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

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

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

From Semantic Games to Provability: The Case of Gödel Logic

We present a semantic game for Gödel logic and its extensions, where the players’ interaction stepwise reduces arbitrary claims about the relative order of truth degrees of complex formulas to atomic ones. The paper builds on a previously developed game for Gödel logic with projection operator in Fermüller et al. (in: M.-J. Lesot, S. Vieira, M.Z. Reformat, J.P. Carvalho, A. Wilbik, B. Bouchon-Meunier, and R.R. Yager, (eds.), Information processing and management of uncertainty in knowledge-based systems, Springer, Cham, 2020, pp. 257–270). This game is extended to cover Gödel logic with involutive negations and constants, and then lifted to a provability game using the concept of disjunctive strategies. Winning strategies in the provability game, with and without constants and involutive negations, turn out to correspond to analytic proofs in a version of SeqGZL (A. Ciabattoni, and T. Vetterlein, Fuzzy Sets and Systems 161(14):1941–1958, 2010) and in a sequent-of-relations calculus (M. Baaz, and Ch.G. Fermüller, in: N.V. Murray, (ed.), Automated reasoning with analytic tableaux and related methods, Springer, Berlin, 1999, pp. 36–51) respectively

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

Dynamic lockstep processors for applications with functional safety relevance

© 2021 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works.Lockstep processing is a recognized technique for helping to secure functional-safety relevant processing against, for instance, single upset errors that might cause faulty execution of code. Lockstepping processors does however bind processing resources in a fashion not beneficial to architectures and applications that would benefit from multi-core/-processors. We propose a novel on-demand synchronizing of cores/processors for lock-step operation featuring post-processing resource release, a concept that facilitates the implementation of modularly redundant core/processor arrays. We discuss the fundamentals of the design and some implementation notes on work achieved to date

On absolute Galois splitting fields of central simple algebras

A splitting field of a central simple algebra is said to be absolute Galois if it is Galois over some fixed subfield of the centre of the algebra. The paper provides an existence theorem for such fields over global fields with enough roots of unity. As an application, all twisted function fields and all twisted Laurent series rings over symbol algebras (or p-algebras) over global fields are crossed products. A closely related statement holds for division algebras over Henselian valued fields with global residue field. The existence of absolute Galois splitting fields in central simple algebras over global fields is equivalent to a suitable generalization of the weak Grunwald-Wang Theorem, which is proved to hold if enough roots of unity are present. In general, it does not hold and counter examples have been used in noncrossed product constructions. This paper shows in particular that a certain computational difficulty involved in the construction of explicit examples of noncrossed product twisted Laurent series rings can not be avoided by starting the construction with a symbol algebra.Comment: 12 pages (A4); to appear in J. Number Theory (2007

Effects of Litter Origin and Weight on Behaviour of Outbred NIH/S Mice in Plus-maze and Staircase Tests

The objective of this study was to investigate the effects of litter and weight on the behavior of mice.  Male outbred NIH/S mice from 8 litters were randomly distributed among 6 cages and subjected to the  plus-maze and staircase tests. The litter from which the animals had originated had a significant effect on  the behavior of mice in the plus-maze test; furthermore addition of the covariates final weight and weight  gain had no effect on significance or explanatory value. It is proposed that litter origin might influence the  adaptation processes, the development of social status and consequently, the behavior of mice. Differences  attributable to litter were not observed in the staircase test, but when both weight parameters were added as  covariates this proved to be significant. Though the source of these litter-related differences remains to be  clarified, these differences do have a significant effect on the behavior of mice. Therefore they need to be  considered since knowledge of the litter where the outbred mice originated can partly explain differences in  the behavior of the animals. The comparison of models showed that incorporation of the natural features of  the animals (as derived from their biological origin) into a calculation can help rationalise the results; and  provide ample opportunities for discussion and understanding of this complex issue.

Prolonged Exposure of Mice to a Nest Box Reduces Locomotor Activity in the Plus-Maze Test

Environmental enrichment (EE) has been associated with many effects on the behavior of laboratory animals.  The term EE is rather vague, often referring to a variety of item combinations as if what is added to  the cage has no significance. EE is indeed housing refinement, and therefore more exact terms should be  used to clarify the situation. This study was designed to assess whether access to a nest box (NB) could  modify behavior of BALB/c mice in the plus-maze test. Two series of experiments were done with an aspen  NB (11 x 11 x 7 cm, wall thickness 1.5 cm, two round holes (d = 3 cm) at opposite sides. Control mice had  no added item in the cage. The plus-maze consisted of two open (8 x 17 cm) and two closed arms (8 x 17  x 30 cm) connected by a central platform (8 x 8 cm). Mice were placed on the central platform facing an  open arm. During five minutes, the numbers of entries made onto the open and into the closed arms were  recorded. From this data, the percentages of entries made onto the open arms, and the percentage of time  spent on the open arms, were calculated. Furthermore, the number of fecal boli left by the mice in the plusmaze,  as a stress indicator, were counted. In the first series of experiments NB was present for one, two  and three weeks but no drugs were administered. NB provided for one or two weeks had no effect on the  behavior of mice. However, exposure to NB for three weeks did decrease the locomotor activity of mice in  the plus-maze test, as reflected in the decline in the total number of entries made in the test. The presence  of NB for one or two weeks resulted in more (p = 0.001) fecal boli voided when compared to the no NB  or NB for three weeks groups. In the second series of experiments we used NB for 10 days and the selective neuronal nitric oxide synthase  (nNOS) inhibitor 1-(2-trifluoromethylphenyl)-imidazole (TRIM) as a pharmacological tool (at doses  of 25.0, 50.0 and 100.0 mg/kg, i.p.). Depending on the dose, the administration of TRIM induced an anxiolytic  (50 mg/kg) or sedative effect (100 mg/kg) as seen in the increase in the percentage of entries made  onto the open arms or a decrease in the total number of entries, respectively. NB for 10 days had no effect  on the behavior of mice or on the effect of TRIM. In conclusion, NB does not appear to interfere with the anxiolytic effect of TRIM in the plus-maze test but  prolonged exposure to NB does reduce the locomotor  activity of mice.