This paper investigates the feasibility and limitations of LLM-powered intelligent Python scripting agents for industrial data analysis. We specifically evaluate the trade-offs of a methodology where the agent is only fed metadata as input, preserving the confidentiality of sensitive industrial data. The evaluation is conducted using a comprehensive benchmark of 80 queries, categorised by complexity, based on two real-world industrial datasets containing variable levels of metadata clarity. Our results indicate that this approach is viable for simple analytical tasks; however, the efficiency of handling more complex queries is contingent on two factors: the clarity of metadata and the selected model’s inherent ability to deal with ambiguity.

Abstract We consider equational theories based on axioms for recursively defining functions, with rules for equality and substitution, but no form of induction—we denote such equational theories as PETS for pure equational theories with substitution. An example is Cook’s system PV without its rule for induction. We show that the Bounded Arithmetic theory $\mathrm {S}^{1}_2$ proves the consistency of PETS. Our approach employs model-theoretic constructions for PETS based on approximate values resembling notions from domain theory in Bounded Arithmetic, which may be of independent interest.

The economic cost of roll refurbishment in the steel-making industry is considerable. In a cold rolling mill, wear and damage of rolls disrupt the industrial environment, so it is critical to predict the remaining useful life early and change the roll without causing disruption to the manufacturing process. However, since cold rolling is a complex process affected by multiple variables which are operated in adverse conditions, it is very challenging to mathematically analyse the roll wear and failure. For this reason, in the present paper, a data-driven solution is proposed to predict the correct time for changing individual rolls. To develop an accurate predictive model, several datasets containing high-resolution production data and roll refurbishment data collected from a UK based steel plant have been acquired and processed in a way that the roll wear is modelled as a Remaining Useful Life (RUL) problem, where the number of coils that a roll is able to process is viewed as the remaining cycles. Then hybrid deep learning models are used to predict the Remaining Useful Life of rolls used in steel making. This novel data-driven approach achieves high prediction accuracy and has been validated on a real-world dataset. The proposed approach not only helps avoiding early failure but also can serve as a critical step towards the design of an optimal, automated maintenance schedule for the roll management.

We introduce a system of Hyper Natural Deduction for Gödel Logic as an extension of Gentzen’s system of Natural Deduction. A deduction in this system consists of a finite set of derivations which uses the typical rules of Natural Deduction, plus additional rules providing means for communication between derivations. We show that our system is sound and complete for infinite-valued propositional Gödel Logic, by giving translations to and from Avron’s Hypersequent Calculus. We provide conversions for normalization extending usual conversions for Natural Deduction and prove the existence of normal forms for Hyper Natural Deduction for Gödel Logic. We show that normal deductions satisfy the subformula property.

We study consistency search problems for Frege and extended Frege proofs—namely the NP search problems of finding syntactic errors in Frege and extended Frege proofs of contradictions. The input is a polynomial time function, or an oracle, describing a proof of a contradiction; the output is the location of a syntactic error in the proof. The consistency search problems for Frege and extended Frege systems are shown to be many-one complete for the provably total NP search problems of the second-order bounded arithmetic theories U 1 2 and V 1 2 , respectively.

We introduce a Hyper Natural Deduction system as an extension of Gentzen's Natural Deduction system. A Hyper Natural Deduction consists of a finite set of derivations which may use, beside typical Natural Deduction rules, additional rules providing means for communication between derivations. We show that our Hyper Natural Deduction system is sound and complete for infinite-valued propositional Gödel Logic, by giving translations to and from Avron's Hyper sequent Calculus. We also provide conversions for normalisation and prove the existence of normal forms for our Hyper Natural Deduction system.

Abstract We introduce the safe recursive set functions based on a Bellantoni–Cook style subclass of the primitive recursive set functions. We show that the functions computed by safe recursive set functions under a list encoding of finite strings by hereditarily finite sets are exactly the polynomial growth rate functions computed by alternating exponential time Turing machines with polynomially many alternations. We also show that the functions computed by safe recursive set functions under a more efficient binary tree encoding of finite strings by hereditarily finite sets are exactly the quasipolynomial growth rate functions computed by alternating quasipolynomial time Turing machines with polylogarithmic many alternations. We characterize the safe recursive set functions on arbitrary sets in definability-theoretic terms. In its strongest form, we show that a function on arbitrary sets is safe recursive if and only if it is uniformly definable in some polynomial level of a refinement of Jensen's J-hierarchy, relativized to the transitive closure of the function's arguments. We observe that safe recursive set functions on infinite binary strings are equivalent to functions computed by infinite-time Turing machines in time less than ωω. We also give a machine model for safe recursive set functions which is based on set-indexed parallel processors and the natural bound on running times.

We consider intermediate predicate logics defined by fixed well-ordered (or dually well-ordered) linear Kripke frames with constant domains where the order-type of the well-order is strictly smaller than ωω. We show that two such logics of different order-type are separated by a first-order sentence using only one monadic predicate symbol. Previous results by Minari, Takano and Ono, as well as the second author, obtained the same separation but relied on the use of predicate symbols of unbounded arity.

