Kruskal's Theorem Research Articles

On unique tensor rank decomposition of 3-tensors

We answer to a question posed recently in reference [Lovitz B, Petrov F. A generalization of Kruskal's theorem on tensor decomposition. Available at arXiv 2103.15633; 2021], proving the conjectured sufficient minimality and uniqueness condition of the 3-tensor decomposition.

Modern perspectives in Proof Theory.

Modern perspectives in Proof Theory.

The uniform Kruskal theorem: between finite combinatorics and strong set existence.

The uniform Kruskal theorem extends the original result for trees to general recursive data types. As shown by Freund, Rathjen and Weiermann(Freund, Rathjen, Weiermann 2022 Adv. Math. 400, 108265 (doi:10.1016/j.aim.2022.108265)), it is equivalent to [Formula: see text]-comprehension, over [Formula: see text] with the ascending descending sequence principle ([Formula: see text]). This result provides a connection between finite combinatorics and abstract set existence. The present article sheds further light on this connection. Firstly, we show that the original Kruskal theorem is equivalent to the uniform version for data types that are finitely generated. Secondly, we prove a dichotomy result for a natural variant of the uniform Kruskal theorem. On the one hand, this variant still implies [Formula: see text]-comprehension over [Formula: see text] extended by the chain antichainprinciple ([Formula: see text]). On the other hand, it becomes weak when [Formula: see text] is removed from the base theory. This article is part of the theme issue 'Modern perspectives in Proof Theory'.

Normal functions and maximal order types

AbstractTransformations of well partial orders induce functions on the ordinals, via the notion of maximal order type. In most examples from the literature, these functions are not normal, in marked contrast with the central role that normal functions play in ordinal analysis and related work from computability theory. The present paper aims to explain this phenomenon. In order to do so, we investigate a rich class of order transformations that are known as $\textsf {WPO}$-dilators. According to a first main result of this paper, $\textsf {WPO}$-dilators induce normal functions when they satisfy a rather restrictive condition, which we call strong normality. Moreover, the reverse implication holds as well, for reasonably well-behaved $\textsf {WPO}$-dilators. Strong normality also allows us to explain another phenomenon: by previous work of Freund, Rathjen and Weiermann, a uniform Kruskal theorem for $\textsf {WPO}$-dilators is as strong as $\varPi ^1_1$-comprehension, while the corresponding result for normal dilators on linear orders is equivalent to the much weaker principle of $\varPi ^1_1$-induction. As our second main result, we show that $\varPi ^1_1$-induction is equivalent to the uniform Kruskal theorem for $\textsf {WPO}$-dilators that are strongly normal.

A Note on Weaker Conditions for Identifying Restricted Latent Class Models for Binary Responses.

Restricted latent class models (RLCMs) are an important class of methods that provide researchers and practitioners in the educational, psychological, and behavioral sciences with fine-grained diagnostic information to guide interventions. Recent research established sufficient conditions for identifying RLCM parameters. A current challenge that limits widespread application of RLCMs is that existing identifiability conditions may be too restrictive for some practical settings. In this paper we establish a weaker condition for identifying RLCM parameters for multivariate binary data. Although the new results weaken identifiability conditions for general RLCMs, the new results do not relax existing necessary and sufficient conditions for the simpler DINA/DINO models. Theoretically, we introduce a new form of latent structure completeness, referred to as dyad-completeness, and prove identification by applying Kruskal's Theorem for the uniqueness of three-way arrays. The new condition is more likely satisfied in applied research, and the results provide researchers and test-developers with guidance for designing diagnostic instruments.

Minimal bad sequences are necessary for a uniform Kruskal theorem

Kruskal's theorem on trees is a classical result of combinatorics with important applications in computer science. The minimal bad sequence argument of Nash-Williams yields a proof that is very elegant but not as elementary as possible, by previous results from mathematical logic. In the present paper, we formulate a uniform Kruskal theorem, which extends the original theorem from trees to general recursive data types. Our main result shows that the minimal bad sequence argument does yield the most elementary proof of this uniform Kruskal theorem. So in the uniform case, the elegant proof coincides with the most elementary one. In more precise and technical terms, we work in the framework of reverse mathematics, where we show that the uniform Kruskal theorem is equivalent to Π11-comprehension, over RCA0 extended by the chain-antichain principle. As a by-product of our investigation, we obtain uniform proofs of several Kruskal-type independence results.

From Kruskal’s theorem to Friedman’s gap condition

AbstractHarvey Friedman’s gap condition on embeddings of finite labelled trees plays an important role in combinatorics (proof of the graph minor theorem) and mathematical logic (strong independence results). In the present paper we show that the gap condition can be reconstructed from a small number of well-motivated building blocks: It arises via iterated applications of a uniform Kruskal theorem.

