Functional Interpretations of Intuitionistic Linear Logic

This article aims to generate all theorems of a given size in the implicational fragment of propositional intuitionistic linear logic. It was presented at the 36th International Conference on Logic Programming (ICLP).

Intuitionistic logic encompasses the principles of reasoning which were used by L. Brouwer in developing his intuitionistic mathematics, beginning in [Bro07]. Intuitionistic logic can be succinctly described as classical logic without the law ϕ ∨ ¬ϕ of excluded middle. Brouwer observed that this law was abstracted from finite situations and its application to statements about infinite collections is not justified. One of the consequences of the rejection of the law of excluded middle is that every intuitionistic proof of an existential sentence can be effectively transformed into an intuitionistic proof of an instance of that sentence. More precisely, if a formula of the form ∃xϕ(x) without free variables is provable in the intuitionistic predicate logic, then there is a term t without free variables such that ϕ(t) is provable. In particular, if ϕ ∨ ψ is provable, then either ϕ or ψ is provable. In that sense intuitionistic logic may provide a logical basis for constructive reasoning. A formal system of intuitionistic logic was proposed in [Hey30]. A relationship between the classical propositional logic PC and intuitionistic propositional logic INT was proved by Glivenko in [Gli29], namely a PC-formula ϕ is PC-provable if and only if ¬ ¬ϕ is INT-provable. Kripke semantics for intuitionistic logic was developed in [Kri65].

Recent development of logic programming languages based on linear logic suggests a successful direction to extend logic programming to be more expressive and more efficient. The treatment of formulasas-resources gives us not only powerful expressiveness, but also efficient access to a large set of data. However, in linear logic, whole resources are kept in one context, and there is no straight way to represent complex data structures as resources. For example, in order to represent an ordered list and time-dependent data, we need to put additional indices for each resource formula. This paper describes a logic programming language, called TLLP, based on intuitionistic temporal linear logic. This logic, an extension of linear logic with some features from temporal logics, allows the use of the modal operators '?'(next-time) and '?'(always) in addition to the operators used in intuitionistic linear logic. The intuitive meaning of modal operators is as follows: ? B means that B can be used exactly once at the next moment in time; ? B means that B can be used exactly once any time; !B means that B can be used arbitrarily many times (including 0 times) at any time. We first give a proof theoretic formulation of the logic of the TLLP language. We then present a series of resource management systems designed to implement not only interpreters but also compilers based on an extension of the standard WAM model.

Intuitionistic linear logic regains the expressive power of intuitionistic logic through the ! (‘of course’) modality. Benton, Bierman, Hyland and de Paiva have given a term assignment system for ILL and an associated notion of categorical model in which the ! modality is modelled by a comonad satisfying certain extra conditions. Ordinary intuitionistic logic is then modelled in a cartesian closed category which arises as a full subcategory of the category of coalgebras for the comonad. This paper attempts to explain the connection between ILL and IL more directly and symmetrically by giving a logic, term calculus and categorical model for a system in which the linear and non-linear worlds exist on an equal footing, with operations allowing one to pass in both directions. We start from the categorical model of ILL given by Benton, Bierman, Hyland and de Paiva and show that this is equivalent to having a symmetric monoidal adjunction between a symmetric monoidal closed category and a cartesian closed category. We then derive both a sequent calculus and a natural deduction presentation of the logic corresponding to the new notion of model.

This article shows how individual Petri nets form models of Girard's intuitionistic linear logic. It explores questions of expressiveness and completeness of linear logic with respect to this interpretation. An aim is to use Petri nets to give an understanding of linear logic and give some appraisal of the value of linear logic as a specification logic for Petri nets. This article might serve as a tutorial, providing one in-road into Girard's linear logic via Petri nets. With this in mind we have added several exercises and their solutions. We have made no attempt to be exhaustive in our treatment, dedicating our treatment to one semantics of intuitionistic linear logic.Completeness is shown for several versions of Girard's linear logic with respect to Petri nets as the class of models. The strongest logic considered is intuitionistic linear logic, with ⊗, ⊸, &, ⊕ and the exponential ! (“of course”), and forms of quantification. This logic is shown sound and complete with respect to atomic nets (these include nets in which every transition leads to a nonempty multiset of places). The logic is remarkably expressive, enabling descriptions of the kinds of properties one might wish to show of nets; in particular, negative properties, asserting the impossibility of an assertion, can also be expressed. A start is made on decidability issues.

