M. H. A. Newman. On theories with a combinatorial definition of equivalence. Annals of Mathematics, 43(2):223--243, 1942.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....For term rewrite systems confluence can be shown using the concept of critical pairs. Critical pairs which can be detected and analyzed statically, represent potential conflicts in a minimal context. If the rewrite system is terminating, confluence follows if all critical pairs can be joined [15]. This theory of critical pairs and confluence has been transferred to transformation systems on term graphs and hyper graphs [17, 18] However, in most applications of graph transformation to visual modelling techniques, attributed graphs are used to represent diagrams with textual, numerical, ....

....X ) contain Boundary , i.e. it satisfies the gluing condition. By Proposition 3 there are transformations G p1 (o # i ) # H # i =# Y with o # i = g k i , thus o # i = o i and hence H # i is isomorphic to H i . As local confluence and termination imply confluence according to Newman s lemma [15], confluence of a terminating consistent graph transformation system can be shown by proving for all critical pairs the property of being confluent. Restricting to vertex preserving transformations and morphisms which are injective up to data vertices, the gluing condition is always satisfied for ....

M. H. A. Newman. On theories with a combinatorial definition of 'equivalence'. In Annals of Mathematics, 43 (2), pages 223--243, 1942.

....1. 4 is locally confluent. 2. is confluent. 3. 4 is confluent. Proof. The first part is proved by inspecting the three critical pairs generated by the rules in the definition of a reduction. The second part follows by the previous lemma and strong normalization of , using Newman s Lemma [New42]. The third part is a consequence of the confluence of a (previous part) and [SS01] using Corollary 4.3. Due to our restriction of reduction, in a a reduction sequence the a steps are always performed before the steps. o The unique a normal form of a term u is denoted by u] Note that the ....

M.H.A. Newman. On theories with a combinatorial definition of equivalence. Annals of Mathematics, 43(2):223 243, 1942.

....let N be the unique normalization function. One can create two different normalization strategies by replacing N(t) by N(t ) and by N(t ) Contradiction. The equivalence between semi completeness and being an isomorphism follows directly from 2.3.1.3 and 2.3.1.4. 2.3.1.7. Newman s lemma [Newman, 1942]. If a term rewriting system is strongly open terminating and weakly open confluent it is strongly open confluent. Proof. We are going to show that in a strongly terminating and weakly confluent term rewriting system there are no terms having more than one different normal form. From this strong ....

M.H.A. Newman, `On Theories with a Combinatorial Definition of Equivalence', Annals of Mathematics 43, 2, 1942, pp. 223-243.

....c there exists an object d such that b d, c d. Gamma is called locally confluent if for every three objects a; b; c such that a b; a c there exists an object d such that b d and c d. The following classical result is sometimes called the Diamond Lemma. Lemma 2. 1 (Newman, [25], see also [9] Every terminating locally confluent rewrite system is confluent. The equivalence relation generated by Gamma , that is the reflexive symmetric transitive closure of Gamma , is denoted by Gamma . An equivalence class of Gamma consists of all vertices of a ....

....which was published only recently. In fact, Squier introduced the 1 skeleton of this complex and a homotopy relation on the set of paths which coincides with the natural homotopy relation induced by K(P) Notice that the idea of considering rewrite systems as graphs goes back to at least Newman [25]. Groves [12] independently introduced the 1 skeleton of K(P) He also gave a definition of a high dimensional complex associated with every 34 string rewriting system. It can be seen that this definition is in some sense dual to ours. A definition of K(P) equivalent to the one presented here ....

M. H. A. Newman. On theories with a combinatorial definition of equivalence. Ann. Math., 43:223--243, 1942.

....for proving the delay monotonicity of T . Lemma 15 Main Lemma. 8 g 2 T : G 1 se G 2 ) 8 f 2 T 9 f 1 ; fn 2 T : f(G 1 ) se fn ffi : ffi f 1 (G 2 ) The following theorem states the desired delay monotonicity result. The reasoning closely resembles the classical Newman Lemma [New42], saying that confluence follows from local confluence if the given relation is wellfounded. Note that monotonicity does not hold. Lemma 16 Delay Monotonicity. T is delay monotone, i.e. 8 f 2 T : G ) 9 f 1 ; fn 2 T : f(G ) se fn ffi : ffi f 1 (G Finally, we have to ....

M.H.A. Newman. On theories with a combinatorial definition of equivalence. Annals of Math., 43,2:223--243, 1942.

....] R r. 2 The significance of this definition becomes evident in the following result: Lemma 2.7 (based on [KB70] A terminating rewrite system R is confluent if each pair of rule instances in R is confluent. Proof: Any terminating rewrite system R is confluent iff it is locally confluent [New42] Hence, it suffices to prove local confluence. Suppose t )R t 1 using rule R 1 = l 1 ) r 1 ) and t )R t 2 using rule R 2 = l 2 ) r 2 ) Thus, there are positions 1 and 2 in t such that t 1 = t 1 [r 1 l 1 ] and t 2 = t 2 [r 2 l 2 ] We obtain the desired t in all possible cases: ....

