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Several type systems have been proposed to statically control the time complexity of lambda-calculus programs and characterize complexity classes such as FPTIME or FEXPTIME. A first line of research stems from linear logic and restricted versions of its !-modality controlling duplication. An instance of this is light linear logic for polynomial time computation [5]. A second approach relies on the idea of tracking the size increase between input and output, and together with a restricted recursion scheme, to deduce time complexity bounds. This second approach is illustrated for instance by non-size-increasing types [8]. However, both approaches suffer from limitations. The first one, that of linear logic, has a limited intensional expressivity, that is to say some natural polynomial time programs are not typable. As to the second approach it is essentially linear, more precisely it does not allow for a non-linear use of functional arguments. In the present work we incorporate both approaches into a common type system, in order to overcome their respective constraints. The source language we consider is a lambda-calculus with data-types and iteration, that is to say a variant of Gödel's system T. Our goal is to design a system for this language allowing both to handle non-linear functional arguments and to keep a good intensional expressivity. We illustrate our methodology by choosing the system of elementary linear logic (ELL) and combining it with a system of linear size types. We discuss the expressivity of this new type system, called sEAL, and prove that it gives a characterization of the complexity classes FPTIME and 2k-FEXPTIME, for k ≥ 0. [ABSTRACT FROM AUTHOR]

When studying sequent calculi, proof theorists often have to prove properties about the systems, whether to show that they are "well-behaved", amenable to automated proof search, complete with respect to another system, consistent, among other reasons. These proofs usually involve many very similar cases, which leads to authors rarely writing them in full detail, only pointing to one or two more complicated cases. Moreover, the amount of details makes them more error-prone for humans. Computers, on the other hand, are very good at handling details and repetitiveness. In this work we have formalized textbook proofs of the meta-theory of sequent calculi for linear logic in Abella. Using the infrastructure developed, the proofs can be easily adapted to other substructural logics. We implemented rules as clauses in an intuitive and straightforward way, similar to logic programming, using operations on multisets for the explicit contexts. Although the proofs are quite big, they use only elementary reasoning principles, which makes the proof techniques fairly portable to other formal reasoning systems. [ABSTRACT FROM AUTHOR]

Abstract: Inspired by the Multiplicative Exponential fragment of Linear Logic, we define a framework called the prismoid of resources where each vertex is a language which refines the -calculus by using a different choice to make explicit or implicit (meta-level) the definition of the contraction, weakening, and substitution operations. For all the calculi in the prismoid we show simulation of -reduction, confluence, preservation of -strong normalisation and strong normalisation for typed terms. Full composition also holds for all the calculi of the prismoid handling explicit substitutions. The whole development of the prismoid is done by making the set of resources a parameter of the formalism, so that all the properties for each vertex are obtained as a particular case of the general abstract proofs. [Copyright &y& Elsevier]

Abstract: The typical AI problem is that of making a plan of the actions to be performed by a robot so that the robot could get into a set of final situations, if it started with a certain initial situation. The planning problem is known to be generally very complex. Even within the case of ‘well-balanced’ actions, strong planning under uncertainty about the effects of actions, or games such as ‘Robot against Nature’, is EXPTIME-complete. As a result, AI planners are very sensitive to the number of the variables involved in making a plan, the inherent symmetry of the problem, and the nature of the logical formalisms being used. This paper shows that linear logic provides a convenient and adequate tool for representing strong and weak planning problems in non-deterministic domains. A particular focus of this paper is on planning problems with an unbounded number of functionally identical objects. We show that for such problems linear logic is especially effective and leads to a dramatic contraction of the search space from exponential to polynomial in size. We employ the ability of linear logic to reason about multisets, which in this instance are created by identifying several distinct objects as being functionally equivalent for the problem at hand (think of a number of balls, each of which must be moved to some new location — the balls are distinct, but are functionally equivalent for the problem). In linear logic terms, we establish a clear syntactic condition that allows us to show that solving a generic planning problem where there is only one generic object, directly implies a solution to the original real planning problem over several real objects, the isomorphic copies of the generic object. Moreover, this correspondence also guarantees to produce a real solution that works in polynomial time. [Copyright &y& Elsevier]

