Proof Nets Research Articles

(Leftmost-Outermost) Beta Reduction is Invariant, Indeed

Slot and van Emde Boas' weak invariance thesis states that reasonable machines can simulate each other within a polynomially overhead in time. Is lambda-calculus a reasonable machine? Is there a way to measure the computational complexity of a lambda-term? This paper presents the first complete positive answer to this long-standing problem. Moreover, our answer is completely machine-independent and based over a standard notion in the theory of lambda-calculus: the length of a leftmost-outermost derivation to normal form is an invariant cost model. Such a theorem cannot be proved by directly relating lambda-calculus with Turing machines or random access machines, because of the size explosion problem: there are terms that in a linear number of steps produce an exponentially long output. The first step towards the solution is to shift to a notion of evaluation for which the length and the size of the output are linearly related. This is done by adopting the linear substitution calculus (LSC), a calculus of explicit substitutions modeled after linear logic proof nets and admitting a decomposition of leftmost-outermost derivations with the desired property. Thus, the LSC is invariant with respect to, say, random access machines. The second step is to show that LSC is invariant with respect to the lambda-calculus. The size explosion problem seems to imply that this is not possible: having the same notions of normal form, evaluation in the LSC is exponentially longer than in the lambda-calculus. We solve such an impasse by introducing a new form of shared normal form and shared reduction, deemed useful. Useful evaluation avoids those steps that only unshare the output without contributing to beta-redexes, i.e. the steps that cause the blow-up in size. The main technical contribution of the paper is indeed the definition of useful reductions and the thorough analysis of their properties.

Proof equivalence in MLL is PSPACE-complete

MLL proof equivalence is the problem of deciding whether two proofs in multiplicative linear logic are related by a series of inference permutations. It is also known as the word problem for star-autonomous categories. Previous work has shown the problem to be equivalent to a rewiring problem on proof nets, which are not canonical for full MLL due to the presence of the two units. Drawing from recent work on reconfiguration problems, in this paper it is shown that MLL proof equivalence is PSPACE-complete, using a reduction from Nondeterministic Constraint Logic. An important consequence of the result is that the existence of a satisfactory notion of proof nets for MLL with units is ruled out (under current complexity assumptions). The PSPACE-hardness result extends to equivalence of normal forms in MELL without units, where the weakening rule for the exponentials induces a similar rewiring problem.

The complexity of interaction

In this paper, we analyze the complexity of functional programs written in the interaction-net computation model, an asynchronous, parallel and confluent model that generalizes linear-logic proof nets. Employing user-defined sized and scheduled types, we certify concrete time, space and space-time complexity bounds for both sequential and parallel reductions of interaction-net programs by suitably assigning complexity potentials to typed nodes. The relevance of this approach is illustrated on archetypal programming examples. The provided analysis is precise, compositional and is, in theory, not restricted to particular complexity classes.

Multiplicative-Additive Focusing for Parsing as Deduction

Spurious ambiguity is the phenomenon whereby distinct derivations in grammar may assign the same structural reading, resulting in redundancy in the parse search space and inefficiency in parsing. Understanding the problem depends on identifying the essential mathematical structure of derivations. This is trivial in the case of context free grammar, where the parse structures are ordered trees; in the case of categorial grammar, the parse structures are proof nets. However, with respect to multiplicatives intrinsic proof nets have not yet been given for displacement calculus, and proof nets for additives, which have applications to polymorphism, are involved. Here we approach multiplicative-additive spurious ambiguity by means of the proof-theoretic technique of focalisation.

Proof nets and the call-by-value λ-calculus

This paper gives a detailed account of the relationship between (a variant of) the call-by-value lambda calculus and linear logic proof nets. The presentation is carefully tuned in order to realize an isomorphism between the two systems: every single rewriting step on the calculus maps to a single step on proof nets, and vice-versa. In this way, we obtain an algebraic reformulation of proof nets. Moreover, we provide a simple correctness criterion for our proof nets, which employ boxes in an unusual way, and identify a subcalculus that is shown to be as expressive as the full calculus.

Disease Severity on Cabbage Protected from Arthropod Pests with Insecticide Treated Agronets in Kenya