Excitation-emission fluorescence-kinetic third-order/four-way data: Determination of bisphenol A and nonylphenol in food-contact plastics

The endocrine disrupting chemicals bisphenol A (BPA) and 4-nonylphenol (NP) were simultaneously quantified through third-order/four-way calibration. Excitation-emission fluorescence matrix-kinetic (EEFM-K) third-order data were generated by measuring the EEFMs of these priority xenoestrogens as a function of reaction time during their Fenton degradation. Third-order/four-way calibration notably improves the sensitivity of the method and provides the required selectivity for quantifying analytes with critically overlapped fluorescence signals. In fact, collinearity between BPA and NP emission spectra prevented their quantification using EEFM second-order data and three-way PARAFAC (parallel factor analysis); however, the addition of a third instrumental mode allowed the correct chemometric modeling with four-way PARAFAC. In this way, the compliance of Kruskal's theorem extended to higher-order data was verified. The method was applied for the determination of the analytes in samples of different plastic materials, which are in contact with food and/or beverages. In these cases, where unmodelled constituents are present, good results for BPA were achieved with four-way PARAFAC, but the predictions for NP using this model were deficient. A better predictive capability for NP in real samples was achieved when either U-PLS/RTL (unfolded partial least-squares combined with residual trilinearization) or MCR-ALS (multivariate curve resolution with alternating least-squares) was applied for data processing, demonstrating the power of these latter models for the resolution of more complex systems.

Analysis of a Double Kruskal Theorem

The strength of an extension of Kruskal's Theorem to certain pairs of cohabitation trees is calibrated.

The Kruskal-Katona Theorem and a Characterization of System Signatures

We show how to determine if a given vector can be the signature of a system on a finite number of components and, if so, exhibit such a system in terms of its structure function. The method employs combinatorial results from the theory of (finite) simplicial complexes, and provides a full characterization of signature vectors using a theorem of Kruskal (1963) and Katona (1968). We also show how the same approach can provide new combinatorial proofs of further results, e.g. that the signature vector of a system cannot have isolated zeroes. Finally, we prove that a signature with all nonzero entries must be a uniform distribution.

The Kruskal-Katona Theorem and a Characterization of System Signatures

We show how to determine if a given vector can be the signature of a system on a finite number of components and, if so, exhibit such a system in terms of its structure function. The method employs combinatorial results from the theory of (finite) simplicial complexes, and provides a full characterization of signature vectors using a theorem of Kruskal (1963) and Katona (1968). We also show how the same approach can provide new combinatorial proofs of further results, e.g. that the signature vector of a system cannot have isolated zeroes. Finally, we prove that a signature with all nonzero entries must be a uniform distribution.

The Kruskal-Katona Theorem and a Characterization of System Signatures

We show how to determine if a given vector can be the signature of a system on a finite number of components and, if so, exhibit such a system in terms of its structure function. The method employs combinatorial results from the theory of (finite) simplicial complexes, and provides a full characterization of signature vectors using a theorem of Kruskal (1963) and Katona (1968). We also show how the same approach can provide new combinatorial proofs of further results, e.g. that the signature vector of a system cannot have isolated zeroes. Finally, we prove that a signature with all nonzero entries must be a uniform distribution.

Well-foundedness for termination of term rewriting systems

Well-foundedness is related to an important property of rewriting systems, namely termination. A well-known technique to prove well-foundedness on term orderings is Kruskal's theorem, which implies that a monotonic term ordering over a finite signature satisfying the subterm property is well-founded. However, it does not seem to work for a number of terminating term rewriting systems. In this paper, it is shown that a term ordering possessing subterm property and decomposability ( in place of monotonicity) does yield a simpler proof of well-foundedness for terminating term rewriting systems than the techniques depending on Kruskal's theorem.