A theorem for embedding a first‐order linear‐time temporal logic LTL into its intuitionistic counterpart ILTL is proved using Baratella‐Masini's temporal extension of the Gödel‐Gentzen negative translation of classical logic into intuitionistic logic. A substructural counterpart LLTL of ILTL is introduced, and a theorem for embedding ILTL into LLTL is proved using a temporal extension of the Girard translation of intuitionistic logic into intuitionistic linear logic. These embedding theorems are proved syntactically based on Gentzen‐type sequent calculi.

We show that the category OS of operator spaces, with complete contractions as morphisms, is locally countably presentable and a model of Intuitionistic Linear Logic in the sense of Lafont. We then describe a model of Classical Linear Logic, based on OS, whose duality is compatible with the Heisenberg-Schrödinger duality of quantum theory. We also show that OS provides a good setting for studying pure state and mixed state quantum information, the interaction between the two, and even higher-order quantum maps such as the quantum switch.

1.1 Relationship between Petri net and linear logic Petri nets were first introduced by Petri in his seminal Ph.D. thesis, and both the theory and the applications of his model have flourished in concurrency theory (Reisig & Rozenberg, 1998a; Reisig & Rozenberg, 1998b). The relationships between Petri nets and linear logics have been studied by many researchers (Engberg & Winskel, 1997; Farwer, 1999; Hirai, 2000; Hirai 1999; Ishihara & Hiraish, 2001; Kamide, 2004, Kamide, 2006; Kanovich, 1995; Kanovich 1994; LarcheyWendling & Galmiche, 1998; Larchey-Wendling & Galmiche, 2000; Lilius, 1992; Mart -Oliet & Meseguer, 1991; Okada, 1998; Tanabe, 1997). A category theoretical investigation of such a relationship was given by Mart -Oliet and Meseguer (Mart -Oliet & Meseguer, 1991), purely syntactical approach using Horn linear logic was established by Kanovich (Kanovich, 1995; Kanovich 1994), a naive phase linear logic for a certain class of Petri nets was given by Okada (Okada, 1998), a linear logical view of object Petri nets were studied by Farwer (Farwer, 1999), and various Petri net interpretations of linear logic using quantale models were obtained by Ishihara and Hiraishi (Ishihara & Hiraish, 2001), Engberg and Winskel (Engberg & Winskel, 1997), Larchey-Wendling and Galmiche (Larchey-Wendling & Galmiche, 1998; Larchey-Wendling & Galmiche, 2000), and Lilius (Lilius, 1992). Petri net interpretations using Kripke semantics for various fragments and extensions of intuitionistic linear logic were studied by Kamide (Kamide, 2004; Kamide, 2006c). In (Kamide, 2004), Petri net interpretations of various fragments of a spatio-temporal soft linear logic were discussed. In (Kamide, 2006c), Petri nets with inhibitor arcs, which were first introduced by Kosaraju (Kosaraju, 1973) to show the limitation of the usual Petri nets, were described using Kripke semantics for intuitionistic linear logic with strong negation. The approarches using Kripke semantics can obtain a very simple correspondence between Petri net and linear logic.

The present article is a brief informal survey o$\textit {computability logic}$ (CoL). This relatively young and still evolving nonclassical logic can be characterized as a formal theory of computability in the same sense as classical logic is a formal theory of truth. In a broader sense, being conceived semantically rather than proof-theoretically, CoL is not just a particular theory but an ambitious and challenging long-term project for redeveloping logic. 
 
 In CoL, logical operators stand for operations on computational problems, formulas represent such problems, and their "truth" is seen as algorithmic solvability. In turn, computational problems – understood in their most general, interactive sense – are defined as games played by a machine against its environment, with "algorithmic solvability" meaning existence of a machine which wins the game against any possible behavior of the environment. With this semantics, CoL provides a systematic answer to the question "What can be computed?", just like classical logic is a systematic tool for telling what is true. Furthermore, as it happens, in positive cases "What can be computed" always allows itself to be replaced by "How can be computed", which makes CoL a problem-solving tool. 
 
