Related papers: Using Duality in Circuit Complexity
Most classical results in circuit complexity theory concern circuits over the Boolean domain. Besides their simplicity and the ease of comparing different languages, the actual architecture of computers is also an important motivating…
This article fits in the area of research that investigates the application of topological duality methods to problems that appear in theoretical computer science. One of the eventual goals of this approach is to derive results in…
We investigate a famous decision problem in automata theory: separation. Given a class of language C, the separation problem for C takes as input two regular languages and asks whether there exists a third one which belongs to C, includes…
Circuit discovery has gradually become one of the prominent methods for mechanistic interpretability, and research on circuit completeness has also garnered increasing attention. Methods of circuit discovery that do not guarantee…
The study of finite automata and regular languages is a privileged meeting point of algebra and logic. Since the work of Buchi, regular languages have been classified according to their descriptive complexity, i.e. the type of logical…
Group languages are regular languages recognized by finite groups, or equivalently by finite automata in which each letter induces a permutation on the set of states. We investigate the separation problem for this class of languages: given…
Separation is a classical problem asking whether, given two sets belonging to some class, it is possible to separate them by a set from a smaller class. We discuss the separation problem for regular languages. We give a Ptime algorithm to…
Separations among the first order logic ${\cal R}ing(0,+,*)$ of finite residue class rings, its extensions with generalized quantifiers, and in the presence of a built-in order are shown, using algebraic methods from class field theory.…
We study Boolean circuits as a representation of Boolean functions and consider different equivalence, audit, and enumeration problems. For a number of restricted sets of gate types (bases) we obtain efficient algorithms, while for all…
$\omega$-regular languages are a natural extension of the regular languages to the setting of infinite words. Likewise, they are recognised by a host of automata models, one of the most important being Alternating Parity Automata (APAs), a…
We refine the complexity landscape for enumeration problems by introducing very low classes defined by using Boolean circuits as enumerators. We locate well-known enumeration problems, e.g., from graph theory, Gray code enumeration, and…
We investigate the duality between algebraic and coalgebraic recognition of languages to derive a generalization of the local version of Eilenberg's theorem. This theorem states that the lattice of all boolean algebras of regular languages…
Explaining why a language model produces a particular output requires local, input-level explanations. Existing methods uncover global capability circuits (e.g., indirect object identification), but not why the model answers a specific…
Inspired by Solomonoffs theory of inductive inference, we propose a prior based on circuit complexity. There are several advantages to this approach. First, it relies on a complexity measure that does not depend on the choice of UTM. There…
Duality methods are used to generate explicit solutions to nonlinear Hodge systems, demonstrate the well-posedness of boundary value problems, and reveal, via the Hodge-B\"acklund transformation, underlying symmetries among superficially…
Separation is a classical problem in mathematics and computer science. It asks whether, given two sets belonging to some class, it is possible to separate them by another set of a smaller class. We present and discuss the separation problem…
Programs with a continuous state space or that interact with physical processes often require notions of equivalence going beyond the standard binary setting in which equivalence either holds or does not hold. In this paper we explore the…
In this work, we propose a class of equational theories for bounded binary circuits that have the finite variant property. These theories could serve as a building block to specify cryptographic primitive implementations and automatically…
Satisfiability of Boolean circuits is among the most known and important problems in theoretical computer science. This problem is NP-complete in general but becomes polynomial time when restricted either to monotone gates or linear gates.…
Bilattices (that is, sets with two lattice structures) provide an algebraic tool to model simultaneously the validity of, and knowledge about, sentences in an appropriate language. In particular, certain bilattices have been used to model…