Inducing syntactic cut-elimination for indexed nested sequents

The key to the proof-theoretic study of a logic is a proof calculus with a subformula property. Many different proof formalisms have been introduced (e.g. sequent, nested sequent, labelled sequent formalisms) in order to provide such calculi for the many logics of interest. The nested sequent formalism was recently generalised to indexed nested sequents in order to yield proof calculi with the subformula property for extensions of the modal logic K by (Lemmon-Scott) Geach axioms. The proofs of completeness and cut-elimination therein were semantic and intricate. Here we show that derivations in the labelled sequent formalism whose sequents are `almost treelike' correspond exactly to indexed nested sequents. This correspondence is exploited to induce syntactic proofs for indexed nested sequent calculi making use of the elegant proofs that exist for the labelled sequent calculi. A larger goal of this work is to demonstrate how specialising existing proof-theoretic transformations alleviate the need for independent proofs in each formalism. Such coercion can also be used to induce new cutfree calculi. We employ this to present the first indexed nested sequent calculi for intermediate logics.Comment: This is an extended version of the conference paper [20

The kinetics, mechanisms, and consequences of HTLV-1 plus-strand expression in naturally-infected T-cell clones

HTLV-1 replication requires the expression of plus-strand-encoded transcriptional transactivator protein Tax. However, Tax protein, a surrogate for HTLV-1 plus-strand expression is seldom detected in freshly isolated infected blood. The kinetics and consequences of plus-strand expression remain poorly understood. I used two fluorescent protein-based Tax reporter systems to study the dynamics and consequences of plus-strand expression and the changes to the host gene expression during plus-strand expression in naturally HTLV-1-infected, non-malignant T-cell clones. Time-lapse live-cell imaging followed by single-cell analysis of two T-cell clones stably transduced with a short-lived enhanced green fluorescent protein Tax reporter system identified five patterns of Tax expression in both clones and the distribution of these patterns was different between the two clones. Mathematical modelling of the experimental data revealed that the mean duration of Tax expression differed between the two clones – 94 and 417 hours, respectively. Host cell transcriptome analysis during successive stages of plus-strand strand expression using a fluorescent timer protein-based Tax reporter system in naturally-infected T-cell clones identified dysregulation in the expression of genes related to multiple cellular processes, including cell cycle, DNA damage response, and apoptosis at the initiation of the plus-strand transcriptional burst. The plus-strand expression showed immediate but transient adverse effects, including reduced proliferation, increased apoptosis, upregulation of a DNA damage marker, and impaired cell cycle progression. In the longer term, the immediate negative consequences of Tax expression were offset by reduced apoptosis and increased proliferation as cells terminated plus-strand expression. Plus-strand expression was also associated with cell-to-cell adhesion and reduced motility. These findings show within and between clone variability in the patterns and duration of HTLV-1 plus-strand expression, changes to the host gene expression during successive stages of the plus-strand expression, and the balance between the beneficial and adverse effects on the host cell associated with the plus-strand expression.Open Acces

Cut-restriction: from cuts to analytic cuts

Cut-elimination is the bedrock of proof theory with a multitude of applications from computational interpretations to proof analysis. It is also the starting point for important meta-theoretical investigations including decidability, complexity, disjunction property, and interpolation. Unfortunately cut-elimination does not hold for the sequent calculi of most non-classical logics. It is well-known that the key to applications is the subformula property (a typical consequence of cut-elimination) rather than cut-elimination itself. With this in mind we introduce cut-restriction, a procedure to restrict arbitrary cuts to analytic cuts (when elimination is not possible). The algorithm applies to all sequent calculi satisfying language-independent and simple-to-check conditions, and it is obtained by adapting age-old cut-elimination. Our work encompasses existing results in a uniform way, and establishes novel analytic subformula properties.Comment: 13 pages, conference preprin

Bounded-analytic sequent calculi and embeddings for hypersequent logics

A sequent calculus with the subformula property has long been recognised as a highly favourable starting point for the proof theoretic investigation of a logic. However, most logics of interest cannot be presented using a sequent calculus with the subformula property. In response, many formalisms more intricate than the sequent calculus have been formulated. In this work we identify an alternative: retain the sequent calculus but generalise the subformula property to permit specific axiom substitutions and their subformulas. Our investigation leads to a classification of generalised subformula properties and is applied to infinitely many substructural, intermediate, and modal logics (specifically: those with a cut-free hypersequent calculus). We also develop a complementary perspective on the generalised subformula properties in terms of logical embeddings. This yields new complexity upper bounds for contractive-mingle substructural logics and situates isolated results on the so-called simple substitution property within a general theory

Immune Moral Models? Pro-Social Rule Breaking as a Moral Enhancement Approach for Ethical AI

The world is heading towards a state in which Artificial Intelligence (AI) based agents make most decisions on behalf of humans. From healthcare decision making to social media censoring, these agents face problems and make decisions that have ethical and societal implications. Hence, ethical behaviour is a critical characteristic of a human-centric AI. A common observation in human-centric industries, like the service industry and healthcare, is that their professionals tend to break rules, if necessary, for pro-social reasons. To make AI agents more human-centric, we argue that there is a need for a mechanism that helps AI agents to identify when and how to break rules set by their designers. In this paper, we examine the when, i.e., conditions under which humans break rules for pro-social reasons. In the presented study, we introduce a 'vaccination strategy dilemma' where one needs to decide whether they would distribute Covid-19 vaccines only to members of a high-risk group (follow the rule) or, in selected cases, administer the vaccine to a few social influencers (break the rule), which might yield an overall greater benefit to society. Results of the empirical study suggest a relationship between stakeholder utilities and pro-social rule breaking (PSRB), which either deontological or utilitarian ethics cannot completely explain. Finally, the paper discusses the design characteristics of an ethical agent capable of PSRB and the future research directions on PSRB in the AI realm.Comment: 15 pages, 2 figure

Cut-restriction: from cuts to analytic cuts

Cut-elimination is the bedrock of proof theory with a multitude of applications from computational interpretations to proof analysis. It is also the starting point for important meta-theoretical investigations into decidability, complexity, disjunction property, interpolation, and more. Unfortunately cut-elimination does not hold for the sequent calculi of most non-classical logics. It is well-known that the key to applications is the subformula property (a typical consequence of cut-elimination) rather than cut-elimination itself. With this in mind, we introduce cut-restriction, a procedure to restrict arbitrary cuts to analytic cuts (when elimination is not possible). The algorithm applies to all sequent calculi satisfying language-independent and simple-to-check conditions, and it is obtained by adapting age-old cut-elimination. Our work encompasses existing results in a uniform way, subsumes Gentzen’s cut-elimination, and establishes new analytic cut properties

Decidability and Complexity in Weakening and Contraction Hypersequent Substructural Logics

We establish decidability for the infinitely many axiomatic extensions of the commutative Full Lambek logic with weakening FLew (i.e. IMALLW) that have a cut-free hypersequent proof calculus. Specifically: every analytic structural rule exten- sion of HFLew. Decidability for the corresponding extensions of its contraction counterpart FLec was established recently but their computational complexity was left unanswered. In the second part of this paper, we introduce just enough on length functions for well-quasi-orderings and the fast-growing complexity classes to obtain complexity upper bounds for both the weakening and contraction extensions. A specific instance of this result yields the first complexity bound for the prominent fuzzy logic MTL (monoidal t-norm based logic) providing an answer to a long- standing open problem