arXiv subjects

Chrystopher L. Nehaniv

Publications and source records attributed to Chrystopher L. Nehaniv.

At least 19 recordsLinked to original sources

arXiv2026

Algebraic Characterizations for Minors of Finite Graphs via Flow Transformation Monoid Division and Embedding

We prove three theorems on the flow monoids of finite graphs. First, we show that a non-empty finite graph G = (V, E) is connected if and only if its flow monoid contains a constant map on V, equivalently, if and only if it contains all constant maps on V. Second, we give a new characterization of graph minors in terms of division of flow transformation monoids, together with an algebraic crossing condition that detects edges between the vertex sets being contracted. Third, we strengthen this to an embedded-copy theorem: a graph M is a minor of G if and only if, subject to analogous crossing conditions, the flow transformation monoid of M is realized as the induced action of a subsemigroup of the ambient flow monoid of G, this subsemigroup being a monoid with a local idempotent identity.

cs.DM↗

arXiv2026

Measuring the Computational Power of Finite Patches of Cellular Automata

Computational power can be measured by assigning an algebraic structure to a computational device. Here, we convert a small patch of Conway's Game of Life into a transformation semigroup. The conversion captures not only time evolution but also interactive operations. In this way, the cellular automaton becomes directly programmable. Once this measurement is made, we apply hierarchical decompositions to the resulting algebraic object as a way of understanding it. These decompositions are based on a macro/micro-state division inspired by statistical mechanics. However, cellular automata have a large number of global states. Therefore, we focus on partitioning the state space and creating morphic images approximations that can serve as macro-level descriptions. The methods developed here are not limited to cellular automata; they apply more generally to discrete dynamical systems.

nlin.CG↗

arXiv2025

Computational Exploration of Finite Semigroupoids

Recent algorithmic advances in algebraic automata theory drew attention to semigroupoids (semicategories). These are mathematical descriptions of typed computational processes, but they have not been studied systematically in the context of automata. Here, we use relational programming to explore finite semigroupoids to improve our mathematical intuition about these models of computation. We implement declarative solutions for enumerating abstract semigroupoids (partial composition tables), finding homomorphisms, and constructing (minimal) transformation representations. We show that associativity and consistent typing are different properties, distinguish between strict and more permissive homomorphisms, and systematically enumerate arrow-type semigroupoids (reified type structures).

cs.FL↗

arXiv2025

Using Dynamical Systems Theory to Quantify Complexity in Asymptotic Lenia

Continuous cellular automata (CCAs) have evolved from discrete lookup tables to continuous partial differential equation (PDE) formulations in the search for novel forms of complexity. Despite innovations in qualitative behavior, analytical methods have lagged behind, reinforcing the notion that emergent complexity defies simple explanation. In this paper, we demonstrate that the PDE formulation of Asymptotic Lenia enables rigorous analysis using dynamical systems theory. We apply the concepts of symmetries, attractors, Lyapunov exponents, and fractal dimensions to characterize complex behaviors mathematically. Our contributions include: (1) a mathematical explanation for the four distinct solution classes (solitons, rotators, periodic and chaotic patterns), (2) conditions for the existence of a global attractor with fractal dimension $>4$, (3) identification of Kaplan-Yorke dimension as an effective complexity measure for CCAs, and (4) an efficient open-source implementation for calculating Lyapunov exponents and the covariant Lyapunov vectors for CCAs. We conclude by identifying the minimal set of properties that enable complex behavior in a broader class of CCAs. This framework provides a foundation for understanding and measuring complexity in artificial life systems.

nlin.PS↗

arXiv2025

Representation Independent Decompositions of Computation

Constructing complex computation from simpler building blocks is a defining problem of computer science. In algebraic automata theory, we represent computing devices as semigroups. Accordingly, we use mathematical tools like products and homomorphisms to understand computation through hierarchical decompositions. To address the shortcomings of some of the existing decomposition methods, we generalize semigroup representations to semigroupoids by introducing types. On the abstraction level of category theory, we describe a flexible, iterative and representation independent algorithm. Moving from the specific state transition model to the abstract composition of arrows unifies seemingly different decomposition methods and clarifies the three algorithmic stages: collapse, copy and compress. We collapse some dynamics through a morphism to the top level; copy the forgotten details into the bottom level; and finally we apply compression there. The hierarchical connections are solely for locating the repeating patterns in the compression. These theoretical findings pave the way for more precise computer algebra tools and allow for understanding computation with other algebraic structures.

