Sharing, freeness, linearity, redundancy, widenings, and cliques

We discuss here different variants of the Sharing abstract domain, including the base domain that captures set-sharing, a variant to capture pairsharing, in which redundant sharing groups (w.r.t. the pair-sharing property) can be eliminated, and an alternative representation based on cliques. The original proposal for using cliques in the non-redundant version of the domain is reviewed, then extended to the base domain. Variants of all the domains including freeness alone, and freeness together with linearity are also studied

A study of set-sharing analysis via cliques

We study the problem of efficient, scalable set-sharing analysis of logic programs. We use the idea of representing sharing information as a pair of abstract substitutions, one of which is a worst-case sharing representation called a clique set, which was previously proposed for the case of inferring pair-sharing. We use the clique-set representation for (1) inferring actual set-sharing information, and (2) analysis within a top-down framework. In particular, we define the abstract functions required by standard top-down analyses, both for sharing alone and also for the case of including freeness in addition to sharing. Our experimental evaluation supports the conclusion that, for inferring set-sharing, as it was the case for inferring pair-sharing, precision losses are limited, while useful efficiency gains are obtained. At the limit, the clique-set representation allowed analyzing some programs that exceeded memory capacity using classical sharing representations.Comment: 15 pages, 0 figure

A correct, precise and efficient integration of set-sharing, freeness and linearity for the analysis of finite and rational tree languages

It is well known that freeness and linearity information positively interact with aliasing information, allowing both the precision and the efficiency of the sharing analysis of logic programs to be improved. In this paper, we present a novel combination of set-sharing with freeness and linearity information, which is characterized by an improved abstract unification operator. We provide a new abstraction function and prove the correctness of the analysis for both the finite tree and the rational tree cases. Moreover, we show that the same notion of redundant information as identified in Bagnara et al. (2000) and Zaffanella et al. (2002) also applies to this abstract domain combination: this allows for the implementation of an abstract unification operator running in polynomial time and achieving the same precision on all the considered observable properties

Enhanced sharing analysis techniques: a comprehensive evaluation

Sharing, an abstract domain developed by D. Jacobs and A. Langen for the analysis of logic programs, derives useful aliasing information. It is well-known that a commonly used core of techniques, such as the integration of Sharing with freeness and linearity information, can significantly improve the precision of the analysis. However, a number of other proposals for refined domain combinations have been circulating for years. One feature that is common to these proposals is that they do not seem to have undergone a thorough experimental evaluation even with respect to the expected precision gains. In this paper we experimentally evaluate: helping Sharing with the definitely ground variables found using Pos, the domain of positive Boolean formulas; the incorporation of explicit structural information; a full implementation of the reduced product of Sharing and Pos; the issue of reordering the bindings in the computation of the abstract mgu; an original proposal for the addition of a new mode recording the set of variables that are deemed to be ground or free; a refined way of using linearity to improve the analysis; the recovery of hidden information in the combination of Sharing with freeness information. Finally, we discuss the issue of whether tracking compoundness allows the computation of more sharing information

Secret Sharing and Shared Information

Secret sharing is a cryptographic discipline in which the goal is to distribute information about a secret over a set of participants in such a way that only specific authorized combinations of participants together can reconstruct the secret. Thus, secret sharing schemes are systems of variables in which it is very clearly specified which subsets have information about the secret. As such, they provide perfect model systems for information decompositions. However, following this intuition too far leads to an information decomposition with negative partial information terms, which are difficult to interpret. One possible explanation is that the partial information lattice proposed by Williams and Beer is incomplete and has to be extended to incorporate terms corresponding to higher order redundancy. These results put bounds on information decompositions that follow the partial information framework, and they hint at where the partial information lattice needs to be improved.Comment: 9 pages, 1 figure. The material was presented at a Workshop on information decompositions at FIAS, Frankfurt, in 12/2016. The revision includes changes in the definition of combinations of secret sharing schemes. Section 3 and Appendix now discusses in how far existing measures satisfy the proposed properties. The concluding section is considerably revise

Three Optimisations for Sharing

In order to improve precision and efficiency sharing analysis should track both freeness and linearity. The abstract unification algorithms for these combined domains are suboptimal, hence there is scope for improving precision. This paper proposes three optimisations for tracing sharing in combination with freeness and linearity. A novel connection between equations and sharing abstractions is used to establish correctness of these optimisations even in the presence of rational trees. A method for pruning intermediate sharing abstractions to improve efficiency is also proposed. The optimisations are lightweight and therefore some, if not all, of these optimisations will be of interest to the implementor.Comment: To appear in Theiry and Practice of Logic Programmin

Algorithms for 3D rigidity analysis and a first order percolation transition

A fast computer algorithm, the pebble game, has been used successfully to study rigidity percolation on 2D elastic networks, as well as on a special class of 3D networks, the bond-bending networks. Application of the pebble game approach to general 3D networks has been hindered by the fact that the underlying mathematical theory is, strictly speaking, invalid in this case. We construct an approximate pebble game algorithm for general 3D networks, as well as a slower but exact algorithm, the relaxation algorithm, that we use for testing the new pebble game. Based on the results of these tests and additional considerations, we argue that in the particular case of randomly diluted central-force networks on BCC and FCC lattices, the pebble game is essentially exact. Using the pebble game, we observe an extremely sharp jump in the largest rigid cluster size in bond-diluted central-force networks in 3D, with the percolating cluster appearing and taking up most of the network after a single bond addition. This strongly suggests a first order rigidity percolation transition, which is in contrast to the second order transitions found previously for the 2D central-force and 3D bond-bending networks. While a first order rigidity transition has been observed for Bethe lattices and networks with ``chemical order'', this is the first time it has been seen for a regular randomly diluted network. In the case of site dilution, the transition is also first order for BCC, but results for FCC suggest a second order transition. Even in bond-diluted lattices, while the transition appears massively first order in the order parameter (the percolating cluster size), it is continuous in the elastic moduli. This, and the apparent non-universality, make this phase transition highly unusual.Comment: 28 pages, 19 figure

On the Optimality of Treating Inter-Cell Interference as Noise in Uplink Cellular Networks

In this paper, we explore the information-theoretic optimality of treating interference as noise (TIN) in cellular networks. We focus on uplink scenarios modeled by the Gaussian interfering multiple access channel (IMAC), comprising $K$ mutually interfering multiple access channels (MACs), each formed by an arbitrary number of transmitters communicating independent messages to one receiver. We define TIN for this setting as a scheme in which each MAC (or cell) performs a power-controlled version of its capacity-achieving strategy, with Gaussian codebooks and successive decoding, while treating interference from all other MACs (i.e. inter-cell interference) as noise. We characterize the generalized degrees-of-freedom (GDoF) region achieved through the proposed TIN scheme, and then identify conditions under which this achievable region is convex without the need for time-sharing. We then tighten these convexity conditions and identify a regime in which the proposed TIN scheme achieves the entire GDoF region of the IMAC and is within a constant gap of the entire capacity region.Comment: Accepted for publication in IEEE Transactions on Information Theor