G. E. Mints. Teorija categorii i teoria dokazatelstv.i. Aktualnye problemy logiki i metodologii nauky, pages 252--278, 1979.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... x: x:fx ) f Similar problems arise if we try to enrich the calculus with extra rewrite rules which may be confluent by themselves, but which when taken in conjunction with j contraction fail to be confluent [16] Recently several researchers [2,15,20,19,22, 49] have adopted older proposals [41,62,68] that j conversion be interpreted as an expansion: t ) x:tx if t : A B and the resulting rewrite relation has been shown confluent. In these works infinite reduction sequences such as: f ) x:fx ) x: y:fy)x ) 1.2) are prohibited by imposing syntactic restrictions to limit the ....

....and then contracting any remaining fi redexes. Historically the use of j expansions, as opposed to j contractions, can be traced back to Prawitz [68] and Huet [41] The formulation of the restrictions on expansion required to recover strong normalisation were originally proposed by Mints [62] although it is only recently that several researchers [2,15,19,22,49] using different proof strategies, have proved confluence and strong normalisation. The Linear Calculus, Although substantial agreement exists on the nature of rewriting for final type constructors such as those ....

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G. E. Mints. Teorija categorii i teoria dokazatelstv.i. Aktualnye problemy logiki i metodologii nauky, pages 252--278, 1979.

....rules, rather than contractions. Building on such ideas, but motivated by the study of coherence problems in category theory, Mints gives a first faulty proof that in the typed framework expansion rules, if handled with care, are weakly normalizing and preserve confluence of the typed calculus [Min79] 3 . This idea of using expansion rules seems to have passed unnoticed for a long time, even if the so called j long normal forms were well known and used in the study of higher order unification problems [Hue76] only in these last years there has been a renewed interest in expansion rules. In ....

G. Mints. Teorija categorii i teoria dokazatelstv.I. Aktualnye problemy logiki i metodologii nauky, pages 252--278, 1979.

....Example 2 2 Fortunately, this inconvenient situation can be overcome by turning the extensional equalities into expansion rules. This idea was suggested by Prawitz [Pra71] in the field of proof theory, and then further studied by Huet [Hue76] in the field of higher order unification, and by Mints [Min79] who was motivated by problems in category theory. Expansions, however, seem to have passed unnoticed for a long time, until recent years, when it has been shown that they can be perfectly combined with typed calculus [CK94, Aka93, Dou93, Jay92, JG95, Cub92] and with many other reduction ....

....section clarifies that the reduction relations E Gamma and b E Gamma are strongly normalizing and confluent. The confluence property is shown by Newman s lemma, that is, by the local confluence and the strong normalization properties. For the strong normalization property, the proof of [Min79] is formalized for the typed calculus: to every reduction relation C 2 fE; b Eg and to every term M , a natural number EC (M) is assigned and it is proven that M C Gamma M 0 implies EC (M) EC (M 0 ) This shows that EC (M) is an upper bound for the length of any C reduction sequence. ....

Grigori Mints. Teorija categorii i teoria dokazatelstv. I. Aktualnye problemy logiki i metodologii nauky, pages 252--278, 1979.

....expansion rule will not be terminating as the following infinite reduction sequence shows: a # c Eta #(a[#] 1) # c Eta #( #(a[#] 1) #] 1) Lemma 16 Strong normalization of d Eta. The reduction # c Eta is strongly normalizing. Proof. Using a decreasing measure as defined in [34]. It is also worth noticing that even if a term a is not a Lambda Abstraction according to definition 13, its W normal form may be one: for example 1[cons W (#1) is not a Lambda Abstraction but W (1[cons W (#1) #1 is. The same happens with the notion of Applied Terms. This remark ....

G. Mints. Teorija categorii i teoria dokazatelstv.I. Aktualnye problemy logiki i metodologii nauky, pages 252--278, 1979.

....form, does not represent any term at the object level. The j expansion reduction rule has been thoroughly studied (see [CK93] Aka93] JG95] Adopting it forces us to restrict the reduction rules in some way if we still want Strong Normalization (these restrictions were rst introduced by [Min79]) Thus the reduction we will consider will not be a congruence (more precisely it will not be compatible with the application) and this will induce slight changes in the usual schemes of the proofs of the Church Rosser and Strong Normalization properties. RR n# 3322 14 Pierre Leleu (fi) Delta ....

G.E. Mints. Teorija categorii i teoria dokazatelstv.l. Aktualnye Problemy logiki i metodologii nauky, pages 252278, 1979.