This article concerns the second-order systems U 1 2 and V 1 2 of bounded arithmetic, which have proof-theoretic strengths corresponding to polynomial-space and exponential-time computation. We formulate improved witnessing theorems for these two theories by using S 1 2 as a base theory for proving the correctness of the polynomial-space or exponential-time witnessing functions. We develop the theory of nondeterministic polynomial-space computation, including Savitch's theorem, in U 1 2 . Kołodziejczyk et al. [2011] have introduced local improvement properties to characterize the provably total NP functions of these second-order theories. We show that the strengths of their local improvement principles over U 1 2 and V 1 2 depend primarily on the topology of the underlying graph, not the number of rounds in the local improvement games. The theory U 1 2 proves the local improvement principle for linear graphs even without restricting to logarithmically many rounds. The local improvement principle for grid graphs with only logarithmically-many rounds is complete for the provably total NP search problems of V 1 2 . Related results are obtained for local improvement principles with one improvement round and for local improvement over rectangular grids.

A propositional proof system is weakly automatizable if there is a polynomial time algorithm that separates satisfiable formulas from formulas that have a short refutation in the system, with respect to a given length bound. We show that if the resolution proof system is weakly automatizable, then parity games can be decided in polynomial time. We give simple proofs that the same holds for depth-1 propositional calculus (where resolution has depth 0) with respect to mean payoff and simple stochastic games. We define a new type of combinatorial game and prove that resolution is weakly automatizable if and only if one can separate, by a set decidable in polynomial time, the games in which the first player has a positional winning strategy from the games in which the second player has a positional winning strategy. Our main technique is to show that a suitable weak bounded arithmetic theory proves that both players in a game cannot simultaneously have a winning strategy, and then to translate this proof into propositional form.

Klaus Ambos-Spies, Arnold Beckmann, Erzsébet Csuhaj-Varjú, Benedikt Löwe; Computability in Europe 2009, Journal of Logic and Computation, Volume 23, Issue

Arnold Beckmann, Benedikt Löwe; Computability in Europe 2008, Journal of Logic and Computation, Volume 22, Issue 2, 1 April 2012, Pages 163–164, https://do

The six papers in this special issue arose from the conference CiE 2009: Mathematical Theory and Computational Practice, held at the Ruprecht-Karls-Universitat Heidelberg, Germany, in July 2009. CiE 2009 was the fifth meeting in the series of conferences associated with the Association for Computability in Europe.

interdisciplinary network Computability in Europe.Computability in Europe (CiE) used to be an informal network of European scientists working on computability theory, including its foundations, technical development, and applications-mainly identified by its successful conference series with

Using appropriate notation systems for proofs, cut-reduction can often be rendered feasible on these notations. Explicit bounds can be given. Developing a suitable notation system for Bounded Arithmetic, and applying these bounds, all the known results on definable functions of certain such theories can be reobtained in a uniform way.

Abstract We investigate the relation between intermediate predicate logics based on countable linear Kripke frames with constant domains and Gödel logics. We show that for any such Kripke frame there is a Gödel logic which coincides with the logic defined by this Kripke frame on constant domains and vice versa. This allows us to transfer several recent results on Gödel logics to logics based on countable linear Kripke frames with constant domains: We obtain a complete characterisation of axiomatisability of logics based on countable linear Kripke frames with constant domains. Furthermore, we obtain that the total number of logics defined by countable linear Kripke frames on constant domains is countable.

The 12 papers in this special issue arose from the conference CiE 2006: Logical Approaches to Computational Barriers, held at the University of Wales Swansea in July, 2006. CiE 2006 was the second of a new series of conferences associated with the interdisciplinary network Computability in Europe. Computability in Europe (CiE) is an informal network of European scientists working on computability theory, including its foundations, technical development and applications. Among the aims of the network is to advance our theoretical understanding of what can and cannot be computed, by any means of computation. Its scientific vision is broad: computations may be performed with discrete or continuous data by all kinds of algorithms, programs and machines. Computations may be made by experimenting with any sort of physical system obeying the laws of a physical theory such as Newtonian mechanics, quantum theory or relativity. Computations may be very general, depending upon the foundations of set theory; or very specific, using the combinatorics of finite structures. CiE also works on subjects intimately related to computation, especially theories of data and information, and methods for formal reasoning about computations. The sources of new ideas and methods include practical developments in areas such as neural networks, quantum computation, natural computation, molecular computation, computational learning. Applications are everywhere, especially, in algebra, analysis and geometry, or data types and programming.

We define the notion of the uniform reduct of a propositional proof system as the set of those bounded formulas in the language of Peano Arithmetic which have polynomial size proofs under the Paris-Wilkie-translation. With respect to the arithmetic complexity of uniform reducts, we show that uniform reducts are Π10-hard and obviously in Σ20. We also show under certain regularity conditions that each uniform reduct is closed under bounded generalisation; that in the case the language includes a symbol for exponentiation, a uniform reduct is closed under modus ponens if and only if it already contains all true bounded formulas; and that each uniform reduct contains all true Π1b(α)-formulas.

Abstract We consider equational theories for functions denned via recursion involving equations between closed terms with natural rules based on recursive definitions of the function symbols. We show that consistency of such equational theories can be proved in the weak fragment of arithmetic S21. In particular this solves an open problem formulated by Takeuti (c.f. [5, p.5 problem 9.]).

Abstract We determine the exact bounds for the length of an arbitrary reduction sequence of a term in the typed λ-calculus with β-, ξ- and η-conversion. There will be two essentially different classifications, one depending on the height and the degree of the term and the other depending on the length and the degree of the term.