Abstract: Full first-order linear logic can be presented as an abstract logic programming language in Miller''s system Forum, which yields a sensible operational interpretation in the ‘proof search as computation’ paradigm. However, Forum still has to deal with syntactic details that would normally be ignored by a reasonable operational semantics. In this respect, Forum improves on Gentzen systems for linear logic by restricting the language and the form of inference rules. We further improve on Forum by restricting the class of formulae allowed, in a system we call G-Forum, which is still equivalent to full first-order linear logic. The only formulae allowed in G-Forum have the same shape as Forum sequents: the restriction does not diminish expressiveness and makes G-Forum amenable to proof theoretic analysis. G-Forum consists of two (big) inference rules, for which we show a cut elimination procedure. This does not need to appeal to finer detail in formulae and sequents than is provided by G-Forum, thus successfully testing the internal symmetries of our system. [Copyright &y& Elsevier]

Abstract: We consider the multiplicative and exponential fragment of linear logic (MELL) and give a geometry of interaction (GoI) semantics for it based on unique decomposition categories. We prove a soundness and finiteness theorem for this interpretation. We show that Girard''s original approach to GoI 1 via operator algebras is exactly captured in this categorical framework. [Copyright &y& Elsevier]

Abstract: Ludics [J.-Y. Girard, Locus solum, Math. Structures in Comput. Sci. 11 (2001) 301–506] is a recent proposal of analysis of interaction, developed by abstracting away from proof-theory. It provides an elegant, abstract setting in which interaction between agents (proofs/programs/processes) can be studied at a foundational level, together with a notion of equivalence from the point of view of the observer. An agent should be seen as some kind of black box. An interactive observation on an agent is obtained by testing it against other agents. In this paper we explore what can be observed interactively in this setting. In particular, we characterize the objects that can be observed in a single test: the primitive observables of the theory. Our approach builds on an analysis of the geometrical properties of the agents, and highlights a deep interleaving between two partial orders underlying the combinatorial structures: the spatial one and the temporal one. [Copyright &y& Elsevier]

It is well known that every proof net of a non-commutative version of MLL (multiplicative fragment of commutative linear logic) can be drawn as a plane Danos–Regnier graph (drawing) satisfying the switching condition of Danos–Regnier . In this paper, we study the reverse direction; we introduce a system MNCLL which is logically equivalent to the multiplicative fragment of cyclic linear logic introduced by Yetter , and show that any plane Danos–Regnier graph drawing with one terminal edge satisfying the switching condition represents a unique non-commutative proof net (i.e., a proof net of MNCLL). In the course of proving this, we also give the characterization of the non-commutative proof nets by means of the notion of strong planarity, as well as the notion of a certain long-trip condition, called the stack-condition, of a Danos–Regnier graph, the latter of which is related to Abrusci''s balanced long-trip condition . [Copyright &y& Elsevier]

The motivating role of linear logic is as a “logic behind logic”. We propose a sibling role for it as a logic of transformational mathematics via the self-dual category of Chu spaces, a generalization of topological spaces. These create a bridge between linear logic and mathematics by soundly and fully completely interpreting linear logic while fully and concretely embedding a comprehensive range of concrete categories of mathematics. Our main goal is to treat each end of this bridge in expository detail. In addition, we introduce the dialectic lambda-calculus, and show that dinaturality semantics is not fully complete for the Chu interpretation of linear logic. [Copyright &y& Elsevier]

Intuitionistic and linear logics can be used to specify the operational semantics of transition systems in various ways. We consider here two encodings: one uses linear logic and maps states of the transition system into formulas, and the other uses intuitionistic logic and maps states into terms. In both cases, it is possible to relate transition paths to proofs in sequent calculus. In neither encoding, however, does it seem possible to capture properties, such as simulation and bisimulation, that need to consider all possible transitions or all possible computation paths. We consider augmenting both intuitionistic and linear logics with a proof theoretical treatment of definitions. In both cases, this addition allows proving various judgments concerning simulation and bisimulation (especially for noetherian transition systems). We also explore the use of infinite proofs to reason about infinite sequences of transitions. Finally, combining definitions and induction into sequent calculus proofs makes it possible to reason more richly about properties of transition systems completely within the formal setting of sequent calculus. [Copyright &y& Elsevier]

Sharing graphs are an implementation of linear logic proof-nets in which a redex is never duplicated. In their usual formulation, sharing graphs present a problem of coherence: if the proof-net N reduces by standard cut-elimination to N′, then, by reducing the sharing graph of N we do not obtain the sharing graph of N′. We solve this problem by changing the way the information is coded into sharing graphs and introducing a new reduction rule (absorption). The rewriting system is confluent and terminating. The proof exploits an algebraic semantics for sharing graphs. [Copyright &y& Elsevier]