Children learn the cardinalities of the first numbers one, two, three and four before they learn how counting tracks cardinality for all numbers. It may be that when children start to understand counting, they also discover how numbers relate to one another in a structured number system. Do children who understand that the cardinality of a set is the last number assigned after counting each item (cardinal principle knowledge) also understand that each number represents the cardinality of the set created by adding one to an empty set for every count it takes to reach that number (a recursive understanding of the successor function)? We tested this by assessing children’s early recursive understanding of the successor function in relation to their cardinality knowledge. Children who were not yet cardinal principle knowers already demonstrated a recursive understanding of the successor function within the limits of their cardinality knowledge. Our findings suggest that children have some structural knowledge of the number system before learning how counting tracks cardinality. We discuss how continued counting practice may eventually allow children to expand this knowledge across all numbers.

Origins of numbers: a shared language-of-thought for arithmetic and geometry?

Large language models (LLMs) now turn their attention to search. Recently, Thought of Search (ToS) proposed defining the search space with code, having an LLM produce that code. ToS requires a human in the loop, collaboratively producing a sound successor function and goal test, achieving impressive 100% accuracy on all the tested datasets. In this work, we automate ToS (AutoToS), completely taking the human out of the loop of solving planning problems. AutoToS guides the language model step by step towards the generation of sound and complete search components, through feedback from both generic and domain specific unit tests. We achieve 100% accuracy, with minimal feedback iterations, using LLMs of various sizes on all evaluated domains.

Additive reasoning is a fundamental mathematical skill that children learn during elementary school. However, previous studies have suggested that children start their learning process in preschool. The current research aims to examine how two additive reasoning skills, successor function and partitioning, emerge in the preschool years. To this purpose, a group of 56 children of 4 and 5 years of age were tested on three additive tasks, a cardinality task, and a counting task. The results show a similar developmental trajectory for children’s performance on the successor function task and the partitioning tasks, with significantly better performance in 5-year-olds. The results also show that children’s cardinality knowledge and counting skills are good predictors in both additive reasoning tasks. These findings suggest that preschool is a critical period for learning the additive structure of the number system and that knowledge of verbal counting boosts this acquisition.

In this paper, we discuss the resonance phenomena in linear and weakly nonlinear impulsive differential equations. Using a topological approach, we obtain the Landesman–Lazer type necessary and sufficient conditions for the existence of periodic solutions and extend the Fredholm alternative for linear equations with impulses. Moreover, we show the coexistence of periodic and unbounded solutions of weakly nonlinear impulsive differential equations through the use of the successor map and a Lyapunov function.

In this paper, a modified Leslie-type predator–prey system with Holling-type III functional response is formulated to investigate the dynamics in the presence of special Allee effect induced by fear factors. First, we calculate the first three focal values by the method of successor function to ensure that the system has an unstable weak focus of multiplicity 3. A focus- or center-type degenerate Bogdanov–Takens singularity of codimension 3 and a weak focus of multiplicity 1 or 2 together with a cusp of codimension 2 are derived for the system. Then multiple bifurcations are explored. It is shown that the system undergoes Hopf bifurcation of codimension 3 and hence three limit cycles are generated. There exists another large stable limit cycle enclosing these three limit cycles by Poincaré–Bendixson theorem. We demonstrate that a degenerate focus-type Bogdanov–Takens bifurcation of codimension 3 can occur by calculating the universal unfolding. It is also shown that a subcritical, supercritical or degenerate Hopf bifurcation occurs together with a Bogdanov–Takens bifurcation. Finally, numerical simulations are used to support the theoretical results. The analysis results reveal that the special Allee effect is beneficial for the persistence and diversity of ecosystem.

In ecological modeling, symmetry can manifest in multiple ways, significantly contributing to our understanding of ecosystem stability, species interactions, and ecological balance. Considering the bloom phenomenon caused by excessive algae in ponds and its impact on aquaculture, we establish a class of algae–fish ecological models with two state-dependent impulse controls. In this paper, we first discuss the existence of positive equilibria in the system without impulses. Second, the existence and asymptotic stability of the order-1 and order-2 periodic solutions are analyzed by applying successor functions and a Similar Poincare Criterion. Finally, numerical simulations demonstrate that the model exhibits order-1 and order-2 periodic solutions, clarifying that the system with two impulses can address the problem more effectively.