 CoL is a conservative extension of classical first order logic but is otherwise much more expressive than the latter, opening a wide range of new application areas. It relates to intuitionistic and linear logics in a similar fashion, which allows us to say that CoL reconciles and unifies the three traditions of logical thought (and beyond) on the basis of its natural and "universal" game semantics. 

With help of a compact Prolog-based theorem prover for Intuitionistic Propositional Logic, we synthesize minimal assumptions under which a given formula formula becomes a theorem. After applying our synthesis algorithm to cover basic abductive reasoning mechanisms, we synthesize conjunctions of literals that mimic rows of truth tables in classical or intermediate logics and we abduce conditional hypotheses that turn the theorems of classical or intermediate logics into theorems in intuitionistic logic. One step further, we generalize our abductive reasoning mechanism to synthesize more expressive sequent premises using a minimal set of canonical formulas, to which arbitrary formulas in the calculus can be reduced while preserving their provability. Organized as a self-contained literate Prolog program, the paper supports interactive exploration of its content and ensures full replicability of our results.

Temporal BI: Proof system, semantics and translations

Nonstandard Reasoning

General and effective methods are required for providing good automation strategies to prove properties of sequent systems. Structural properties such as admissibility, invertibility, and permutability of rules are crucial in proof theory, and they can be used for proving other key properties such as cut-elimination. However, finding proofs for these properties requires inductive reasoning over the provability relation, which is often quite elaborated, exponentially exhaustive, and error prone. This paper aims at developing automatic techniques for proving structural properties of sequent systems. The proposed techniques are presented in the rewriting logic metalogical framework, and use rewrite- and narrowing-based reasoning. They have been fully mechanized in Maude and achieve a great degree of automation when used on several sequent systems, including intuitionistic and classical logics, linear logic, and normal modal logics.

This paper introduces the enriched eect calculus , which extends established type theories for computational eects with primitives from linear logic. The new calculus provides a formalism for expressing linear aspects of computational eects; for example, the linear usage of imperative features such as state and/or continuations. The enriched eect calculus is implemented as an extension of a basic eect calculus without linear primitives, which is closely related to Moggi’s computational metalanguage, Filinski’s eect PCF and Levy’s call-by-push-value. We present syntactic results showing: the delity of the behaviour of the linear connectives of the enriched eect calculus; the conservativity of the enriched eect calculus over its non-linear core (the eect calculus); and the non-conservativity of intuitionistic linear logic when considered as an extension of the enriched eect calculus. The second half of the paper investigates models for the enriched eect calculus, based on enriched category theory. We give several examples of such models, relating them to models of standard eect calculi (such as those based on monads), and to models of intuitionistic linear logic. We also prove soundness and completeness.

Girard's linear logic [Gir87] is an extension of propositional logic with new connectives providing a logical treatment of resource control. As a first hint, consider the linear λ-terms, which are the λ-terms defined with the following restriction: when an abstraction λ x.M is formed, then x occurs exactly once in M . Linear λ-terms are normalized in linear time, that is, the number of reduction steps to their normal form is proportional to their size: a linear β-redex (λ x.M ) N involves no duplication of the argument N . Thus all the complexity of normalization comes from non-linearity. Linear logic pushes the limits of constructivity much beyond intuitionistic logic. A proper proof theoretical introduction to linear logic is beyond the scope of this book. In this chapter, we content ourselves with a semantic introduction. By doing so, we actually follow the historical thread: the connectives of linear logic arose from the consideration of (a particularly simple version of) the stable model. In section 13.1, we examine stable functions between coherence spaces, and discover two decompositions. First the function space E → E ′ is isomorphic to a space (! E )⊸ E ′, where ⊸ constructs the space of linear functions, and where! is a constructor which allows reuse of data. Intuitively, linear functions, like linear terms, can use their input only once. On the other hand, the explicit declaration of reusability,!, allows us to recover all functions and terms.