True Concurrency of Deep Inference Proofs

The deep inference presentation of multiplicative exponential linear logic (MELL) benefits from a rich combinatoric analysis with many more proofs in comparison to its sequent calculus presentation. In the deep inference setting, all the sequent calculus proofs are preserved. Moreover, many other proofs become available, and some of these proofs are much shorter. However, proof search in deep inference is subject to a greater nondeterminism, and this nondeterminism constitutes a bottleneck for applications. To this end, we address the problem of reducing nondeterminism in MELL by refining and extending our technique that has been previously applied to multiplicative linear logic and classical logic. We show that, besides the nondeterminism in commutative contexts, the nondeterminism in exponential contexts can be reduced in a proof theoretically clean manner. The method conserves the exponential speed-up in proof construction due to deep inference, exemplified by Statman tautologies. We validate the improvement in accessing the shorter proofs by experiments with our implementations.

A cornerstone of the theory of proof nets for unit-free multiplicative linear logic (MLL) is the abstract representation of cut-free proofs modulo inessential rule commutation. The only known extension to additives, based on monomial weights, fails to preserve this key feature: a host of cut-free monomial proof nets can correspond to the same cut-free proof. Thus, the problem of finding a satisfactory notion of proof net for unit-free multiplicative-additive linear logic (MALL) has remained open since the inception of linear logic in 1986. We present a new definition of MALL proof net which remains faithful to the cornerstone of the MLL theory.

We derive two novel theorems regarding pre-fixed points of non-monotonic functions and demonstrate that they have immediate applications in logic programming and formal grammars. In particular, the results of the paper lead to a natural generalization of the model intersection theorem for definite logic programs, to the more general class of normal logic programs. Moreover, the obtained results also offer the first to our knowledge model intersection result for Boolean grammars.

This paper presents a simple and intuitive syntax for proof nets of the multiplicative cyclic fragment (McyLL) of linear logic (LL). The main technical achievement of this work is to propose a correctness criterion that allows for sequentialization (recovering a proof from a proof net) for all McyLL proof nets, including those containing cut links. This is achieved by adapting the idea of contractibility (originally introduced by Danos to give a quadratic time procedure for proof nets correctness) to cyclic LL. This paper also gives a characterization of McyLL proof nets for Lambek Calculus and thus a geometrical (i.e., non-inductive) way to parse phrases or sentences by means of Lambek proof nets.

Proof nets, garbage, and computations

The Mozart consortium ( German Research Center for Artificial Intelligence (DFKI), Universit~it des Saarlandes (UdS), Swedish Institute of Computer Science (SICS), Universit6 catholique de Louvain (UCL)) is pleased to announce the release of Mozart 1.0, an extensive development platform for open fault-tolerant distributed applications, constraint programming applications, and logic programming applications. The system is available at http://www.mozart-oz.org and runs on many Unix-based platforms and Windows 95/98/NT. The platform implements the Oz language and is the fruit of a decade of research. The platform is released without charge with a very liberal license (XI 1-style) that allows both commercial and non-commercial use. Full source code is available. The platform includes a full-fledged interactive development environment with many libraries and tools. There is extensive documentation including tutorials covering all below mentioned application areas, a complete set of reference manuals, and many demos. The distribution, constraint, and logic programming abilities are the subject of ongoing research. 2 Constraint and logic programming

Connectedness and synchronization

In this paper, we introduce the notion of expansion tree for linear logic. As in Miller's original work, we have a shallow reading of an expansion tree that corresponds to the conclusion of the proof, and a deep reading which is a formula that can be proved by propositional rules. We focus our attention to MLL2, and we also present a deep inference system for that logic. This allows us to give a syntactic proof to a version of Herbrand's theorem.

We analyse the reduction of differential interaction nets from the point of view of so-called ‘true concurrency,’ that is, employing a non-interleaving model of parallelism. More precisely, we associate with each differential interaction net an event structure describing its reduction. We show how differential interaction nets are only able to generate confusion-free event structures, and we argue that this is a serious limitation in terms of the concurrent behaviours they may express. In fact, confusion is an extremely elementary phenomenon in concurrency (for example, it already appears in CCS with just prefixing and parallel composition) and we show how its presence is preserved by any encoding respecting the degree of distribution and the reduction semantics. We thus infer that no reasonably expressive process calculus may be satisfactorily encoded in differential interaction nets. We conclude with an analysis of one such encoding proposed by Ehrhard and Laurent, and argue that it does not contradict our claims, but rather supports them.

