A Labelled Sequent Calculus for BBI: Proof Theory and Proof Search

We present a labelled sequent calculus for Boolean BI, a classical variant of O'Hearn and Pym's logic of Bunched Implication. The calculus is simple, sound, complete, and enjoys cut-elimination. We show that all the structural rules in our proof system, including those rules that manipulate labels, can be localised around applications of certain logical rules, thereby localising the handling of these rules in proof search. Based on this, we demonstrate a free variable calculus that deals with the structural rules lazily in a constraint system. A heuristic method to solve the constraints is proposed in the end, with some experimental results

Syntactic Interpolation for Tense Logics and Bi-Intuitionistic Logic via Nested Sequents

We provide a direct method for proving Craig interpolation for a range of modal and intuitionistic logics, including those containing a "converse" modality. We demonstrate this method for classical tense logic, its extensions with path axioms, and for bi-intuitionistic logic. These logics do not have straightforward formalisations in the traditional Gentzen-style sequent calculus, but have all been shown to have cut-free nested sequent calculi. The proof of the interpolation theorem uses these calculi and is purely syntactic, without resorting to embeddings, semantic arguments, or interpreted connectives external to the underlying logical language. A novel feature of our proof includes an orthogonality condition for defining duality between interpolants

Proving the monotonicity criterion for a plurality vote-counting program as a step towards verified vote-counting

We show how modern interactive verification tools can be used to prove complex properties of vote-counting software. Specifically, we give an ML implementation of a votecounting program for plurality voting; we give an encoding of this program into the higher-order logic of the HOL4 theorem prover; we give an encoding of the monotonicity property in the same higher-order logic; we then show how we proved that the encoding of the program satisfies the encoding of the monotonicity property using the interactive theorem prover HOL4. As an aside, we also show how to prove the correctness of the vote-counting program. We then discuss the robustness of our approach

Well-Founded Unions

Given two or more well-founded (terminating) binary relations, when can one be sure that their union is likewise well-founded? We suggest new conditions for an arbitrary number of relations, generalising known conditions for two relations. We also provide counterexamples to several potential weakenings. All proofs have been machine checked.J. Dawson—Supported by Australian Research Council Discovery Project DP140101540

On the correspondence between display postulates and deep inference in nested sequent calculi for tense logics

We consider two styles of proof calculi for a family of tense logics, presented in a formalism based on nested sequents. A nested sequent can be seen as a tree of traditional single-sided sequents. Our first style of calculi is what we call "shallow calculi", where inference rules are only applied at the root node in a nested sequent. Our shallow calculi are extensions of Kashima's calculus for tense logic and share an essential characteristic with display calculi, namely, the presence of structural rules called "display postulates". Shallow calculi enjoy a simple cut elimination procedure, but are unsuitable for proof search due to the presence of display postulates and other structural rules. The second style of calculi uses deep-inference, whereby inference rules can be applied at any node in a nested sequent. We show that, for a range of extensions of tense logic, the two styles of calculi are equivalent, and there is a natural proof theoretic correspondence between display postulates and deep inference. The deep inference calculi enjoy the subformula property and have no display postulates or other structural rules, making them a better framework for proof search

An Introduction to Voting Rule Verification

We give an introduction to deductive verification methods that can be used to formally prove that voting rules and their implementations satisfy specified properties and conform to the desired democratic principles. In the first part of the paper we explain the basic principles: We describe how first-order logic with theories can be used to formalise the desired properties. We explain the difference between (1) proving that one set of properties implies another property, (2) proving that a voting rule implementation has a certain property, and (3) proving that a voting rule implementation is a refinement of an executable specification. And we explain the different technologies: (1) SMT-based testing, (2) bounded program verification, (3) relational program verification, and (4) symmetry breaking. In this first part of the paper, we also explain the difference between verifying functional and relational properties (such as symmetries). In the second part, we present case studies, including (1) the specification and verification of semantic properties for an STV rule used for electing the board of trustees for a major international conference and (2) the deduction-based computation of election margins for the Danish national parliamentary elections

CancerNet: a unified deep learning network for pan‑cancer diagnostics

Article states that despite remarkable advances in cancer research, cancer remains one of the leading causes of death worldwide. The author's proposed framework for cancer diagnostics detects cancers and their tissues of origin using a unified model of cancers encompassing 33 cancers represented in The Cancer Genome Atlas. Their model exploits the learned features of different cancers reflected in the respective dysregulated epigenomes, holding a great promise in early cancer detection