A Linear/Producer/Consumer Model of Classical Linear Logic

This paper defines a new proof- and category-theoretic framework for classical linear logic that separates reasoning into one linear regime and two persistent regimes corresponding to ! and ?. The resulting linear/producer/consumer (LPC) logic puts the three classes of propositions on the same semantic footing, following Benton's linear/non-linear formulation of intuitionistic linear logic. Semantically, LPC corresponds to a system of three categories connected by adjunctions reflecting the linear/producer/consumer structure. The paper's metatheoretic results include admissibility theorems for the cut and duality rules, and a translation of the LPC logic into category theory. The work also presents several concrete instances of the LPC model.Comment: In Proceedings LINEARITY 2014, arXiv:1502.0441

A theory of effects and resources: adjunction models and polarised calculi

We consider the Curry-Howard-Lambek correspondence for effectful computation and resource management, specifically proposing polarised calculi together with presheaf-enriched adjunction models as the starting point for a comprehensive semantic theory relating logical systems, typed calculi, and categorical models in this context. Our thesis is that the combination of effects and resources should be considered orthogonally. Model theoretically, this leads to an understanding of our categorical models from two complementary perspectives: (i) as a linearisation of CBPV (Call-by-Push-Value) adjunction models, and (ii) as an extension of linear/non-linear adjunction models with an adjoint resolution of computational effects. When the linear structure is cartesian and the resource structure is trivial we recover Levy’s notion of CBPV adjunction model, while when the effect structure is trivial we have Benton’s linear/nonlinear adjunction models. Further instances of our model theory include the dialogue categories with a resource modality of Melliès and Tabareau, and the [E]EC ([Enriched] Effect Calculus) models of Egger, Møgelberg and Simpson. Our development substantiates the approach by providing a lifting theorem of linear models into cartesian ones. artesian ones. To each of our categorical models we systematically associate a typed term calculus, each of which corresponds to a variant of the sequent calculi LJ (Intuitionistic Logic) or ILL (Intuitionistic Linear Logic). The adjoint resolution of effects corresponds to polarisation whereby, syntactically, types locally determine a strict or lazy evaluation order and, semantically, the associativity of cuts is relaxed. In particular, our results show that polarisation provides a computational interpretation of CBPV in direct style. Further, we characterise depolarised models: those where the cut is associative, and where the evaluation order is unimportant. We explain possible advantages of this style of calculi for the operational semantics of effects.G. Munch-Maccagnoni was supported by ERC ECSYM; M. Fiore partially so.This is the author accepted manuscript. The final version is available from the Association for Computing Machinery via http://dx.doi.org/10.1145/2837614.283765

Non-neutrality of economic policy: An application of the Tinbergen-Theil’s approach to a strategic context

Issues of policy effectiveness and neutrality are widespread in the economic literature. They have been increasingly raised in specific contexts within the class of LQ (linear-quadratic) policy games in the last 20 years, notably with reference to monetary policy. The more general conditions ensuring nonneutrality in a strategic environment remain however to be inquired. We fill this gap by applying the classical theory of economic policy to a strategic context. This is also useful to highlight some existence conditions for policy game solutions. We restrict ourselves to the common LQ-games in a static perfect information framework, but our simple logic can be extended to other more general situations.LQ-policy games, policy ineffectiveness, controllability.

Price Discrimination between Retailers with and without Market Power

Some retail markets are more competitive than others. A manufacturer with market power in the wholesale market who sells his product to competing retailers in cities and monopolistic ones in each of various towns must set the wholesale price difference between towns and cities to be smaller than the transportation cost to prevent “grey market” arbitrage. If he uses linear pricing, the town retail price will be even higher than under single-retailer double marginalization. Two-part tariffs do not solve the problem as they would if there were a single retailer, because the wholesale unit price must be higher than marginal cost to prevent arbitrage to the cities. If transportation costs are low, price discrimination is difficult and two- part tariffs come to resemble inefficient linear monopoly pricing. High transportation costs allow greater efficiency in contracting, and this can outweigh the negative direct effect on welfare.price discrimination, double marginalization, retail network, transportation costs, two-part tariffs, vertical restraints

