A Brief History of Strahler Numbers

Esparza, Javier; Luttenberger, Michael; Schlund, Maximilian

doi:10.1007/978-3-319-04921-2_1

Cited by 20 publications

(26 citation statements)

References 33 publications

(31 reference statements)

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“…An interesting parameter of a tree t is its Horton-Strahler number or Strahler number, see [15] for a recent survey. It can be defined as the value t I under the interpretation I over N which interprets constant symbols a ∈ F 0 by a I = 0 and each symbol f ∈ F n with n > 0 as follows: Let a 1 , .…”

Section: Polynomial Time Solvable Evaluation Problemsmentioning

confidence: 99%

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Tree Compression Using String Grammars

et al. 2017

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We study the compressed representation of a ranked tree by a (string) straight-line program (SLP) for its preorder traversal, and compare it with the well-studied representation by straight-line context free tree grammars (which are also known as tree straight-line programs or TSLPs). Although SLPs turn out to be exponentially more succinct than TSLPs, we show that many simple tree queries can still be performed efficiently on SLPs, such as computing the height and Horton-Strahler number of a tree, tree navigation, or evaluation of Boolean expressions. Other problems on tree traversals turn out to be intractable, e.g. pattern matching and evaluation of tree automata.

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Section: Polynomial Time Solvable Evaluation Problemsmentioning

confidence: 99%

“…, a n ) = a + 1. The Strahler number was first defined in hydrology, but also has many applications in computer science [15] , e.g. to calculate the minimum number of registers required to evaluate an arithmetic expression [17].…”

Section: Polynomial Time Solvable Evaluation Problemsmentioning

confidence: 99%

Tree Compression Using String Grammars

et al. 2017

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“…For instance, using the knowledge that the McCarthy 91-function has dimension at most 2 would allow us to restrict the proof of any program property relating to successful derivations to the program P [2] where P is the set of clauses for the McCarthy 91-function. The notion of dimension of a tree has a long history in science (starting with Geology) which has been detailed by Esparza et al [8]. However, the use of dimension for program verification is more recent.…”

Section: Examplementioning

confidence: 99%

Decomposition by tree dimension in Horn clause verification

Kafle

Gallagher

Ganty

2015

Electron. Proc. Theor. Comput. Sci.

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In this paper we investigate the use of the concept of tree dimension in Horn clause analysis and verification. The dimension of a tree is a measure of its non-linearity -for example a list of any length has dimension zero while a complete binary tree has dimension equal to its height. We apply this concept to trees corresponding to Horn clause derivations. A given set of Horn clauses P can be transformed into a new set of clauses P ≤k , whose derivation trees are the subset of P's derivation trees with dimension at most k. Similarly, a set of clauses P >k can be obtained from P whose derivation trees have dimension at least k + 1. In order to prove some property of all derivations of P, we systematically apply these transformations, for various values of k, to decompose the proof into separate proofs for P ≤k and P >k (which could be executed in parallel). We show some preliminary results indicating that decomposition by tree dimension is a potentially useful proof technique. We also investigate the use of existing automatic proof tools to prove some interesting properties about dimension(s) of feasible derivation trees of a given program.

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“…Intuitively, hardness measures the height of the biggest full binary tree which can be embedded into each tree-like resolution refutation of the formula. This is also known as the Horton-Strahler number of a tree (see [60,24]). In the context of resolution this measure was first introduced in [44,48].…”

Section: Tree-hardnessmentioning

confidence: 99%

Unified Characterisations of Resolution Hardness Measures

Beyersdorff

Kullmann

2014

Lecture Notes in Computer Science

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Abstract. Various "hardness" measures have been studied for resolution, providing theoretical insight into the proof complexity of resolution and its fragments, as well as explanations for the hardness of instances in SAT solving. In this paper we aim at a unified view of a number of hardness measures, including different measures of width, space and size of resolution proofs. Our main contribution is a unified game-theoretic characterisation of these measures. As consequences we obtain new relations between the different hardness measures. In particular, we prove a generalised version of Atserias and Dalmau's result on the relation between resolution width and space from [5].

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A Brief History of Strahler Numbers

Cited by 20 publications

References 33 publications

Tree Compression Using String Grammars

Tree Compression Using String Grammars

Decomposition by tree dimension in Horn clause verification

Unified Characterisations of Resolution Hardness Measures

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