### Working paper

## On noncommutative extensions of linear logic

This paper proposes a definition of categorical model of the deep inference system BV, defined by Guglielmi. Deep inference introduces the idea of performing a deduction in the interior of a formula, at any depth. Traditional sequent calculus rules only see the roots of formulae. However in these new systems, one can rewrite at any position in the formula tree. Deep inference in particular allows the syntactic description of logics for which there is no sequent calculus. One such system is BV, which extends linear logic to include a noncommutative self-dual connective. This is the logic our paper proposes to model. Our definition is based on the notion of a *linear functor*, due to Cockett and Seely. A BV-*category* is a linearly distributive category, possibly with negation, with an additional tensor product which, when viewed as a bivariant functor, is linear with a degeneracy condition. We show that this simple definition implies all of the key isomorphisms of the theory. We consider Girard’s category of *probabilistic coherence spaces* and show that it contains a self-dual monoidal structure in addition to the *-autonomous structure exhibited by Girard. This structure makes the category a BV-category. We believe this structure is also of independent interest, as well-behaved noncommutative operators generally are.

In their seminal paper:

Lincoln, P., Mitchell, J., Scedrov, A. and Shankar, N. (1992). Decision problems for propositional linear logic. Annals of Pure and Applied Logic 56 (1–3) 239–311,

LMSS have established an extremely surprising result that propositional linear logic is undecidable. Their proof is very complex and involves numerous nested inductions of different kinds. Later an alternative proof for the LL undecidability has been developed based on simulation Minsky machines in linear logic: Kanovich, M. (1995). The direct simulation of Minsky machines in linear logic. In: Girard, J.-Y., Lafont, Y. and Regnier, L. (eds.) Advances in Linear Logic, London Mathematical Society Lecture Notes, volume 222, Cambridge University Press 123–145. Notice that this direct simulation approach has been successfully applied for a large number of formal systems with resolving a number of open problems in computer science and even computational linguistics, e.g.,

James Brotherston, Max I. Kanovich: Undecidability of Propositional Separation Logic and Its Neighbours. J. ACM 61(2): 14:1-14:43 (2014), Max Kanovich, Stepan Kuznetsov, Andre Scedrov: Undecidability of the Lambek Calculus with a Relevant Modality. FG 2016: 240-256. Nevertheless, recently the undecidability of linear logic is questioned by some people. They claim that they have found lacunae in the LMSS 1992 paper, and, moreover, they have a proof that propositional linear logic is decidable!!! I have been asked to submit a paper, as clear as possible, to the Journal, in order to sort out such a confusing problem, once and for all.

Here, we give a fully self-contained, easy-to-follow, but fully detailed, direct and constructive proof of the undecidability of a very simple Horn-like fragment of linear logic, the proof is accessible to a wide range of people. Namely, we show that there is a direct correspondence between terminated computations of a Minsky machine M and cut-free linear logic derivations for a Horn-like sequent of the form \Phi_M, l1 |- l0 where \Phi_M consists only of Horn-like implications of the very simple forms. Neither negation, nor &, nor constants, nor embedded implications/bangs are used here. Furthermore, our particular correspondence constructed above provides decidability for some smaller Horn-like fragments along with the complexity bounds that come from the proof.

This paper proposes a definition of categorical model of the deep inference system BV, defined by Guglielmi. Deep inference introduces the idea of performing a deduction in the interior of a formula, at any depth. Traditional sequent calculus rules only see the roots of formulae. However in these new systems, one can rewrite at any position in the formula tree. Deep inference in particular allows the syntactic description of logics for which there is no sequent calculus. One such system is BV, which extends linear logic to include a noncommutative self-dual connective. This is the logic our paper proposes to model. Our definition is based on the notion of a *linear functor*, due to Cockett and Seely. A BV-*category* is a linearly distributive category, possibly with negation, with an additional tensor product which, when viewed as a bivariant functor, is linear with a degeneracy condition. We show that this simple definition implies all of the key isomorphisms of the theory. We consider Girard’s category of *probabilistic coherence spaces* and show that it contains a self-dual monoidal structure in addition to the *-autonomous structure exhibited by Girard. This structure makes the category a BV-category. We believe this structure is also of independent interest, as well-behaved noncommutative operators generally are.

The Lambek calculus can be considered as a version of non-commutative intuitionistic linear logic. One of the interesting features of the Lambek calculus is the so-called “Lambek’s restriction,” that is, the antecedent of any provable sequent should be non-empty. In this paper we discuss ways of extending the Lambek calculus with the linear logic exponential modality while keeping Lambek’s restriction. We present several versions of the Lambek calculus extended with exponential modalities and prove that those extensions are undecidable, even if we take only one of the two divisions provided by the Lambek calculus.

Let k be a field of characteristic zero, let G be a connected reductive algebraic group over k and let g be its Lie algebra. Let k(G), respectively, k(g), be the field of k- rational functions on G, respectively, g. The conjugation action of G on itself induces the adjoint action of G on g. We investigate the question whether or not the field extensions k(G)/k(G)^G and k(g)/k(g)^G are purely transcendental. We show that the answer is the same for k(G)/k(G)^G and k(g)/k(g)^G, and reduce the problem to the case where G is simple. For simple groups we show that the answer is positive if G is split of type A_n or C_n, and negative for groups of other types, except possibly G_2. A key ingredient in the proof of the negative result is a recent formula for the unramified Brauer group of a homogeneous space with connected stabilizers. As a byproduct of our investigation we give an affirmative answer to a question of Grothendieck about the existence of a rational section of the categorical quotient morphism for the conjugating action of G on itself.

Let G be a connected semisimple algebraic group over an algebraically closed field k. In 1965 Steinberg proved that if G is simply connected, then in G there exists a closed irreducible cross-section of the set of closures of regular conjugacy classes. We prove that in arbitrary G such a cross-section exists if and only if the universal covering isogeny Ĝ → G is bijective; this answers Grothendieck's question cited in the epigraph. In particular, for char k = 0, the converse to Steinberg's theorem holds. The existence of a cross-section in G implies, at least for char k = 0, that the algebra k[G]G of class functions on G is generated by rk G elements. We describe, for arbitrary G, a minimal generating set of k[G]G and that of the representation ring of G and answer two Grothendieck's questions on constructing generating sets of k[G]G. We prove the existence of a rational (i.e., local) section of the quotient morphism for arbitrary G and the existence of a rational cross-section in G (for char k = 0, this has been proved earlier); this answers the other question cited in the epigraph. We also prove that the existence of a rational section is equivalent to the existence of a rational W-equivariant map T- - - >G/T where T is a maximal torus of G and W the Weyl group.

This proceedings publication is a compilation of selected contributions from the “Third International Conference on the Dynamics of Information Systems” which took place at the University of Florida, Gainesville, February 16–18, 2011. The purpose of this conference was to bring together scientists and engineers from industry, government, and academia in order to exchange new discoveries and results in a broad range of topics relevant to the theory and practice of dynamics of information systems. Dynamics of Information Systems: Mathematical Foundation presents state-of-the art research and is intended for graduate students and researchers interested in some of the most recent discoveries in information theory and dynamical systems. Scientists in other disciplines may also benefit from the applications of new developments to their own area of study.