math.GR↗

arXiv2025

On Constructing Finite Automata by Relational Programming

We consider ways to construct a transducer for a given set of input word to output symbol pairs. This is motivated by the need for representing game playing programs in a low-level mathematical format that can be analyzed by algebraic tools. This is different from the classical applications of finite state automata, thus the usual optimization techniques are not directly applicable. Therefore, we use relational programming tools to find minimal transducers realizing a given set of input-output pairs.

cs.FL↗

arXiv2024

From Relation to Emulation and Interpretation: Computer Algebra Implementation of the Covering Lemma for Finite Transformation Semigroups

We give a practical computer algebra implementation of the Covering Lemma for finite transformation semigroups. The lemma states that given a surjective relational morphism $(X,S)\twoheadrightarrow(Y,T)$, we can establish emulation by a cascade product (subsemigroup of the wreath product): $(X,S)\hookrightarrow (Y,T)\wr (Z,U)$. The dependent component $(Z,U)$ contains the kernel of the morphism, the information lost in the map. The implementation complements the existing tools for the holonomy decomposition algorithm. It gives an incremental method to get a coarser decomposition when computing the complete skeleton for holonomy is not feasible. Here, we describe a simplified and generalized algorithm for the lemma and compare it to the holonomy method. Incidentally, the kernel-based method could be the easiest way of understanding the hierarchical decompositions of transformation semigroups and thus the celebrated Krohn-Rhodes theory.

math.GR↗

arXiv2024

Self-Reproduction and Evolution in Cellular Automata: 25 Years after Evoloops

The year of 2024 marks the 25th anniversary of the publication of evoloops, an evolutionary variant of Chris Langton's self-reproducing loops which proved constructively that Darwinian evolution of self-reproducing organisms by variation and natural selection is possible within deterministic cellular automata. Over the last few decades, this line of Artificial Life research has since undergone several important developments. Although it experienced a relative dormancy of activities for a while, the recent rise of interest in open-ended evolution and the success of continuous cellular automata models have brought researchers' attention back to how to make spatio-temporal patterns self-reproduce and evolve within spatially distributed computational media. This article provides a review of the relevant literature on this topic over the past 25 years and highlights the major accomplishments made so far, the challenges being faced, and promising future research directions.

nlin.CG↗

arXiv2023

A Personalized Household Assistive Robot that Learns and Creates New Breakfast Options through Human-Robot Interaction

For robots to assist users with household tasks, they must first learn about the tasks from the users. Further, performing the same task every day, in the same way, can become boring for the robot's user(s), therefore, assistive robots must find creative ways to perform tasks in the household. In this paper, we present a cognitive architecture for a household assistive robot that can learn personalized breakfast options from its users and then use the learned knowledge to set up a table for breakfast. The architecture can also use the learned knowledge to create new breakfast options over a longer period of time. The proposed cognitive architecture combines state-of-the-art perceptual learning algorithms, computational implementation of cognitive models of memory encoding and learning, a task planner for picking and placing objects in the household, a graphical user interface (GUI) to interact with the user and a novel approach for creating new breakfast options using the learned knowledge. The architecture is integrated with the Fetch mobile manipulator robot and validated, as a proof-of-concept system evaluation in a large indoor environment with multiple kitchen objects. Experimental results demonstrate the effectiveness of our architecture to learn personalized breakfast options from the user and generate new breakfast options never learned by the robot.

cs.RO↗

arXiv2023

How Do Human Users Teach a Continual Learning Robot in Repeated Interactions?