Pomset logic introduced by Retor\'e is an extension of linear logic with a self-dual noncommutative connective. The logic is defined by means of proof-nets, rather than a sequent calculus. Later a deep inference system BV was developed with an eye to capturing Pomset logic, but equivalence of system has not been proven up to now. As for a sequent calculus formulation, it has not been known for either of these logics, and there are convincing arguments that such a sequent calculus in the usual sense simply does not exist for them. In an on-going work on semantics we discovered a system similar to Pomset logic, where a noncommutative connective is no longer self-dual. Pomset logic appears as a degeneration, when the class of models is restricted. Motivated by these semantic considerations, we define in the current work a semicommutative multiplicative linear logic}, which is multiplicative linear logic extended with two nonisomorphic noncommutative connectives (not to be confused with very different Abrusci-Ruet noncommutative logic). We develop a syntax of proof-nets and show how this logic degenerates to Pomset logic. However, a more interesting problem than just finding yet another noncommutative logic is to find a sequent calculus for this logic. We introduce decorated sequents, which are sequents equipped with an extra structure of a binary relation of reachability on formulas. We define a decorated sequent calculus for semicommutative logic and prove that it is cut-free, sound and complete. This is adapted to "degenerate" variations, including Pomset logic. Thus, in particular, we give a variant of sequent calculus formulation for Pomset logic, which is one of the key results of the paper.

Constant-only multiplicative linear logic is NP-complete

A logical characterization of well branching event structures

We design a proof system for propositional classical logic that integrates two languages for Boolean functions: standard conjunction-disjunction-negation and binary decision trees. We give two reasons to do so. The first is proof-theoretical naturalness: The system consists of all and only the inference rules generated by the single, simple, linear scheme of the recently introduced subatomic logic. Thanks to this regularity, cuts are eliminated via a natural construction. The second reason is that the system generates efficient proofs. Indeed, we show that a certain class of tautologies due to Statman, which cannot have better than exponential cut-free proofs in the sequent calculus, have polynomial cut-free proofs in our system. We achieve this by using the same construction that we use for cut elimination. In summary, by expanding the language of propositional logic, we make its proof theory more regular and generate more proofs, some of which are very efficient. That design is made possible by considering atoms as superpositions of their truth values, which are connected by self-dual, non-commutative connectives. A proof can then be projected via each atom into two proofs, one for each truth value, without a need for cuts. Those projections are semantically natural and are at the heart of the constructions in this article. To accommodate self-dual non-commutativity, we compose proofs in deep inference.

Standardization is a fundamental notion for connecting programming languages and rewriting calculi. Since both programming languages and calculi rely on substitution for defining their dynamics, explicit substitutions (ES) help further close the gap between theory and practice. This paper focuses on standardization for the linear substitution calculus, a calculus with ES capable of mimicking reduction in lambda-calculus and linear logic proof-nets. For the latter, proof-nets can be formalized by means of a simple equational theory over the linear substitution calculus. Contrary to other extant calculi with ES, our system can be equipped with a residual theory in the sense of Lévy, which is used to prove a left-to-right standardization theorem for the calculus with ES but without the equational theory. Such a theorem, however, does not lift from the calculus with ES to proof-nets, because the notion of left-to-right derivation is not preserved by the equational theory. We then relax the notion of left-to-right standard derivation, based on a total order on redexes, to a more liberal notion of standard derivation based on partial orders. Our proofs rely on Gonthier, Lévy, and Melliès' axiomatic theory for standardization. However, we go beyond merely applying their framework, revisiting some of its key concepts: we obtain uniqueness (modulo) of standard derivations in an abstract way and we provide a coinductive characterization of their key abstract notion of external redex. This last point is then used to give a simple proof that linear head reduction --a nondeterministic strategy having a central role in the theory of linear logic-- is standard.

We introduce graph reduction technology that implements functional languages with control, such as Scheme with call/cc, where continuations can be manipulated explicitly as values, and can be optimally reduced in the sense of LEvy. The technology is founded on proofnets for multiplicative-exponential linear logic, extending the techniques originally proposed by Lamping, where we adapt the continuation-passing style transformation to yield a new understanding of sharable values. Confluence is maintained by returning multiple answers to a (shared) continuation. Proofnets provide a concurrent version of linear logic proofs, eliminating structurally irrelevant sequentialization, and ignoring asymmetric distinctions between inputs and outputs--dually, expressions and continuations. While Lamping's graphs and their variants encode an embedding of intuitionistic logic into linear logic, our construction implicitly contains an embedding of classical logic into linear logic. We propose a family of translations, produced uniformly by beginning with a continuation-passing style semantics for the languages, employing standard codings into proofnets using call-by-value, call-by-name--or hybrids of the two--to locate proofnet boxes, and converting the proofnets to direct style. The resulting graphs can be reduced simply (cut elimination for linear logic), have a consistent semantics that is preserved by reduction (geometry of interaction, via the so-called context semantics), and allow shared, incremental evaluation of continuations (optimal reduction).