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We study Brill-Noether existence on a finite graph using methods from polyhedral geometry and lattices. We start by formulating analogues of the Brill-Noether conjectures (both the existence and non-existence parts) for $\mathbb{R}$-divisors, i.e. divisors with real coefficients, on a graph. We then reformulate the Brill-Noether existence conjecture for $\mathbb{R}$-divisors on a graph in geometric terms, that we refer to as the covering radius conjecture and we show a weak version, in support of it. Using this, we show an approximate version of the Brill-Noether existence conjecture for divisors on a graph. As applications, we derive upper bounds on the gonality of a graph and its $\mathbb{R}$-divisor analogue., Comment: 34 Pages, substantial revisions based on the referee's comments, in Selecta Mathematica, New Series

We consider the decidability of the verification problem of programs \emph{modulo axioms} --- that is, verifying whether programs satisfy their assertions, when the functions and relations it uses are assumed to interpreted by arbitrary functions and relations that satisfy a set of first-order axioms. Unfortunately, verification of entirely uninterpreted programs (with the empty set of axioms) is already undecidable. A recent work introduced a subclass of \emph{coherent} uninterpreted programs, and showed that they admit decidable verification \cite{coherence2019}. We undertake a systematic study of various natural axioms for relations and functions, and study the decidability of the coherent verification problem. Axioms include relations being reflexive, symmetric, transitive, or total order relations, %and their combinations, functions restricted to being associative, idempotent or commutative, and combinations of such axioms as well. Our comprehensive results unearth a rich landscape that shows that though several axiom classes admit decidability for coherent programs, coherence is not a panacea as several others continue to be undecidable.

We identify a decidable synthesis problem for a class of programs of unbounded size with conditionals and iteration that work over infinite data domains. The programs in our class use uninterpreted functions and relations, and abide by a restriction called coherence that was recently identified to yield decidable verification. We formulate a powerful grammar-restricted (syntax-guided) synthesis problem for coherent uninterpreted programs, and we show the problem to be decidable, identify its precise complexity, and also study several variants of the problem.

The extended minimum of Solar Cycle 23, the extremely quiet solar-wind conditions prevailing, and the mini-maximum of Solar Cycle 24 drew global attention and many authors have since attempted to predict the amplitude of the upcoming Solar Cycle 25, which is predicted to be the third successive weak cycle; it is a unique opportunity to probe the Sun during such quiet periods. Earlier work has established a steady decline, over two decades, in solar photospheric fields at latitudes above $45^{\circ}$ and a similar decline in solar-wind micro-turbulence levels as measured by interplanetary scintillation (IPS) observations. However, the relation between the photospheric magnetic fields and those in the low corona/solar-wind are not straightforward. Therefore, in the present article, we have used potential-field source-surface (PFSS) extrapolations to deduce global magnetic-fields using synoptic magnetograms observed with National Solar Observatory (NSO), Kitt Peak, USA (NSO/KP) and Solar Optical Long-term Investigation of the Sun (NSO/SOLIS) instruments during 1975-2018. Furthermore, we have measured the normalized scintillation index [m] using the IPS observations carried out at the Institute of Space Earth Environment Research (ISEE), Japan during 1983-2017. From these observations, we have found that, since the mid-1990s, the magnetic-field over different latitudes at 2.5 $\rm R_{\odot}$ and 10 $\rm R_{\odot}$(extrapolated using PFSS method) has decreased by $\approx 11.3-22.2 \%$. In phase with the declining magnetic-fields, the quantity m also declined by $\approx 23.6 \%$. These observations emphasize the inter-relationship between the global magnetic-field and various turbulence parameters in the solar corona and solar wind., Comment: Accepted for publication in Solar Physics, 16 pages, 5 Figures, 1 Table

We consider the problem of combining machine learning models to perform higher-level cognitive tasks with clear specifications. We propose the novel problem of Visual Discrimination Puzzles (VDP) that requires finding interpretable discriminators that classify images according to a logical specification. Humans can solve these puzzles with ease and they give robust, verifiable, and interpretable discriminators as answers. We propose a compositional neurosymbolic framework that combines a neural network to detect objects and relationships with a symbolic learner that finds interpretable discriminators. We create large classes of VDP datasets involving natural and artificial images and show that our neurosymbolic framework performs favorably compared to several purely neural approaches., Comment: Published at IJCAI 2022