Continual learning (CL) has emerged as an important avenue of research in recent years, at the intersection of Machine Learning (ML) and Human-Robot Interaction (HRI), to allow robots to continually learn in their environments over long-term interactions with humans. Most research in continual learning, however, has been robot-centered to develop continual learning algorithms that can quickly learn new information on static datasets. In this paper, we take a human-centered approach to continual learning, to understand how humans teach continual learning robots over the long term and if there are variations in their teaching styles. We conducted an in-person study with 40 participants that interacted with a continual learning robot in 200 sessions. In this between-participant study, we used two different CL models deployed on a Fetch mobile manipulator robot. An extensive qualitative and quantitative analysis of the data collected in the study shows that there is significant variation among the teaching styles of individual users indicating the need for personalized adaptation to their distinct teaching styles. The results also show that although there is a difference in the teaching styles between expert and non-expert users, the style does not have an effect on the performance of the continual learning robot. Finally, our analysis shows that the constrained experimental setups that have been widely used to test most continual learning techniques are not adequate, as real users interact with and teach continual learning robots in a variety of ways. Our code is available at https://github.com/aliayub7/cl_hri.

cs.RO↗

arXiv2023

Continual Learning through Human-Robot Interaction: Human Perceptions of a Continual Learning Robot in Repeated Interactions

For long-term deployment in dynamic real-world environments, assistive robots must continue to learn and adapt to their environments. Researchers have developed various computational models for continual learning (CL) that can allow robots to continually learn from limited training data, and avoid forgetting previous knowledge. While these CL models can mitigate forgetting on static, systematically collected datasets, it is unclear how human users might perceive a robot that continually learns over multiple interactions with them. In this paper, we developed a system that integrates CL models for object recognition with a Fetch mobile manipulator robot and allows human participants to directly teach and test the robot over multiple sessions. We conducted an in-person study with 60 participants that interacted with our system in 300 sessions (5 sessions per participant). We conducted a between-subject study with three different CL models to understand human perceptions of continual learning robots over multiple sessions. Our results suggest that participants' perceptions of trust, competence, and usability of a continual learning robot significantly decrease over multiple sessions if the robot forgets previously learned objects. However, the perceived task load on participants for teaching and testing the robot remains the same over multiple sessions even if the robot forgets previously learned objects. Our results also indicate that state-of-the-art CL models might perform unreliably when applied on robots interacting with human participants. Further, continual learning robots are not perceived as very trustworthy or competent by human participants, regardless of the underlying continual learning model or the session number.

cs.RO↗

arXiv2022

Don't Forget to Buy Milk: Contextually Aware Grocery Reminder Household Robot

Assistive robots operating in household environments would require items to be available in the house to perform assistive tasks. However, when these items run out, the assistive robot must remind its user to buy the missing items. In this paper, we present a computational architecture that can allow a robot to learn personalized contextual knowledge of a household through interactions with its user. The architecture can then use the learned knowledge to make predictions about missing items from the household over a long period of time. The architecture integrates state-of-the-art perceptual learning algorithms, cognitive models of memory encoding and learning, a reasoning module for predicting missing items from the household, and a graphical user interface (GUI) to interact with the user. The architecture is integrated with the Fetch mobile manipulator robot and validated in a large indoor environment with multiple contexts and objects. Our experimental results show that the robot can adapt to an environment by learning contextual knowledge through interactions with its user. The robot can also use the learned knowledge to correctly predict missing items over multiple weeks and it is robust against sensory and perceptual errors.

cs.RO↗

arXiv2021

Cascade Product of Permutation Groups

Motivated by computational efficiency in algebraic automata theory here we define the cascade product of permutation groups as an external product, as a generic extension. It is the most general hierarchical product that uses arbitrary total functions to combine a linearly ordered set of permutation groups. Algebraically speaking, cascade products are explicitly constructed substructures of the iterated wreath product. We show how direct, semidirect and wreath products can be described as cascade products and we also discuss examples of composite groups that can only be constructed exactly as generic extensions by cascade products. The cascade construction naturally generalizes to the transformation semigroup case by leaving out the details of defining inverse operations.

