### Working paper

## Linear logic with idempotent exponential modalities: a note

In this article, we investigate the logical structure of memory models of theoretical and practical interest. Our main interest is in “the logic behind a fixed memory model”, rather than in “a model of any kind behind a given logical system”. As an effective language for reasoning about such memory models, we use the formalism of separation logic. Our main result is that for any concrete choice of heap-like memory model, validity in that model is *undecidable* even for purely propositional formulas in this language.

The main novelty of our approach to the problem is that we focus on validity in specific, concrete memory models, as opposed to validity in general classes of models.

Besides its intrinsic technical interest, this result also provides new insights into the nature of their decidable fragments. In particular, we show that, in order to obtain such decidable fragments, either the formula language must be severely restricted or the valuations of propositional variables must be constrained.

In addition, we show that a number of propositional systems that approximate separation logic are undecidable as well. In particular, this resolves the open problems of decidability for Boolean BI and Classical BI.

Moreover, we provide one of the simplest undecidable propositional systems currently known in the literature, called “Minimal Boolean BI”, by combining the purely positive implication-conjunction fragment of Boolean logic with the laws of multiplicative *-conjunction, its unit and its adjoint implication, originally provided by intuitionistic multiplicative linear logic. Each of these two components is individually decidable: the implication-conjunction fragment of Boolean logic is co-NP-complete, and intuitionistic multiplicative linear logic is NP-complete.

All of our undecidability results are obtained by means of a direct encoding of Minsky machines.

Key Words and Phrases: Separation logic, undecidability, memory models, bunched logic

This volume contains the proceedings of the Joint Meeting of the Twenty-Third Annual EACSL Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/ IEEE Symposium on Logic in Computer Science (LICS). CSL is the annual meeting of the European Association for Computer Science Logic (EACSL) intended for computer scientists whose research activities involve logic, as well as for logicians working on issues significant for computer science. LICS is an annual international forum on theoretical and practical topics in computer science that relate to logic. Every 3--4 years, LICS has been part of the Federated Logic Conference (FLoC). Given that FLoC was to be held as part of the Vienna Summer of Logic (VSL) during July 2014, the organizers of CSL and LICS have chosen to merge the 2014 editions of these meetings into a single event within FLoC and VSL. Thus, in 2014, the joint meeting had one program committee, one program, and one proceedings.

We investigate the satisfiability problem for Horn fragments of the Halpern-Shoham interval temporal logic depending on the type (box or diamond) of the interval modal operators, the type of the underlying linear order (discrete or dense), and the type of semantics for the interval relations (reflexive or irreflexive). For example, we show that satisfiability of Horn formulas with diamonds is undecidable for any type of linear orders and semantics. On the contrary, satisfiability of Horn formulas with boxes is tractable over both discrete and dense orders under the reflexive semantics and over dense orders under the irreflexive semantics but becomes undecidable over discrete orders under the irreflexive semantics. Satisfiability of binary Horn formulas with both boxes and diamonds is always undecidable under the irreflexive semantics.

It is well-known that the Dolev–Yao adversary is a powerful adversary. Besides acting as the network, intercepting, decomposing, composing and sending messages, he can remember as much information as he needs. That is, his memory is unbounded. We recently proposed a weaker Dolev–Yao like adversary, which also acts as the network, but whose memory is bounded. We showed that this Bounded Memory Dolev–Yao adversary, when given enough memory, can carry out many existing protocol anomalies. In particular, the known anomalies arise for bounded memory protocols, where although the total number of sessions is unbounded, there are only a bounded number of concurrent sessions and the honest participants of the protocol cannot remember an unbounded number of facts or an unbounded number of nonces at a time. This led us to the question of whether it is possible to infer an upper-bound on the memory required by the Dolev–Yao adversary to carry out an anomaly from the memory restrictions of the bounded protocol. This paper answers this question negatively (Theorem 8).

The Markov School is the school of mathematics (mostly mathematical logic) arisen in St.Petersburg in the 20th century. Historically, the Markov School has been largely connected with the Steklov Mathematical Institute in St. Petersburg. Today, many Markov School mathematicians have taken up positions in different countries; of course, we do not attempt to restrict this survey only to those who have stayed in St. Petersburg. However, all mathematicians whose work we describe studied in St. Petersburg and, at least for several years and often for decades, worked at the Steklov Mathematical Institute. In this paper, we primarily discuss what has happened over the last two decades, so a certain bias to computational complexity is to be expected.

This volume contains the proceedings of the 17th International Conference on the Foundations of Software Science and Computation Structures, FOSSACS 2014, held in Grenoble, France, 5–13 April 2014. FOSSACS is an event of the Joint European Conferences on Theory and Practice of Software (ETAPS).

FOSSACS presents original papers on the foundations of software science. The conference invited submissions on theories and methods to support analysis, synthesis, transformation, and verification of programs and software systems.

We received 128 abstracts and 106 full paper s ubmissions; of these, 28 were selected for presentation at FOSSACS and inclusion in the proceedings.

The PC members, and the external experts they consulted, wrote over 320 paper reviews, and the discussion phase of the meeting included a 3-day author rebuttal phase. The competition was very strong, and unfortunately many good papers could not be accepted.

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 article deals with the problem of translations. It covers the history of translation in linguistics and analyzes peculiarities and role of translation in logic. Moreover, the article contains typical examples of embedding operations in terms of dierent logical theories.

A model for organizing cargo transportation between two node stations connected by a railway line which contains a certain number of intermediate stations is considered. The movement of cargo is in one direction. Such a situation may occur, for example, if one of the node stations is located in a region which produce raw material for manufacturing industry located in another region, and there is another node station. The organization of freight traﬃc is performed by means of a number of technologies. These technologies determine the rules for taking on cargo at the initial node station, the rules of interaction between neighboring stations, as well as the rule of distribution of cargo to the ﬁnal node stations. The process of cargo transportation is followed by the set rule of control. For such a model, one must determine possible modes of cargo transportation and describe their properties. This model is described by a ﬁnite-dimensional system of diﬀerential equations with nonlocal linear restrictions. The class of the solution satisfying nonlocal linear restrictions is extremely narrow. It results in the need for the “correct” extension of solutions of a system of diﬀerential equations to a class of quasi-solutions having the distinctive feature of gaps in a countable number of points. It was possible numerically using the Runge–Kutta method of the fourth order to build these quasi-solutions and determine their rate of growth. Let us note that in the technical plan the main complexity consisted in obtaining quasi-solutions satisfying the nonlocal linear restrictions. Furthermore, we investigated the dependence of quasi-solutions and, in particular, sizes of gaps (jumps) of solutions on a number of parameters of the model characterizing a rule of control, technologies for transportation of cargo and intensity of giving of cargo on a node station.

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.