We investigate the decidability of automatic program verification for programs that manipulate heaps, and in particular, decision procedures for proving memory safety for them. We extend recent work that identified a decidable subclass of uninterpreted programs to a class of alias-aware programs that can update maps. We apply this theory to develop verification algorithms for memory safety--- determining if a heap-manipulating program that allocates and frees memory locations and manipulates heap pointers does not dereference an unallocated memory location. We show that this problem is decidable when the initial allocated heap forms a forest data-structure and when programs are streaming-coherent, which intuitively restricts programs to make a single pass over a data-structure. Our experimental evaluation on a set of library routines that manipulate forest data-structures shows that common single-pass algorithms on data-structures often fall in the decidable class, and that our decision procedure is efficient in verifying them., Comment: StreamVerif tool for automata-based verification of uninterpreted programs can be found at https://github.com/umangm/streamverif

We associate two modules, the $G$-parking critical module and the toppling critical module, to an undirected connected graph $G$. The $G$-parking critical module and the toppling critical module are canonical modules (with suitable twists) of quotient rings of the well-studied $G$-parking function ideal and the toppling ideal, respectively. For each critical module, we establish a Tutte-like short exact sequence relating the modules associated to $G$, an edge contraction $G/e$ and an edge deletion $G \setminus e$ ($e$ is a non-bridge). We obtain purely combinatorial consequences of Tutte short exact sequences. For instance, we reprove a theorem of Merino that the critical polynomial of a graph is an evaluation of its Tutte polynomial, and relate the vanishing of certain combinatorial invariants (the number of acyclic orientations on connected partition graphs satisfying a unique sink property) of $G/e$ to the equality of the corresponding invariants of $G$ and $G \setminus e$., Comment: Journal version, 39 pages, 3 figures

Travelling Salesman Problem (TSP) is one of the unsolved problems in computer science. TSP is NP Hard. Till now the best approximation ratio found for symmetric TSP is three by two by Christofides Algorithm more than forty years ago. There are different approaches to solve this problem. These range from methods based on neural networks, genetic algorithm, swarm optimization, ant colony optimization etc. The bound is further reduced from three by two but for graphic TSP. A factor of thirteen by nine was found for Graphic TSP. A newly proposed heuristic called 2 RNN is considered here. It seems from experimental results that five by four is the approximation ratio. Upper bound analysis for approximation ratio is done for this heuristic and it confirms experimental bound of five by four., Comment: 8 pages, 1 figure, 1 table

Proteins are the workhorses of life and gaining insight on their functions is of paramount importance for applications such as drug design. However, the experimental validation of functions of proteins is highly-resource consuming. Therefore, recently, automated protein function prediction (AFP) using machine learning has gained significant interest. Many of these AFP tools are based on supervised learning models trained using existing gold-standard functional annotations, which are known to be incomplete. The main challenge associated with conducting systematic testing on AFP software is the lack of a test oracle, which determines passing or failing of a test case; unfortunately, due to the incompleteness of gold-standard data, the exact expected outcomes are not well defined for the AFP task. Thus, AFP tools face the \emph{oracle problem}. In this work, we use metamorphic testing (MT) to test nine state-of-the-art AFP tools by defining a set of metamorphic relations (MRs) that apply input transformations to protein sequences. According to our results, we observe that several AFP tools fail all the test cases causing concerns over the quality of their predictions., Comment: 8 pages, First IEEE International Conference On Artificial Intelligence Testing

We propose a novel logic, called Frame Logic (FL), that extends first-order logic (with recursive definitions) using a construct Sp(.) that captures the implicit supports of formulas -- the precise subset of the universe upon which their meaning depends. Using such supports, we formulate proof rules that facilitate frame reasoning elegantly when the underlying model undergoes change. We show that the logic is expressive by capturing several data-structures and also exhibit a translation from a precise fragment of separation logic to frame logic. Finally, we design a program logic based on frame logic for reasoning with programs that dynamically update heaps that facilitates local specifications and frame reasoning. This program logic consists of both localized proof rules as well as rules that derive the weakest tightest preconditions in FL., Comment: This manuscript is an extended and revised version of the publication with the same title that appeared at ESOP 2022 (https://doi.org/10.1007/978-3-030-44914-8_19). It is currently under review

Knowing the lengthscales at which turbulent fluctuations dissipate is key to understanding the nature of weakly compressible magnetohydrodynamic turbulence. We use radio wavelength interferometric imaging observations which measure the extent to which distant cosmic sources observed against the inner solar wind are scatter-broadened. We interpret these observations to determine that the dissipation scales of solar wind density turbulence at heliocentric distances of 2.5 -- 20.27 $R_{\odot}$ range from $\approx$ 13500 to 520 m. Our estimates from $\approx$ 10--20 $R_{\odot}$ suggest that the dissipation scale corresponds to the proton gyroradius. They are relevant to in-situ observations to be made by the Parker Solar Probe, and are expected to enhance our understanding of solar wind acceleration., Comment: Accepted for publication in The Astrophysical Journal (8 pages, 4 figures, 1 table)

We study the problem of completely automatically verifying uninterpreted programs---programs that work over arbitrary data models that provide an interpretation for the constants, functions and relations the program uses. The verification problem asks whether a given program satisfies a postcondition written using quantifier-free formulas with equality on the final state, with no loop invariants, contracts, etc. being provided. We show that this problem is undecidable in general. The main contribution of this paper is a subclass of programs, called coherent programs that admits decidable verification, and can be decided in PSPACE. We then extend this class of programs to classes of programs that are $k$-coherent, where $k \in \mathbb{N}$, obtained by (automatically) adding $k$ ghost variables and assignments that make them coherent. We also extend the decidability result to programs with recursive function calls and prove several undecidability results that show why our restrictions to obtain decidability seem necessary.

