-t-reachability in undirected graphs. The main result of this paper is a new lower bound on undirected s-t-reachability. We show that while pointer programs with counting can decide this problem using Reingold's algorithm, the problem cannot be decided by pointer programs with iteration. As a corollary we obtain that Deterministic Transitive Closure logic on locally ordered graphs cannot express undirected s-t-reachability.

This beginning graduate textbook describes both recent achievements and classical results of computational complexity theory. Requiring essentially no background apart from mathematical maturity, the book can be used as a reference for self-study for anyone interested in complexity, including physicists, mathematicians, and other scientists, as well as a textbook for a variety of courses and seminars. More than 300 exercises are included with a selected hint set. The book starts with a broad introduction to the field and progresses to advanced results. Contents include: definition of Turing machines and basic time and space complexity classes, probabilistic algorithms, interactive proofs, cryptography, quantum computation, lower bounds for concrete computational models (decision trees, communication complexity, constant depth, algebraic and monotone circuits, proof complexity), average-case complexity and hardness amplification, derandomization and pseudorandom constructions, and the PCP theorem.

Geometry of Interaction is a transcendental syntax developed in the framework of operator algebras. This fifth installment of the program takes place inside a von Neumann algebra, the hyperfinite factor. It provides a built-in interpretation of cut-elimination as well as an explanation for light, i.e., complexity sensitive, logics.

Pointer programs are a model of structured computation within LOGSPACE. They capture the common description of LOGSPACE algorithms as programs that take as input some structured data (e.g. a graph) and that store in memory only a constant number of pointers to the input (e.g. to the graph nodes). In this paper we study undirected s-t-reachability for a class of pure pointer programs in which one can work with a constant number of abstract pointers, but not with arbitrary data, such as memory registers of logarithmic size. In earlier work we have formalised this class as a programming language PURPLE that features a for all-loop for iterating over the input structure and thus subsumes other formalisations of pure pointer programs, such as Jumping Automata on Graphs JAGs and Deterministic Transitive Closure logic (DTC-logic) for locally ordered graphs. In this paper we show that PURPLE cannot decide undirected s-t-reachability, even though there does exist a LOGSPACE-algorithm for this problem by Reingold's theorem. As a corollary we obtain that DTC-logic for locally ordered graphs cannot express undirected s-t-reachability.

Incluye índice Incluye bibliografía Contenido: Espacios Hilbert; Operadores en Espacios Hilbert; Espacios Benach; Espacios localmente convexos; Topologís débiles; Operadores lineales en espacios Benach; Algebras Benach y teoría espectral para operadores en un espacio Benach; C*- Algebras; Operadores normales en un espacio Hilbert; Operadores no delimitados; Teoría Fredholm.

A number of complexity classes, most notably PTIME, have been characterised by sub-systems of linear logic. In this paper we show that the functions computable in logarithmic space can also be characterised by a restricted version of linear logic. We introduce stratified bounded affine logic (SBAL), a restricted version of bounded linear logic, in which not only the modality, but also the universal quantifier is bounded by a resource polynomial. We show that the proofs of certain sequents in SBAL represent exactly the functions computable logarithmic space. The proof that SBAL-proofs can be compiled to LOGSPACE functions rests on modelling computation by interaction dialogues in the style of game semantics. We formulate the compilation of SBAL-proofs to space-efficient programs as an interpretation in a realisability model, in which realisers are taken from a geometry of interaction situation.

We study the relationship between proof nets for mutiplicative linear logic (with unbounded fan-in logical connectives) and Boolean circuits. We give simulations of each other in the style of the proofs-as-programs correspondence; proof nets correspond to Boolean circuits and cut-elimination corresponds to evaluation. The depth of a proof net is defined to be the maximum logical depth of cut formulas in it, and it is shown that every unbounded fan-in Boolean circuit of depth n, possibly with stC0NN<sub>2</sub> gates, is polynomially simulated by a proof net of depth O(n) and vice versa. Here, stC0NN<sub>2</sub> stands for st-connectivity gates for undirected graphs of degree 2. Let APN<sup>i</sup> be the class of languages for which there is a polynomial size, log<sup>i</sup>-depth family of proof nets. We then have APN<sup>i</sup> = AC<sup>i</sup>(stCONN<sub>2</sub>).

It is shown that nondeterministic space s ( n ) is
closed under complementation for s ( n ) greater than or
equal to log n . It immediately follows that the
context-sensitive languages are closed under complementation. The proof
is an offshoot of work in first-order expressibility

Introduction Think of elementary linear logic as an idealized functional programming language with a severe typing mechanism. Definition by recursion is, of course, forbidden, but some sort of iteration still is possible and the purpose of this paper is to show that enough computing power remains so that elementary recursive functions can be implemented. Actually, the whole paper can be considered an exercise in programming elegantly with a rather desolate language. To zero in on an interesting class of functions, one usually tries to weaken in the given logic whatever corresponds to induction or iteration. Here we're following a di#erent strand, rather specific to the linear logic decomposition of the implication as !A#B, by fiddling with the rules handling `!'. The standard rules are enough to embed the full power of intuitionistic computations. So the game is to find a sensible way to make them harder to use than in full linear logic. There ar

Linearity and ramification constraints have been widely used to weaken higher-order (primitive) recursion in such a way that the class of representable functions equals the class of polytime functions. We show that fine-tuning these two constraints leads to different expressive strengths, some of them lying well beyond polynomial time. This is done by introducing a new semantics, called algebraic context semantics. The framework stems from Gonthier's original work and turns out to be a versatile and powerful tool for the quantitative analysis of normalization in presence of constants and higher-order recursion. Comment: 23 pages, extended version of a paper appearing in LICS 2005 proceedings

Theory of Operator Algebras 1, Encyclopedia of Mathematical Sciences

- M Takesaki

Takesaki, M. (2001) Theory of Operator Algebras 1, Encyclopedia of Mathematical Sciences, volume 124, Springer.

Theory of Operator Algebras 3, Encyclopedia of Mathematical Sciences Proof nets and boolean circuits

- M Takesaki

Takesaki, M. (2003b) Theory of Operator Algebras 3, Encyclopedia of Mathematical Sciences, volume 127, Springer. Terui, K. (2004) Proof nets and boolean circuits. In: Proceedings of the 19th Annual IEEE Symposium on Logic in Computer Science, IEEE Computer Society 182–191.

Bounded linear logic, revisited', Typed Lambda Calculi and Applications

- U Dal Lago
- M Hofmann

Dal Lago, U. & Hofmann, M. (2009), 'Bounded linear logic, revisited', Typed Lambda Calculi and
Applications pp. 80-94.

The geometry of linear higher-order recursion, in 'Logic in Computer Science

- U Dal Lago

Dal Lago, U. (2005), The geometry of linear higher-order recursion, in 'Logic in Computer Science,
2005. LICS 2005. Proceedings. 20th Annual IEEE Symposium on', IEEE, pp. 366-375.

- M Takesaki

Takesaki, M. (2001), Theory of Operator Algebras 1, Vol. 124 of Encyclopedia of Mathematical Sciences, Springer.

- M Takesaki

Takesaki, M. (2003a), Theory of Operator Algebras 2, Vol. 125 of Encyclopedia of Mathematical Sciences, Springer.

Interaction graphs: Additives. Arxiv preprint abs/1205

- T Seiller

The method of forcing for nondeterministic automata

- Szelepscényi