M. H. A. Newman. On theories with a combinatorial definition of equivalence. Annals of Mathematics, 43(2):223--243, 1942.

....of . This is at least the case if all pairs of direct derivations are independent. G is called confluent, if 8G 1 ; G 2 ; G 3 2 L Sigma : G 2 G 1 G 3 = 9H : G 2 H G 3 . G is called convergent if it is terminating and confluent [DJ90] Such systems deliver a unique normal form [New42] Graph rewrite systems with several normal forms are called non deterministic. 2.2 Termination by edge addition A set of graph rewrite rules S terminates on a finite axiom Z if every derivation step adds some edges to a particular relation. At last when this relation is complete the derivation ....

....direct derivations are always independent. Additionally, if each rule does not form overlaps with itself (this is at least the case if its redexes do not overlap) all pairs of direct derivations from rules in the stratum may be interchanged directly. Thus the stratum is strong confluent and with [New42] yields a unique normal form. If all strata of G yield a unique normal form, and if the strata are computed in stratification order, also the whole derivation process delivers a single normal form. The above heuristic is a generalization of the stratification of Datalog : to graph rewrite ....

M.H.A. Newman. On theories with a combinatorial definition of equivalence. Annals of Mathematics, 43(2):223--243, 1942.

.... is confluent if, for any terms a; b; c such that a b and a c, there exists a term d such that b d and c d. 23 is locally confluent if, for any terms a; b; c such that a b and a c, there exists a term d such that b d and c d. Lemma 6. 2 (Newman) New42] If a relation is strongly normalizable and locally confluent then it is confluent. Definition 6.3 (Critical pair) Given two rewrite rules l 1 r 1 and l 2 r 2 with no variable in common, the critical pair of l 2 r 2 on l 1 r 1 at a non variable position p 2 P os(l 1 ) if it ....

M. H. A. Newman. On theories with A combinatorial definition of equivalence. In Annals of Math, volume 43, pages 223--243, 1942. 31

....reduction property, that is, preservation of types under reductions. The proof is an easy induction on the structure of the reduced term. Fact 16 (Subject reduction) If a : s and a Gamma a 0 then a 0 : s. Full confluence is proved after strong normalization, using Newman s Lemma [23], that is, a strongly normalizable relation that is locally confluent is confluent, and assuming that (i) the first order (algebraic) rewrite rules are confluent, and (ii) there are no critical pairs between any two higher order rules, and between the higher order rules and the first order rules. ....

M. H. A. Newman. On theories with a combinatorial definition of equivalence. In Annals of Math, volume 43, pages 223--243, 1942.

....test for confluence of a terminating program: Definition 5.5. A CHR program is called terminating, if there are no infinite computations. Corollary 5.1. A terminating CHR program is confluent iff all its critical pairs are joinable. Proof. Immediately from Theorem 5. 1 and Newman s lemma [New42]. Our notion of confluence subsumes the notion of determinacy as used by Maher [Mah87] and Saraswat [Sar93] for (concurrent) constraint (logic) programs. In a determinate program, guards of rules for the same predicate are mutually exclusive. Thus they are trivially confluent, since no critical ....

M. H. A. Newman, On Theories with a Combinatorial Definition of Equivalence, Annals of Math, Vol. 43, pp 223--243, 1942.

....of TRS s. 4. 4 Termination A TRS R is said to be terminating iff there exist no infinite rewriting sequences t 0 R t 1 R Delta Delta Delta R t n R Delta Delta Delta When R is terminating the concepts of confluence and local confluence coincide; this result is due to Newman [62] and the proof presented here is due to Barendregt [3] Lemma 4.42 (Newman [62] If R is terminating then R is confluent iff it is locally confluent. Proof. The only if part is trivial. For the if part if R is not confluent then there there exist terms r , s , and t such that r R t R s ....

....no infinite rewriting sequences t 0 R t 1 R Delta Delta Delta R t n R Delta Delta Delta When R is terminating the concepts of confluence and local confluence coincide; this result is due to Newman [62] and the proof presented here is due to Barendregt [3] Lemma 4. 42 (Newman [62]) If R is terminating then R is confluent iff it is locally confluent. Proof. The only if part is trivial. For the if part if R is not confluent then there there exist terms r , s , and t such that r R t R s and r 6 # s . We prove that in these conditions there exist terms r 0 , s , and ....

M. H. A. Newman. On theories with a combinatorial definition of 'equivalence'. Annals of Mathematics, 43:223--243, 1942.

....Local confluence is like confluence except that we replace t t 1 and t t 2 in the above by t t 1 and t t 2 , i.e. t 1 and t 2 can be obtained by exactly one reduction from t. The properties of confluence and termination are undecidable in themselves. However, Newman s theorem [New42] shows that a terminating term rewriting system is confluent if, and only if, it is locally confluent. A procedure exists, due to Knuth and Bendix [KB70] which checks a rule set for local confluence, and if the set is not locally confluent, adds rules to try to make it locally confluent. Used in ....