We investigate the decidability of first-order logic extensions. For example, it is established in A. S. Zolotov’s works that a logic with a unary transitive closure operator for the one successor theory is decidable. We show that in a similar case, a logic with a unary partial fixed point operator is undecidable. For this purpose, we reduce the halting problem for the counter machine to the problem of truth of the underlying formula. This reduction uses only one unary non-nested partial fixed operator that is applied to a universal or existential formula.

On Undecidability of Unary Nonnested PFP Operators for One Successor Function Theory

This paper studies analogical proportions in monounary algebras consisting only of a universe and a single unary function, where we analyze the role of congruences, and we show that the analogical proportion relation is characterized in the infinite monounary algebra formed by the natural numbers together with the successor function via difference proportions.

The article is dedicated to the properties of 1-types in small ordered theories, specifically focusing on a particular generalization of the concept of a successor function on integers known as a quasi-successor. The article includes examples of 1-types, present in both discrete and dense orders, on which the 2-formula φ(x, y) acts, which is a quasi-successor. For the first time, the concept of quasisuccessor was introduced by the authors in their previous works. The problems of properties and counting the number of countable models of theories with a definable linear order have been studied by many scientists, among them L. Mayer, S.V. Sudoplatov, B.Sh. Kulpeshov, B.S. Baizhanov, A. Alibek, T.S. Zambarnaya, S. Moconja and P. Tanovic. Formulas with properties of succession play an important role in the study of the case of maximal countable models. Using the compactness theorem for a non-isolated 1-type, it is possible to construct a model in which there is an infinite discrete order of the type ω* + ω. On this discrete order, a complete 1-type with the quasi-successor property can be distinguished. A complete description of the subclass of theories having the maximal countable spectrum opens p the possibility of describing a possible countable spectrum for theories with a definable linear order.

The paper offers a thorough study of multiorders and their applications to measure-preserving actions of countable amenable groups. By a multiorder on a countable group, we mean any probability measure \nu on the collection \widetilde{\mathcal O} of linear orders of type \mathbb Z on G , invariant under the natural action of G on such orders. Multiorders exist on any countable amenable group (and only on such groups) and every multiorder has the Følner property, meaning that almost surely the order intervals starting at the unit form a Følner sequence. Every free measure-preserving G -action (X,\mu,G) has a multiorder (\widetilde{\mathcal O},\nu,G) as a factor and has the same orbits as the \mathbb Z -action (X,\mu,S) , where S is the successor map determined by the multiorder factor. Moreover, the sub-sigma-algebra \Sigma_{\widetilde{\mathcal O}} associated with the multiorder factor is invariant under S , which makes the corresponding \mathbb Z -action (\widetilde{\mathcal O},\nu,\widetilde{S}) a factor of (X,\mu,S) . We prove that the entropy of any G -process generated by a finite partition of X , conditional with respect to \Sigma_{\widetilde{\mathcal O}} , is preserved by the orbit equivalence with (X,\mu,S) . Furthermore, this entropy can be computed in terms of the so-called random past, by a formula analogous to h(\mu,T,\mathcal P)=H(\mu,\mathcal P|\mathcal P^-) known for \mathbb Z -actions. The above fact is then applied to prove a variant of a result by Rudolph and Weiss (2000). The original theorem states that orbit equivalence between free actions of countable amenable groups preserves conditional entropy with respect to a sub-sigma-algebra \Sigma , as soon as the “orbit change” is measurable with respect to \Sigma . In our variant, we replace the measurability assumption by a simpler one: \Sigma should be invariant under both actions and the actions on the resulting factor should be free. In conclusion, we provide a characterization of the Pinsker sigma-algebra of any G -process in terms of an appropriately defined remote past arising from a multiorder. The paper has an appendix in which we present an explicit construction of a particularly regular (uniformly Følner) multiorder based on an ordered dynamical tiling system of G .

