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Simple Gap Termination for Term Graph Rewriting Systems
 In Theory of Rewriting Systems and Its Application
, 1995
"... This paper proves the extension of KruskalFriedman theorem, which is an extension of the ordinary Kruskal's theorem with gapcondition, on !trees (Main theorem 1 in section 3). Based on the theorem, a new termination criteria for cyclic term graph rewriting systems, named simple gap terminati ..."
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This paper proves the extension of KruskalFriedman theorem, which is an extension of the ordinary Kruskal's theorem with gapcondition, on !trees (Main theorem 1 in section 3). Based on the theorem, a new termination criteria for cyclic term graph rewriting systems, named simple gap termination (Main theorem 2 in section 4), is proposed where the naive extension of simple termination (based on [Lav78]) does not work well. 1 Introduction A term graph rewriting system (TGRS) has been commonly used from efficiency reasons in implementations of a term rewriting system (TRS), such as CLEAN 1 . A TGRS can be regarded as a TRS with addresses  i.e., a variable in a rule of a TRS is regarded as an address in a TGRS. Thus, subterms will be shared in each reduction step of a TGRS, whereas each reduction step of a TRS simply copies. Theoretical basis for a TGRS has been extensively worked [MSvE94], but the most works have been devoted to a acyclic TGRS. For a cyclic TGRS which can simulate i...
A Gap Tree Theorem for QuasiOrdered Labels 1
"... Given a quasiordering of labels, a labelled ordered tree s is embedded with gaps in another tree t if there is an injection from the nodes of s into those of t that maps each edge in s to a unique disjoint path in t with greaterorequivalent labels, and which preserves the order of children. We sh ..."
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Given a quasiordering of labels, a labelled ordered tree s is embedded with gaps in another tree t if there is an injection from the nodes of s into those of t that maps each edge in s to a unique disjoint path in t with greaterorequivalent labels, and which preserves the order of children. We show that finite trees are wellquasiordered with respect to gap embedding when labels are taken from an arbitrary wellquasiordering provided every tree path can be partitioned into a bounded number of subpaths of comparable nodes. This extends Kˇríˇz’s combinatorial result [7] to partially ordered labels.
Strong WQO phase transitions Preliminary report 1
"... We elaborate Weiermannstyle phase transitions for Gordeev wellquasiordering (called wellpartialordering below, abbr.: wpo) results with respect to nested finite sequences and nested finite trees under the homeomorphic embedding with symmetrical gap condition. For every nested partial ordering i ..."
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We elaborate Weiermannstyle phase transitions for Gordeev wellquasiordering (called wellpartialordering below, abbr.: wpo) results with respect to nested finite sequences and nested finite trees under the homeomorphic embedding with symmetrical gap condition. For every nested partial ordering in question, �, we fix a natural extension of Peano Arithmetic, T, that proves the corresponding 2order sentence SPQ(�). Furthermore, we consider the appropriate parametrized 1order slow wellpartialordering sentence SWPO (�,r) with r ranging over computable reals and show that for some computable real α, the following holds. 1. If r < α then SWPO (�,r) is provable in PA. 2. If r> α then SWPO (�,r) is not provable in T. In the limit cases we replace computable reals r by computable functions f: N → R and prove analogous theorems. In whole generality, these results strengthen both KruskalFriedmanKriz wqo theorems and Weiermann phase transition results concerning KruskalFriedmanSchütteSimpson cases. 2 Preliminaries (1D case) 2.1 Partial and linear well orderings • By � and ≤ we denote partial and linear countable well orderings (abbr.: wpo and wo), respectively. A wo O = (W, ≤) is called a linearization of a wpo W = (W,�) iff (∀x,y ∈ W) (x � y → x ≤ y). A wpo W = (W,�) is called enumerated iff it is supplied with a bijection, also called enumeration, ν: N → W. For any enumerated wpo W = (W,ν,�) we fix its lexicographical linearization Wν = (W, ≤ν) that is defined as follows W × W ∋ x ≤ν y: ⇔ (∀i ∈ N) (ν (i) � x ← → ν (i) � y) ∨ (∃i ∈ N)
www.math.ohiostate.edu/~friedman/ TABLE OF CONTENTS
"... this paper as reaxiomatizations of set theory. A vital feature of the standard set theories associated with the axiomatizations presented here is that they are missing the axiom of choice. This is an essential feature. For instance, ZF does not prove the existence of a standard model of each theorem ..."
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this paper as reaxiomatizations of set theory. A vital feature of the standard set theories associated with the axiomatizations presented here is that they are missing the axiom of choice. This is an essential feature. For instance, ZF does not prove the existence of a standard model of each theorem of ZFC; in fact, ZF does not prove the existence of a standard model of Zermelo set theory with the axiom of choice. Thus in this paper, we relate our axiomatizations to extensions of ZF by large cardinal axioms. In each case, we have chosen an appropriate version of the large cardinal axiom so that if ZF is replaced by ZFC then the resulting system is equivalent to a system which is familiar in the set theory literature. But one would like to know the relationship between the system with ZF and the system with ZFC. This relationship cannot be gauged by considering standard models. {PAGE } The normal way of gauging this relationship is through
KruskalFriedman Gap . . .
, 2002
"... We investigate new extensions of the KruskalFriedman theorems concerning wellquasiordering of finite trees with the gap condition. For two labelled trees s and t we say that s is embedded with gap into t if there is an injection from the vertices of s into t which maps each edge in s to a unique ..."
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We investigate new extensions of the KruskalFriedman theorems concerning wellquasiordering of finite trees with the gap condition. For two labelled trees s and t we say that s is embedded with gap into t if there is an injection from the vertices of s into t which maps each edge in s to a unique path in t with greaterorequal labels. We show that finite trees are wellquasiordered with respect to the gap embedding when the labels are taken from an arbitrary wellquasiordering and each tree path can be partitioned into k ∈ N or less comparable subpaths. This result generalizes both [Kˇrí89] and [OT87], and is also optimal in the sense that unbounded partiality over tree paths yields a counter example.
Phase transitions in Proof Theory
"... Using standard methods of analytic combinatorics we elaborate critical points (thresholds) of phase transitions from provability to unprovability of arithmetical wellpartialordering assertions in several familiar theories occurring in the reverse mathematics program. ..."
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Using standard methods of analytic combinatorics we elaborate critical points (thresholds) of phase transitions from provability to unprovability of arithmetical wellpartialordering assertions in several familiar theories occurring in the reverse mathematics program.