Jeff Polakow

Monadic concurrent linear logic programming

Lolli is a logic programming language based on the asynchronous propositions of intuitionistic linear logic. It uses a backward chaining, backtracking operational semantics. In this paper we extend Lolli with the remaining connectives of intuitionistic linear logic restricted to occur inside a monad, an idea taken from the concurrent logical framework (CLF). The resulting language, called LolliMon, has a natural forward chaining, committed choice operational semantics inside the monad, while retaining Lollis semantics outside the monad. LolliMon thereby cleanly integrates both concurrency and saturation with logic programming search. We illustrate its expressive power through several examples including an implementation of the pi-calculus, a call-by-need lambda-calculus, and several saturating algorithms presented in logical form.

Natural Deduction for Intuitionistic Non-communicative Linear Logic

We present a system of natural deduction and associated term calculus for intuitionistic non-commutative linear logic (INCLL) as a conservative extension of intuitionistic linear logic. We prove subject reduction and the existence of canonical forms in the implicational fragment.

Relating Natural Deduction and Sequent Calculus for Intuitionistic Non-Commutative Linear Logic

Abstract We present a sequent calculus for intuitionistic non-commutative linear logic (INCLL), show that it satisfies cut elimination, and investigate its relationship to a natural deduction system for the logic. We show how normal natural deductions correspond to cut-free derivations, and arbitrary natural deductions to sequent derivations with cut. This gives us a syntactic proof of normalization for a rich system of non-commutative natural deduction and its associated λ-calculus. INCLL conservatively extends linear logic with means to express sequencing, which has applications in functional programming, logical frameworks, logic programming, and natural language parsing.

System E: Expansion Variables for Flexible Typing with Linear and Non-linear Types and Intersection Types

Types are often used to control and analyze computer programs. Intersection types give great flexibility, but have been difficult to implement. The ! operator, used to distinguish between linear and non-linear types, has good potential for better resource-usage tracking, but has not been as flexible as one might want and has been difficult to use in compositional analysis. We introduce System E, a type system with expansion variables, linear intersection types, and the ! type constructor for creating non-linear types. System E is designed for maximum flexibility in automatic type inference and for ease of automatic manipulation of type information. Expansion variables allow postponing the choice of which typing rules to use until later constraint solving gives enough information to allow making a good choice. System E removes many difficulties that expansion variables had in the earlier System I and extends expansion variables to work with ! in addition to the intersection type constructor. We present subject reduction for call-by-need evaluation and discuss program analysis in System E.

Linearity Constraints as Bounded Intervals in Linear Logic Programming

We present a new system of resource management for linear logic programming which ensures linearity constraints are satisfied solely by manipulating individual formula tags. In our system, tags are rational numbers, and a single bounded interval suffices to characterize all of the available formulas at any point in the proof. This system, which we prove correct directly with respect to the efficient resource management system of Cervesato et al., simplifies and improves upon the tag-frame system of Hodas et al.

Implementing efficient resource management for linear logic programming

The Tag-Frame system of resource management [1] reunited two divergent threads of linear logic programming research by achieving the efficient proof search behaviour of abstract systems, such as [2], while using a low-level tag-based approach, as in [3], suitable for specifying an abstract machine. However, Tag-Frame relies on set operations which are linear in the size of the sets, and is not as efficient, in general, as it could be. We present a new tag-based derivation system which relies solely on low-level concepts to implement efficient resource management, where most linear time operations have been replaced by constant time ones. Though motivated and informed by the Tag-Frame system, we derive our system directly from, and prove its correctness with respect to the system of Cervesato et al. [2]. An abstract machine based on the new system has been implemented by Tamura and Banbara, and its performance compared to their previous machine.

A formalization of an Ordered Logical Framework in Hybrid with applications to continuation machines

We report on work in progress devoted to the formalization of an Ordered Logical Framework (OLF) [16] based on a two-levels architecture [10] in the Hybrid system [2]. OLF here is a second-order version of ordered linear logic to be used as a meta-language for the verification of the (meta) theory of deductive systems. It is implemented roughly as a meta-interpreter on top of the Hybrid system, which provides the full HOAS language. We apply the framework to the formal verification of type preservation of a simple continuation machine for Mini-ML.