M.H.A. Newman. On Theories with a Combinatorial Definition of Equivalence. Annals of Mathematics, 43(2):223--243, 1942.

....confluence proofs in the literature. Here, an ARS A = hA; i (or ) is said to be terminating if there exists no infinite reduction sequence a 0 a 1 a 2 : of elements of A) A (or ) is called complete (or convergent) if it is confluent and terminating. Theorem 2. 3 (Newman s Lemma, [New42]) A terminating ARS A = hA; i is confluent if and only if it is locally confluent. 2.2 Term Rewriting Systems A term rewriting system R (over some set T (F ; V) of terms) may be viewed as the ARS hT (F ; V) R i where R or simply is the rewrite relation induced by the rewrite rules l r 2 ....

M.H.A. Newman. On theories with a combinatorial definition of equivalence. Annals of Mathematics, 43(2):223--242, 1942.

....for proving the delay monotonicity of T . Lemma 4.10 (Main Lemma) 8 g 2 T : G 1 g se G 2 ) 8 f 2 T 9 f 1 ; fn 2 T : f(G 1 ) se f n ffi: ffif 1 (G 2 ) The following theorem states the desired delay monotonicity result. The reasoning closely resembles the classical Newman Lemma [17], saying that confluence follows from local confluence if the given relation is wellfounded. Note that monotonicity does not hold. Lemma 4.11 (Delay Monotonicity) T is delay monotone, i.e. 8 f 2 T : G 0 se G 00 ) 9 f 1 ; fn 2 T : f(G 0 ) se f n ffi : ffi f 1 (G 00 ....

M.H.A. Newman. On theories with a combinatorial definition of equivalence. Annals of Mathematics, 43,2:223--243, 1942.

.... R u R u 2 implies u 1 # u 2 , it is locally confluent if u 1 R u R u 2 implies u 1 # u 2 . The Church Rosser property is equivalent to confluence: for all u 1 ; u 2 2 Sigma ; u 1 R u 2 implies u 1 # u 2 . If R is noetherian and locally confluent, then it is also confluent [New42]. A semi Thue system R is canonical if R is confluent and noetherian, then each word u 2 Sigma has an unique irreducible normalform u. If we view a system R as a Thue system as well as a semi Thue system, they define the same monoid MR , i.e. R = R . Therefore the word problem of a ....

....all critical pairs are joinable within their minimal solutions. if: Let P (u) be valid if and only if 8u 1 ; u 2 with u 1 R u R u 2 there exists u 0 2 Sigma such that u 1 # u 0 u 2 . We show that P (u) is valid for all u 2 Sigma by noetherian induction on Sigma , see e.g. [New42]. Figure 2 turns out the existence of u 0 , provided that w 1 R u R w 2 implies 9w 2 Sigma with w 1 # w w 2 . Let us assume that we used the rules (r) 8 n i=1 u i = v i : u 0 v 0 and (r 0 ) 8 n 0 i=1 u 0 i = v 0 i : u 0 0 v 0 0 to reduce u to w 1 resp. w 2 . There are ....

M. H. A. Newman. On theories with a combinatorial definition of equivalence. Annals of Mathematics, 43(2):223--243, 1942.

....to check the following property: for any words X, Y , Z such that X Y , X Z, there exists a word W such that Y W , Z W . Figure 1: u u u u X Y Z A B u u u u X Y Z A B Rewriting systems satisfying ( are called locally confluent. There is a classical result due to Newman [New]: any terminating rewriting system is confluent if and only if it is locally confluent. Let us give some convenient conditions which are equivalent to local confluence. These conditions are well known (see [DJ] Sq] GuPr] C1) If X, Y , Z, A, B are words, Y is nonempty, XY; A) 2 R, Y Z; ....

M.H.A. Newman. On theories with a combinatorial definition of equivalence, Ann. of Math., v. 43 (1943) pp. 223--243.

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M. H. A. Newman. On theories with a combinatorial definition of equivalence. Annals of Mathematics, 43(2):223--243, 1942.

No context found.

M.H.A. Newman. On theories with a combinatorial definition of equivalence. Annals of Mathematics, 43(2):223--243, 1942.

No context found.

M. H. A. Newman, On Theories with a Combinatorial Definition of Equivalence, Annals of Math, Vol. 43, pp 223--243, 1942.

No context found.

M. H. A. Newman. On theories with a combinatorial definition of equivalence. Annals of Mathematics, 43(2):223--243, 1942.

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Newman, M.H.A. (1942). On Theories with a Combinatorial Definition of Equivalence. Annals of Mathematics 43(2), pp. 223--243.

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M.H.A. Newman. On Theories with a Combinatorial Definition of Equivalence. Annals of Mathematics, 43(2):223--243, 1942.

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Newman, M.H.A. (1942). On Theories with a Combinatorial Definition of Equivalence. Annals of Mathematics 43(2), pp. 223--243.

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New42. M. H. A. Newman. On theories with a combinatorial definition of equivalence. In Annals of Math, volume 43, pages 223--243, 1942.

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