Cabbage (Brassica oleracea var. capitata L.) is one of the most important vegetables in Kenya, mainly grown by smallholder farmers for food and source of income. Its production is hampered by insect pests and diseases, leading to economic yield losses. Row cover insect proof nets have been used in the past to manage insect pests of cabbage and were recently introduced to Kenya. This study planned to document disease incidence and severity levels on cabbage grown under insecticide impregnated Agronets as a tool to manage insect pests and microclimate. The study was conducted at KALRO-Kabete and PTC (Practical Training Centre)-Thika, Kenya both at the nursery and in the field. Treatments included 1) use of insecticide (alpha cypermethrin) impregnated Agronets with (0.9 mm mesh), 2) untreated 0.4 mm mesh Agronet, 3) grass shading as a farmer practice (at nursery) and 4) control where there was no Agronet use. The treatments had five replications in a completely randomized block design. Plots measured 2 x 6 m, with a 1 m path between consecutive plots and 2 m between consecutive blocks. Three major diseases were recorded: damping off (Rhizoctonia solani), downy mildew (Peronospora parasitica) and black rot (Xanthomonas campestris pv. campestris). Treatments showed significant (P<0.05) differences on severity of damping off, which was lowest on seedlings covered with Agronets compared with the other treatments at nursery. Comparably, the downy mildew disease severity was higher on seedlings covered with Agronets than those with no cover or shading at the nursery. The findings show that frequent disease scouting is crucial when using Agronets for cabbage pests management. This would allow farmers to take action when diseases are noticed. Further supportive studies to understand threshold limits that can trigger growers to employ disease control application may be needed. (Resume d'auteur)

Proof nets and semi-star-autonomous categories

In this paper, it is proved that Girard's proof nets for multiplicative linear logic characterize free semi-star-autonomous categories.

Multiparty Sessions based on Proof Nets

We interpret Linear Logic Proof Nets in a term language based on Solos calculus. The system includes a synchronisation mechanism, obtained by a conservative extension of the logic, that enables to define non-deterministic behaviours and multiparty sessions.

Evaluation of susceptibility level of pear cultivars to fire blight (Erwinia amylovora) in the Czech Republic

Thirty-three pear cultivars and selections of potential interest to pear producers and plant breeders in the Czech Republic were tested for relative field susceptibility to the fire blight (Erwinia amylovora) over five years. Level of fire blight susceptibility was evaluated according to the extent of lesion development on the shoot tips after artificial inoculation in experimental plots under insect proof nets. Old cvs Alexander Lucas (as resistant standard), Conference (as moderately resistant standard) and Beurr&amp;eacute; Bosc (as susceptible standard) were included in the tests. The 33 pear cultivars and selections were tested, only one of them was highly resistant (3.0%), 3.0% were evaluated as resistant, 12.1% moderately resistant, 36.4% moderately susceptible, 18.2% susceptible and 27.3% highly susceptible. Breeding selection US 625-63-10 was the only highly resistant pear genotype (necrosis of shoots of 0&amp;ndash;7.0%). Resistant genotype group (necrosis 7.1&amp;ndash;13.0%) comprised US 625-63-4. Moderately resistant genotypes (necrosis 13.1&amp;ndash;25.0%) included cvs Alexander Lucas, Alfa, Bohemica and HL 31-50-31. Highly susceptible genotypes (necrosis more than 80.1%) included cvs Vonka, Karina, Bona, Decora, Elektra, Milka, Regina, Alice and TE 4763. The remaining genotypes were moderately susceptible (necrosis 26.1&amp;ndash;60.0%) and susceptible (necrosis 60.1&amp;ndash;80.0%). &amp;nbsp; &amp;nbsp;

Proof nets and the call-by-value lambda-calculus

This paper gives a detailed account of the relationship between (a variant of) the call-by-value lambda calculus and linear logic proof nets. The presentation is carefully tuned in order to realize a strong bisimulation between the two systems: every single rewriting step on the calculus maps to a single step on the nets, and viceversa. In this way, we obtain an algebraic reformulation of proof nets. Moreover, we provide a simple correctness criterion for our proof nets, which employ boxes in an unusual way.

Evaluating functions as processes

A famous result by Milner is that the lambda-calculus can be simulated inside the pi-calculus. This simulation, however, holds only modulo strong bisimilarity on processes, i.e. there is a slight mismatch between beta-reduction and how it is simulated in the pi-calculus. The idea is that evaluating a lambda-term in the pi-calculus is like running an environment-based abstract machine, rather than applying ordinary beta-reduction. In this paper we show that such an abstract-machine evaluation corresponds to linear weak head reduction, a strategy arising from the representation of lambda-terms as linear logic proof nets, and that the relation between the two is as tight as it can be. The study is also smoothly rephrased in the call-by-value case, introducing a call-by-value analogous of linear weak head reduction.

Interaction graphs: Multiplicatives

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, Girard, 2001 [11]) related to game semantics (Hyland and Ong, 2000 [17], Abramsky et al., 1994 [2]) and the Danos–Regnier interpretation of GoI operators as paths in proof nets (Asperti et al., 1994 [3], Danos and Regnier, 1995 [4]). We show how we can retrieve from this locative framework both a categorical semantics for Multiplicative Linear Logic (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 Girardʼs geometry of interaction (Girard, 2011 [14]). This shows that this restriction of our framework gives a combinatorial approach to J.-Y. Girardʼs geometry of interaction in the hyperfinite factor, while using only graph-theoretical notions.