Making Sense of Non-Binding Retail-Price Recommendations

This paper provides a theoretical rationale for non-binding retail price recommendations (RPRs) in vertical supply relations. Analyzing a bilateral manufacturer-retailer relationship with repeated trade, we show that linear relational contracts can implement the surplusmaximizing outcome. If the manufacturer has private information about production costs or consumer demand, RPRs may serve as a communication device from manufacturer to retailer. We characterize the properties of efficient bilateral relational contracts with RPRs and discuss extensions to settings where consumer demand is affected by RPRs, and where there are multiple retailers or competing supply chains.vertical relationships, relational contracts, asymmetric information, price recommendations

Linear/non-Linear Types For Embedded Domain-Specific Languages

Domain-specific languages are often embedded inside of general-purpose host languages so that the embedded language can take advantage of host-language data structures, libraries, and tools. However, when the domain-specific language uses linear types, existing techniques for embedded languages fall short. Linear type systems, which have applications in a wide variety of programming domains including mutable state, I/O, concurrency, and quantum computing, can manipulate embedded non-linear data via the linear type !σ. However, prior work has not been able to produce linear embedded languages that have full and easy access to host-language data, libraries, and tools. This dissertation proposes a new perspective on linear, embedded, domain-specific languages derived from the linear/non-linear (LNL) interpretation of linear logic. The LNL model consists of two distinct fragments---one with linear types and another with non-linear types---and provides a simple categorical interface between the two. This dissertation identifies the linear fragment with the linear embedded language and the non-linear fragment with the general-purpose host language. The effectiveness of this framework is illustrated via a number of examples, implemented in a variety of host languages. In Haskell, linear domain-specific languages using mutable state and concurrency can take advantage of the monad that arises from the LNL model. In Coq, the QWIRE quantum circuit language uses linearity to enforce the no-cloning axiom of quantum mechanics. In homotopy type theory, quantum transformations can be encoded as higher inductive types to simplify the presentation of a quantum equational theory. These examples serve as case studies that prove linear/non-linear type theory is a natural and expressive interface in which to embed linear domain-specific languages

Simple, safe, and efficient memory management using linear pointers

Efficient and safe memory management is a hard problem. Garbage collection promises automatic memory management but comes with the cost of increased memory footprint, reduced parallelism in multi-threaded programs, unpredictable pause time, and intricate tuning parameters balancing the program's workload and designated memory usage in order for an application to perform reasonably well. Existing research mitigates the above problems to some extent, but programmer error could still cause memory leak by erroneously keeping memory references when they are no longer needed. We need a methodology for programmers to become resource aware, so that efficient, scalable, predictable and high performance programs may be written without the fear of resource leak. Linear logic has been recognized as the formalism of choice for resource tracking. It requires explicit introduction and elimination of resources and guarantees that a resource cannot be implicitly shared or abandoned, hence must be linear. Early languages based on linear logic focused on Curry-Howard correspondence. They began by limiting the expressive powers of the language and then reintroduced them by allowing controlled sharing which is necessary for recursive functions. However, only by deviating from Curry-Howard correspondence could later development actually address programming errors in resource usage. The contribution of this dissertation is a simple, safe, and efficient approach introducing linear resource ownership semantics into C++ (which is still a widely used language after 30 years since inception) through linear pointer, a smart pointer inspired by linear logic. By implementing various linear data structures and a parallel, multi-threaded memory allocator based on these data structures, this work shows that linear pointer is practical and efficient in the real world, and that it is possible to build a memory management stack that is entirely leak free. The dissertation offers some closing remarks on the difficulties a formal system would encounter when reasoning about a concurrent linear data algorithm, and what might be done to solve these problems