Deep Proof Search in MELL

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.

Proof nets, garbage, and computations

Constant-only multiplicative linear logic is NP-complete

Each Multiplicative Exponential Linear Logic (MELL) proof-net can be expanded into a differential net, which is its Taylor expansion. We prove that two different MELL proof-nets have two different Taylor expansions. As a corollary, we prove a completeness result for MELL: We show that the relational model is injective for MELL proof-nets, i.e. the equality between MELL proof-nets in the relational model is exactly axiomatized by cut-elimination.

Recently, linear Logic has been used to specify sequent calculus proof systems in such a way that the proof search in linear logic can yield proof search in the specified logic. Furthermore, the meta-theory of linear logic can be used to draw conclusions about the specified sequent calculus. For example, derivability of one proof system from another can be decided by a simple procedure that is implemented via bounded logic programming-style search. Also, simple and decidable conditions on the linear logic presentation of inference rules, called homogeneous and coherence, can be used to infer that the initial rules can be restricted to atoms and that cuts can be eliminated. In the present paper we introduce Llinda, a logical framework based on linear logic augmented with inference rules for definition (fixed points) and induction. In this way, the above properties can be proved entirely inside the framework. To further illustrate the power of Llinda, we extend the definition of coherence and provide a new, semi-automated proof of cut-elimination for Girard’s Logic of Unicity (LU).

Temporal BI: Proof system, semantics and translations

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.

Subexponential logic is a variant of linear logic with a family of exponential connectives – called subexponentials – that are indexed and arranged in a pre-order. Each subexponential has or lacks associated structural properties of weakening and contraction. We show that a classical propositional multiplicative subexponential logic (MSEL) with one unrestricted and two linear subexponentials can encode the halting problem for two register Minsky machines, and is hence undecidable. We then show how the additive connectives can be directly simulated by giving an encoding of propositional multiplicative additive linear logic (MALL) in an MSEL with one unrestricted and four linear subexponentials.

We give a general proof-theoretic method for proving Craig interpolation for displayable logics, based on an analysis of the individual proof rules of their display calculi. Using this uniform method, we prove interpolation for a spectrum of display calculi differing in their structural rules, including those for multiplicative linear logic, multiplicative additive linear logic and ordinary classical logic. Our analysis of proof rules also provides new insights into why interpolation fails, or seems likely to fail, in many substructural logics. Specifically, contraction appears particularly problematic for interpolation except in special circumstances.

Permutative logic (PL) is a noncommutative variant of multiplicative linear logic (MLL) arising from recent investigations concerning the topology of linear proofs. Permutative sequents are structured as oriented surfaces with boundary whose topological complexity is able to encode some information about the exchange in sequential proofs. In this paper we provide a complete permutative sequent calculus by extending that one of PL with rules for additives and exponentials. This extended system, here called permutative linear logic (PLL), is shown to be a conservative extension of linear logic and able to enjoy cut-elimination. Moreover, some basic isomorphisms are pointed out.

Probably the most significant result concerning cut-free sequent calculus proofs in linear logic is the completeness of focused proofs. This completeness theorem has a number of proof theoretic applications — e.g. in game semantics, Ludics, and proof search — and more computer science applications — e.g. logic programming, call-by-name/value evaluation. Andreoli proved this theorem for first-order linear logic 15 years ago. In the present paper, we give a new proof of the completeness of focused proofs in terms of proof transformation. The proof of this theorem is simple and modular: it is first proved for MALL and then is extended to full linear logic. Given its modular structure, we show how the proof can be extended to larger systems, such as logics with induction. Our analysis of focused proofs will employ a proof transformation method that leads us to study how focusing and cut elimination interact. A key component of our proof is the construction of a focalization graph which provides an abstraction over how focusing can be organized within a given cut-free proof. Using this graph abstraction allows us to provide a detailed study of atomic bias assignment in a way more refined that is given in Andreoli’s original proof. Permitting more flexible assignment of bias will allow this completeness theorem to help establish the completeness of a number of other automated deduction procedures. Focalization graphs can be used to justify the introduction of an inference rule for multifocus derivation: a rule that should help us better understand the relations between sequentiality and concurrency in linear logic.

We give an event structures based true-concurrency characterization of deep inference proofs. The method is general to all deep inference systems that can be expressed as term rewriting systems. This delivers three consequences in a spectrum from theoretical to practical: the event structure characterization i provides a qualification of proof identity akini¾?to proof nets for multiplicative linear logic and to atomic flows for classical logic; ii provides a concurrency theoretic interpretation for applications in logic programming; iii reduces the length of the proofs, and thereby extends the margin of proof search applications.

We propose a notion of symmetric reduction for a system of proof-nets for Multiplicative Affine Logic with Mix (MAL + Mix) (namely, multiplicative linear logic with the mix-rule the unrestricted weakening-rule). We prove that such a reduction has the strong normalization and Church-Rosser properties. A notion of irrelevance in a proof-net is defined and the possibility of cancelling the irrelevant parts of a proof-net without erasing the entire net is taken as one of the correctness conditions; therefore purely local cut-reductions are given, minimizing cancellation and suggesting a paradigm of computation without garbage collection. Reconsidering Ketonen and Weyhranch's decision procedure for affine logic [15, 4], the use of the mix-rule is related to the non-determinism of classical proof-theory. The question arises, whether these features of classical cut-elimination are really irreducible to the familiar paradigm of cut-elimination for intuitionistic and linear logic.

We describe in full detail a solution to the problem of proving the cut elimination theorem for FILL, a variant of (multiplicative and exponential-free) Linear Logic<br />introduced by Hyland and de Paiva. Hyland and de Paiva's work used a term assignment<br />system to describe FILL and barely sketched the proof of cut elimination. In this paper, as well as correcting a small mistake in their paper and extending the<br />system to deal with exponentials, we introduce a different formal system describing the intuitionistic character of FILL and we provide a full proof of the cut elimination<br />theorem. The formal system is based on a notion of dependence between formulae within a given proof and seems of independent interest. The procedure for<br />cut elimination applies to (classical) multiplicative Linear Logic, and we can (with care) restrict our attention to the subsystem FILL. The proof, as usual with cut<br />elimination proofs, is a little involved and we have not seen it published anywhere.

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.