math.GR↗

arXiv2021

Hierarchical Complexity of Finite Groups

What are simplest ways to construct a finite group from its atomic constituents? To understand part-whole relations between finite simple groups and the global structure of finite groups, we axiomatize complexity measures on finite groups. From the Jordan-Hölder theorem and Frobenius-Lagrange embedding in an iterated wreath product, any finite group $G$ can be constructed from a unique collection of simple groups, its Jordan-Hölder factors, each with well-defined multiplicities through iterated extension. What is the least number of levels needed in such a hierarchical construction if a level is allowed to include several of these atomic pieces? To answer this question rigorously, we give a natural set of hierarchical complexity axioms for finite groups, and prove these axioms are satisfied by a unique maximal complexity function $\mathbf{cx}$. We prove this function is the same as the minimal number of "spans of gems" (direct products of simple groups) in a subnormal series with all factors of this type. This hierarchical complexity is thus effectively computable, and bounded below by all other complexity measures satisfying the axioms, including generalizations of derived length and Fitting height. For solvable groups, the unique maximal group complexity measure satisfying the axioms agrees with the restriction of the one for all finite groups, and in addition satisfies an embedding axiom. In both cases, the complexity of a group is bounded above and below by various natural functions. In particular, hierarchical complexity is sharply bounded above by socle length, with a canonical decomposition. Examples illustrate applications of the bounds and axiomatic methods in determining complexity of groups. We show also that minimal decompositions need not be unique in terms of what components occur nor their ordering. The complexity axioms are also shown to be independent.

math.GM↗

arXiv2020

Spatial Iterated Prisoner's Dilemma as a Transformation Semigroup

The prisoner's dilemma (PD) is a game-theoretic model studied in a wide array of fields to understand the emergence of cooperation between rational self-interested agents. In this work, we formulate a spatial iterated PD as a discrete-event dynamical system where agents play the game in each time-step and analyse it algebraically using Krohn-Rhodes algebraic automata theory using a computational implementation of the holonomy decomposition of transformation semigroups. In each iteration all players adopt the most profitable strategy in their immediate neighbourhood. Perturbations resetting the strategy of a given player provide additional generating events for the dynamics. Our initial study shows that the algebraic structure, including how natural subsystems comprising permutation groups acting on the spatial distributions of strategies, arise in certain parameter regimes for the pay-off matrix, and are absent for other parameter regimes. Differences in the number of group levels in the holonomy decomposition (an upper bound for Krohn-Rhodes complexity) are revealed as more pools of reversibility appear when the temptation to defect is at an intermediate level. Algebraic structure uncovered by this analysis can be interpreted to shed light on the dynamics of the spatial iterated PD.

math.DS↗

arXiv2020

Algebraic Structure of the Varikon Box

The 15-Puzzle is a well studied permutation puzzle. This paper explores the group structure of a three-dimensional variant of the 15-Puzzle known as the Varikon Box, with the goal of providing a heuristic that would help a human solve it while minimizing the number of moves. First, we show by a parity argument which configurations of the puzzle are reachable. We define a generating set based on the three dimensions of movement, which generates a group that acts on the puzzle configurations, and we explore the structure of this group. Finally, we show a heuristic for solving the puzzle by writing an element of the symmetry group as a word in terms of a generating set, and we compute the shortest possible word for each puzzle configuration.

math.GR↗

arXiv2020

Algebraic Structure and Complexity of Bootstrap Percolation with External Inputs

In this paper a modification of the standard Bootstrap Percolation model is introduced. In our modification a discrete time update rule is constructed that allows for non-monotonicity - unlike its classical counterpart. External inputs to drive the system into desirable states are also included in the model. The algebraic structure and complexity properties of the system are inferred by studying the system's holonomy decomposition. We introduce methods of inferring the pools of reversibility for the system. Dependence of system complexity on process parameters is presented and discussed.

math.DS↗

arXiv2020

Exploring Tetris as a Transformation Semigroup

Tetris is a popular puzzle video game, invented in 1984. We formulate two versions of the game as a transformation semigroup and use this formulation to view the game through the lens of Krohn-Rhodes theory. In a variation of the game upon which it restarts if the player loses, we find permutation group structures, including the symmetric group $S_5$ which contains a non-abelian simple group as a subgroup. This implies, at least in a simple case, that iterated Tetris is finitarily computationally universal.

math.GR↗