Labelled calculi of resources

Levy’s labelled λ-calculus has played an important role in the understanding of the Geometry of Interaction and its applications to the implementation of λ-evaluators: labels relate to the multiplicative information of paths. In this paper, we generalise the structure of labels, and the underlying term structure, in order to keep track of exponential information too. We first define two labelled calculi with explicit substitutions and resource management, where labels are in close correspondence with paths in call-by-value and call-by-name translations of the λ-calculus into linear logic proof nets, respectively. We observe a tight relationship between labels and the dynamics of substitutions; this will then guide us through the design of a third calculus that combines the advantages of the previous two, where labels fully reflect the dynamics of substitutions.

Canonical proof nets for classical logic

Proof nets provide abstract counterparts to sequent proofs modulo rule permutations; the idea being that if two proofs have the same underlying proof-net, they are in essence the same proof. Providing a convincing proof-net counterpart to proofs in the classical sequent calculus is thus an important step in understanding classical sequent calculus proofs. By convincing, we mean that (a) there should be a canonical function from sequent proofs to proof nets, (b) it should be possible to check the correctness of a net in polynomial time, (c) every correct net should be obtainable from a sequent calculus proof, and (d) there should be a cut-elimination procedure which preserves correctness.Previous attempts to give proof-net-like objects for propositional classical logic have failed at least one of the above conditions. In Richard McKinley (2010) [22], the author presented a calculus of proof nets (expansion nets) satisfying (a) and (b); the paper defined a sequent calculus corresponding to expansion nets but gave no explicit demonstration of (c). That sequent calculus, called LK⁎ in this paper, is a novel one-sided sequent calculus with both additively and multiplicatively formulated disjunction rules. In this paper (a self-contained extended version of Richard McKinley (2010) [22]), we give a full proof of (c) for expansion nets with respect to LK⁎, and in addition give a cut-elimination procedure internal to expansion nets – this makes expansion nets the first notion of proof-net for classical logic satisfying all four criteria.

A coding theoretic study of MLL proof nets

In this paper we propose a novel approach for analysing proof nets of Multiplicative Linear Logic (MLL) using coding theory. We define families of proof structures called PS-families and introduce a metric space for each family. In each family: (1)an MLL proof net is a true code element; and(2)a proof structure that is not an MLL proof net is a false (or corrupted) code element. The definition of our metrics elegantly reflects the duality of the multiplicative connectives. We show that in our framework one-error-detection is always possible but one-error-correction is always impossible. We also demonstrate the importance of our main result by presenting two proof-net enumeration algorithms for a given PS-family: the first searches proof nets naively and exhaustively without help from our main result, while the second uses our main result to carry out an intelligent search. In some cases, the first algorithm visits proof structures exponentially, while the second does so only polynomially.

An approach to innocent strategies as graphs

This paper proposes an approach for extending to graphs the close relation between proofs and innocent strategies. We work in the setting of L-nets, introduced by Faggian and Maurel as a game model of concurrent interaction. We show how L-nets satisfying an additional condition, which we call LS-nets, can be sequentialized into traditional tree-like strategies. Conversely, sequential strategies can be relaxed into more asynchronous ones.We develop an algebra of constructors and destructors that serve to build and decompose graph strategies, and to describe a class of minimally sequential graph strategies, which can be seen as an abstract kind of multiplicative–additive proof nets.

Sublogarithmic uniform Boolean proof nets

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.

Isomorphic formulae in classical propositional logic

AbstractIsomorphism between formulae is defined with respect to categories formalizing equality of deductions in classical propositional logic and in the multiplicative fragment of classical linear propositional logic caught by proof nets. This equality is motivated by generality of deductions. Characterizations are given for pairs of isomorphic formulae, which lead to decision procedures for this isomorphism.

A new graphical calculus of proofs

We offer a simple graphical representation for proofs of intuitionistic logic, which is inspired by proof nets and interaction nets (two formalisms originating in linear logic). This graphical calculus of proofs inherits good features from each, but is not constrained by them. By the Curry-Howard isomorphism, the representation applies equally to the lambda calculus, offering an alternative diagrammatic representation of functional computations.

Intuitionistic differential nets and lambda-calculus

We define pure intuitionistic differential proof nets, extending Ehrhard and Regnier’s differential interaction nets with the exponential box of Linear Logic. Normalization of the exponential reduction and confluence of the full one is proved. These results are directed and adjusted to give a translation of Boudol’s untyped λ-calculus with resources extended with a linear–nonlinear reduction à la Ehrhard and Regnier’s differential λ-calculus. Such reduction comes in two flavours: baby-step and giant-step β-reduction. The translation, based on Girard’s encoding A→B∼!A⊸B and as such extending the usual one for λ-calculus into proof nets, enjoys bisimulation for giant-step β-reduction. From this result we also derive confluence of both reductions.