Related papers: Subsystems of Open Induction
We introduce and study logic programs whose clauses are built out of monotone constraint atoms. We show that the operational concept of the one-step provability operator generalizes to programs with monotone constraint atoms, but the…
Separation Logic (SL) with inductive definitions is a natural formalism for specifying complex recursive data structures, used in compositional verification of programs manipulating such structures. The key ingredient of any automated…
We study variants of Buss's theories of bounded arithmetic axiomatized by induction schemes disallowing the use of parameters, and closely related induction inference rules. We put particular emphasis on $\hat\Pi^b_i$ induction schemes,…
This paper presents the first in a series of results that allow us to develop a theory providing finer control over the complexity of normalisation, and in particular of cut elimination. By considering atoms as self-dual non-commutative…
We define a bi-directional embedding between hypersequent calculi and a subclass of systems of rules (2-systems). In addition to showing that the two proof frameworks have the same expressive power, the embedding allows for the recovery of…
This work presents a novel systematic methodology to analyse the capabilities and limitations of Large Language Models (LLMs) with feedback from a formal inference engine, on logic theory induction. The analysis is complexity-graded w.r.t.…
Implicit Computational Complexity makes two aspects implicit, by manipulating programming languages rather than models of com-putation, and by internalizing the bounds rather than using external measure. We survey how automata theory…
We study substitutive systems generated by nonprimitive substitutions and show that transitive subsystems of substitutive systems are substitutive. As an application we obtain a complete characterisation of the sets of words that can appear…
This article introduces the concept of the algorithmic unconscious to designate the set of structural determinations that operate within large language models (LLMs) without being accessible either to the model's own reflexivity or to that…
A sound and complete embedding of conditional logics into classical higher-order logic is presented. This embedding enables the application of off-the-shelf higher-order automated theorem provers and model finders for reasoning within and…
An inductive logic can be formulated in which the elements are not propositions or probability distributions, but information systems. The logic is complete for information systems with binary hypotheses, i.e., it applies to all such…
In this paper, we shall present a topological approach for the computation of some supremal sublanguages, often specified by language equations, which arise from the study of the supervisory control theory. The basic idea is to identify the…
We present new induction principles for the syntax of dependent type theories, which we call relative induction principles. The result of the induction principle relative to a functor F into the syntax is stable over the codomain of F. We…
We present a system for generating parsers based directly on the metaphor of parsing as deduction. Parsing algorithms can be represented directly as deduction systems, and a single deduction engine can interpret such deduction systems so as…
The problem of "what is 'system'?" is in the very foundations of modern quantum mechanics. Here, we point out the interest in this topic in the information-theoretic context. E.g., we point out the possibility to manipulate a pair of…
We propose various methods for combining or amalgamating propositional languages and deductive systems. We make heavy use of quantales and quantale modules in the wake of previous works by the present and other authors. We also describe…
Recently we presented a concise survey of the formulation of the induction and coinduction principles, and some concepts related to them, in programming languages type theory and four other mathematical disciplines. The presentation in type…
We use automated theorem provers to significantly shorten a formal development in higher order set theory. The development includes many standard theorems such as the fundamental theorem of arithmetic and irrationality of square root of…
Despite recent advances in automating theorem proving in full first-order theories, inductive reasoning still poses a serious challenge to state-of-the-art theorem provers. The reason for that is that in first-order logic induction requires…
Inductive theorem proving is an important long-standing challenge in computer science. In this extended abstract, we first summarize the recent developments of proof by induction for Isabelle/HOL. Then, we propose united reasoning, a novel…