Bioinformatics software plays a very important role in making critical decisions within many areas including medicine and health care. However, most of the research is directed towards developing tools, and little time and effort is spent on testing the software to assure its quality. In testing, a test oracle is used to determine whether a test is passed or failed during testing, and unfortunately, for much of bioinformatics software, the exact expected outcomes are not well defined. Thus, the main challenge associated with conducting systematic testing on bioinformatics software is the oracle problem. Metamorphic testing (MT) is a technique used to test programs that face the oracle problem. MT uses metamorphic relations (MRs) to determine whether a test has passed or failed and specifies how the output should change according to a specific change made to the input. In this work, we use MT to test LingPipe, a tool for processing text using computational linguistics, often used in bioinformatics for bio-entity recognition from biomedical literature. First, we identify a set of MRs for testing any bio-entity recognition program. Then we develop a set of test cases that can be used to test LingPipe's bio-entity recognition functionality using these MRs. To evaluate the effectiveness of this testing process, we automatically generate a set of faulty versions of LingPipe. According to our analysis of the experimental results, we observe that our MRs can detect the majority of these faulty versions, which shows the utility of this testing technique for quality assurance of bioinformatics software.

Production distributed systems are challenging to formally verify, in particular when they are based on distributed protocols that are not rigorously described or fully understood. In this paper, we derive models and properties for two core distributed protocols used in eventually consistent production key-value stores such as Riak and Cassandra. We propose a novel modeling called certified program models, where complete distributed systems are captured as programs written in traditional systems languages such as concurrent C. Specifically, we model the read-repair and hinted-handoff recovery protocols as concurrent C programs, test them for conformance with real systems, and then verify that they guarantee eventual consistency, modeling precisely the specification as well as the failure assumptions under which the results hold., Comment: 15 pages, 2 figures

We design learning algorithms for synthesizing invariants using Horn implication counterexamples (Horn-ICE), extending the ICE-learning model. In particular, we describe a decision-tree learning algorithm that learns from Horn-ICE samples, works in polynomial time, and uses statistical heuristics to learn small trees that satisfy the samples. Since most verification proofs can be modeled using Horn clauses, Horn-ICE learning is a more robust technique to learn inductive annotations that prove programs correct. Our experiments show that an implementation of our algorithm is able to learn adequate inductive invariants and contracts efficiently for a variety of sequential and concurrent programs.

We propose a framework for synthesizing inductive invariants for incomplete verification engines, which soundly reduce logical problems in undecidable theories to decidable theories. Our framework is based on the counter-example guided inductive synthesis principle (CEGIS) and allows verification engines to communicate non-provability information to guide invariant synthesis. We show precisely how the verification engine can compute such non-provability information and how to build effective learning algorithms when invariants are expressed as Boolean combinations of a fixed set of predicates. Moreover, we evaluate our framework in two verification settings, one in which verification engines need to handle quantified formulas and one in which verification engines have to reason about heap properties expressed in an expressive but undecidable separation logic. Our experiments show that our invariant synthesis framework based on non-provability information can both effectively synthesize inductive invariants and adequately strengthen contracts across a large suite of programs.

We propose a fragment of many-sorted second order logic called EQSMT and show that checking satisfiability of sentences in this fragment is decidable. EQSMT formulae have an $\exists^*\forall^*$ quantifier prefix (over variables, functions and relations) making EQSMT conducive for modeling synthesis problems. Moreover, EQSMT allows reasoning using a combination of background theories provided that they have a decidable satisfiability problem for the $\exists^*\forall^*$ FO-fragment (e.g., linear arithmetic). Our decision procedure reduces the satisfiability of EQSMT formulae to satisfiability queries of $\exists^*\forall^*$ formulae of each individual background theory, allowing us to use existing efficient SMT solvers supporting $\exists^*\forall^*$ reasoning for these theories; hence our procedure can be seen as effectively quantified SMT (EQSMT) reasoning. Errata: We have modified the transformation step-2 (page 9) to correct for a slight error. Also, the description above Theorem 10 is different from the published version.