Mini-workshop: Logic, Combinatorics and Independence Results

The mini-workshop “Logic, Combinatorics and Independence results” took place on November 25–December 2, 2006. The workshop was organized by Andrey Bovykin, Lorenzo Carlucci and Andreas Weiermann and attended by 16 participants: There were 16 one-hour talks, two problem sessions and many one-to-one and small group discussions. The workshop brought together researchers specialising in several connected disciplines: Reverse Mathematics (Alberto Marcone, Joseph Mileti, Antonio Montalbán), Proof Theory (Lev Beklemishev, Wilfried Buchholz, Lev Gordeev, Michael Rathjen, Sergei Tupailo, Stanley Wainer, Andreas Weiermann), WQO theory (Alberto Marcone, Antonio Montalbán, Lev Gordeev, Andreas Weiermann), Models of Arithmetic (Andrey Bovykin, Henryk Kotlarski, Konrad Zdanowski, Alan Woods), weak arithmetics (Lev Beklemishev, Alan Woods, Konrad Zdanowski), logical aspects of finite Ramsey Theory (Andrey Bovykin, Lorenzo Carlucci, Henryk Kotlarski, Joseph Mileti, Eran Omri, Andreas Weiermann). However, the central theme of the workshop was first-order unprovable statements and statements of large logical strength. The subject originated in the late 1970s in the work of several mathematicians, most notably Jeff Paris and Harvey Friedman and attracted a large community of researchers at that time. The discoveries of the Paris–Harrington Principle and unprovability of Kruskal's Theorem provided, fourty years after Gödel's theorems, the first examples of mathematically natural unprovable statements. Since then, many other examples were found in Ramsey Theory, Graph Theory, well-quasi-order theory and other subjects. One of the main objectives of the workshop was to revive research in this area, especially in view of some spectacular recent developments. These developments revealed deep connections between the study of logical strength and several mathematical disciplines: Analytic Combinatorics, Graph Theory, Tauberian Theory, Number Theory, Dynamical Systems. Another objective of the workshop was to stimulate communication and joint research between researchers from different sub-areas of the subject (Ordinal Analysis, Reverse Mathematics, Models of Arithmetic). All these different areas were represented at the workshop by leading researchers. The workshop was very successful in setting grounds for comparison and interaction of methods from these areas. The two open problem sessions resulted in a list of problems of common interest and in better understanding of possible directions for future research. The talks varied from reports on recent results and proposals of new general approaches to discussions of new strategies to tackle long-standing open problems. Time allowed for free informal discussion and research in small groups that will eventually result in publications. We would like to thank the Oberwolfach Institute for the wonderful opportunity to hold a meeting there and for providing NSF travel grants to some participants. We also thank all the staff working in the institute for the pleasant experience we all had during our week in Oberwolfach Institute.

Algorithm for finding structures and obstructions of tree ideals

Let I be any topological minor closed class of trees (a tree ideal). A classical theorem of Kruskal [Well-quasi-ordering, the Tree Theorem, and Vazsonyi's conjecture, Trans. Am. Math. Soc. 95 (1960) 210–223] states that the set O ( I ) of minimal non-members of I is finite. On the other hand, a finite structural description S ( I ) is developed by Robertson, et al. [Structural descriptions of lower ideals of trees, Contemp. Math. 147 (1993) 525–538]. Given either of the two finite characterizations of I , we present an algorithm that computes the other.

Phase transition thresholds for some Friedman‐style independence results

AbstractWe classify the phase transition thresholds from provability to unprovability for certain Friedman‐style miniaturizations of Kruskal's Theorem and Higman's Lemma. In addition we prove a new and unexpected phase transition result for ε0. Motivated by renormalization and universality issues from statistical physics we finally state a universality hypothesis. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Andreas Weiermann. Complexity bounds for some finite forms of Kruskal's Theorem. Journal of Symbolic Computation, vol. 18 (1994), pp. 463–448. - Andreas Weiermann. Termination proofs for term rewriting systems with lexicographic path ordering imply multiply recursive derivation lengths. Theoretical Computer Science, vol. 139 (1995), pp. 355–362. - Andreas Weiermann. Bounding derivation lengths with functions from the slow growing hierarchy. Archive

Andreas Weiermann. Complexity bounds for some finite forms of Kruskal's Theorem. Journal of Symbolic Computation, vol. 18 (1994), pp. 463–448. - Andreas Weiermann. Termination proofs for term rewriting systems with lexicographic path ordering imply multiply recursive derivation lengths. Theoretical Computer Science, vol. 139 (1995), pp. 355–362. - Andreas Weiermann. Bounding derivation lengths with functions from the slow growing hierarchy. Archive

Nonexistence of a Kruskal-Katona Type Theorem for Subword Orders

We consider the poset SO(n) of all words over an n-element alphabet ordered by the subword relation. It is known that SO(2) falls into the class of Macaulay posets, i. e. there is a theorem of Kruskal–Katona type for SO(2). As the corresponding linear ordering of the elements of SO(2) the vip-order can be chosen.Daykin introduced the V-order which generalizes the vip-order to the n≥2 case. He conjectured that the V-order gives a Kruskal–Katona type theorem for SO(n).We show that this conjecture fails for all n≥3 by explicitly giving a counterexample. Based on this, we prove that for no n≥3 the subword order SO(n) is a Macaulay poset.

Finite Functions and the Necessary Use of Large Cardinals

Finite Functions and the Necessary Use of Large Cardinals

Ordinals II: Some applications and a functorial approach

The first part of this paper is a presentation of some common applications of ordinals: definition of a system of ordinal notations for ordinals less than Γ0, direct connection between Kruskal's theorem and Γ0, consistency proofs in proof theory (such as the consistency of Peano arithmetic by means of transfinite induction up to e0). In the second part of the paper, a functorial construction of ordinals and in particular of the Veblen hierarchy is explained. This approach, introduced by Girard (theory of dilators), allows the construction of ordinals greater than Γ0 to be pursued in a more natural way than if the Bachmann hierarchy is used.