We investigate number and arithmetic learning among a Bolivian indigenous people, the Tsimane', for whom formal schooling is comparatively recent in history and variable in both extent and consistency. We first present a large-scale meta-analysis on child number development involving over 800 Tsimane' children. The results emphasize the impact of formal schooling: Children are only found to be full counters when they have attended school, suggesting the importance of cultural support for early mathematics. We then test especially remote Tsimane' communities and document the development of specialized arithmetical knowledge in the absence of direct formal education. Specifically, we describe individuals who succeed on arithmetic problems involving the number five-which has a distinct role in the local economy-even though they do not succeed on some lower numbers. Some of these participants can perform multiplication with fives at greater accuracy than addition by one. These results highlight the importance of cultural factors in early mathematics and suggest that psychological theories of number where quantities are derived from lower numbers via repeated addition (e.g., a successor function) are unlikely to explain the diversity of human mathematical ability.

An undecidability result for Separation Logic with theory reasoning

Abstract In [Appl. Comput. Harmon. Anal., 46 (2019), 664673] O. Christensen and M. Hasannasab observed that assuming the existence of an operator T sending en to e n +1 for all n ∈ ℕ (where (e n ) n ∈ℕ is a sequence of vectors) guarantees that (e n ) n ∈ℕ is linearly independent if and only if dim{e n } n ∈ℕ = ∞. In this article, we recover this result as a particular case of a general order-theory-based model-theoretic result. We then return to the context of vector spaces to show that, if we want to use a condition like T(e i ) = e ϕ ( i ) for all i ∈ I where I is countable as a replacement of the previous one, the conclusion will only stay true if ϕ : I → I is conjugate to the successor function succ : n ↦n + 1 defined on ℕ. We finally prove a tentative generalization of the result, where we replace the condition T(e i ) = e ϕ ( i ) for all i ∈ I where ϕ is conjugate to the successor function with a more sophisticated one, and to which we have not managed to find a new application yet.

Given an undirected graph G on n nodes and m edges in the form of a data stream we study the problem of finding an Euler tour in G. Our main result is the first one-pass streaming algorithm computing an Euler tour of G in the form of an edge successor function with only mathcal O(nlog (n)) RAM, which is optimal for this setting (e.g. Sun and Woodruff (2015)). Since the output size can be much larger, we use a write-only tape to gradually output the solution. The previously best-known result for finding Euler tours in data streams is implicitly given by the W-stream algorithm of Demetrescu et al. (2010) using mathcal O(m/n) passes under the same RAM limitation. Our approach is to partition the edges into edge-disjoint cycles and to merge the cycles until a single Euler tour is achieved. In the streaming environment such a merging is far from being obvious as the limited RAM allows the processing of only a constant number of cycles at once. This enforces merging of cycles that partially are no longer present in RAM. We solve this problem with a new edge swapping technique, for which storing two certain edges per node is sufficient to merge tours without having all tour edges in RAM. The mathematical key is to model tours and their merging in an algebraic way, where certain equivalence classes represent subtours. This quite general approach might be of interest also in other routing problems.

Arithmetic thinking as the basis of children's generative number concepts

A multi-state dependent model is proposed for integrated pest management, which adopts different control methods at different thresholds. The sufficient conditions for the existence of order one periodic solution to the system are obtained by using differential equation geometry theory and successor function. Furthermore, we have discussed the existence of order-k (k ≥ 2) periodic solution by using series convergence. Besides we have proved the order one periodic solution is orbitally asymptotically stable under certain conditions with analogue of the Poincare criterion. Finally, numerical simulations are given to show the feasibility of our main results. Especially, the proved process of the existence of order one periodic solution shows that our method used in this paper is easier than the existing methods.

Based on the practice of controlling water quality, a kind of nutrition-phytoplankton model with nutrient cycle and state feedback control was established on the basis of the chemostat model. Firstly, the qualitative and stability of the continuous model were studied, and the conditions for the existence and stability of equilibria were given.Then, by using the theory of semi-continuous dynamical system and the method of successor function, the conditions under which the model has an unique and asymptotically stable periodic solution were obtained. Numerical simulations verified the theoretical results. The results showed that the phytoplankton such as algae can be controlled below the given threshold under the state feedback control.

In this paper, a general predator–prey model with state-dependent impulse is considered. Based on the geometric analysis and Poincaré map or successor function, we construct three typical types of Bendixson domains to obtain some sufficient conditions for the existence of order-1 periodic solutions. At the same time, the existing domains are discussed with respect to the system parameters. Moreover, the Analogue of Poincaré Criterion is used to obtain the asymptotic stability of the periodic solutions. Finally, to illustrate the results, an example is presented and some numerical simulations are carried out.