....f : 1 1 and : 1 and with rewrite rule fx ) then ) is confluent. However, as the above counterexample shows shows, the combination of ) with the contractive j rewrite rule fails to be confluent see [5] for further details. Recently several authors [1, 4, 6, 14] have accepted the old proposal [13, 16, 17] that j conversion be interpreted as an expansion f ) x:fx and the resulting rewrite relation has been shown confluent. Infinite reduction sequences such as x:t ) y: x:t)y ) y:t[y=x] j x:t tu ) x:tx)u ) tu (1) are avoided by imposing syntactic restrictions to limit the possibilities for ....

G. E. Mints. Teorija categorii i teoria dokazatelstv.i. Aktualnye problemy logiki i metodologii nauky, pages 252--278, 1979.

....rules, rather than contractions. Building on such ideas, but motivated by the study of coherence problems in category theory, Mints gives a first faulty proof that in the typed framework expansion rules, if handled with care, are weakly normalizing and preserve confluence of the typed calculus [Min79] 3 . This idea of using expansion rules seems to have passed unnoticed for a long time, even if the so called j long normal forms were well known and used in the study of higher order unification problems [Hue76] only in these last years there has been a renewed interest in expansion rules. In ....

Gregory Mints. Teorija categorii i teoria dokazatelstv.I. Aktualnye problemy logiki i metodologii nauky, pages 252--278, 1979.

.... ) This relation is confluent, but when taken together with j contractions confluence is lost as equation 2 demonstrates for a detailed discussion the reader should consult [3] These deficiencies in j contractions have recently led several authors [1, 4, 6, 12] to reconsider the old proposal [11, 13, 14] that j conversion be interpreted as an expansion f ) x : T:fx if f : T T 0 and x 6 2FV(f) and the resulting rewrite relation has been shown confluent. In these works infinite reduction sequences such as f ) x : T:fx ) x : T: y : T:fy)x ) are avoided by imposing syntactic restrictions ....

G. E. Mints. Teorija categorii i teoria dokazatelstv.i. Aktualnye problemy logiki i metodologii nauky, pages 252--278, 1979.

....example, if we regard 1 as a base type with constants f : 1 1 and : 1 and with rewrite rule fx ) then ) is confluent while the divergence above shows that the combination of ) with the contractive j rewrite rule is not confluent. Recently several authors [5, 6, 13] have accepted the old proposal [12, 14, 15] that j conversion be interpreted as an expansion f ) x : T:fx (where T is the type of x) and the resulting rewrite relation has been shown confluent. In these works infinite reduction sequences such as f ) x : T:fx ) x : T: y : T:fy)x ) are avoided by imposing syntactic restrictions to ....

G. E. Mints. Teorija categorii i teoria dokazatelstv.i. Aktualnye problemy logiki i metodologii nauky, pages 252--278, 1979.

....# equality the other way around into an expansion instead of a contraction (# e ) M # #x.Mx if M : A # B and x ## FV (M) the previous counterexample simply goes away as it becomes f #e # #x.fx alg # #x.a. The expansive interpretation of extensional equalities was pioneered by Mints [Min79] and in the last years there has been an increasing interest in this reading of extensional rules, see for example [Aka93, Dou93, DCK94b, Cub92, JG92] where the applicability of the # e rule to a subterm M is restricted in order to guarantee strong normalization, as follows (see [DCK94b] for a ....

Gregory Mints. Teorija categorii i teoria dokazatelstv.I. Aktualnye problemy logiki i metodologii nauky, pages 252--278, 1979.

....closed. This inconvenient can be fortunately overcome in the LWPP calculus by turning the extensional equalities into expansion rules. The idea was suggested by Prawitz [Pra71] in the field of proof theory, and further studied by Huet [Hue76] in the field of higher order unification, and by Mints [Min79] who was motivated by problems in category theory. But expansions seem however to have passed unnoticed for a long time, until these last two years, where many extensional calculi (without patterns) have been investigated [Jay92, JG92, Cub92, Dou93, Aka93, DCK93a] Mainly inspired by [Aka93, ....

....of the relation B = in terms of the confluence property of the relation I. 2 3.2 Strong Normalization and Confluence of E = We show in this section that the reduction relation E = is strongly normalizing and confluent. For the strong normalization property, we adapt the proof of [Min79] for the typed calculus. We assign to every term T a natural number E(T ) and we prove that T E = T 0 implies E(T ) E(T 0 ) which shows that E(T ) is an upper bound for the length of any E = reduction sequence. We note jAj the number of operations and base types in the type A. For ....

Grigori Mints. Teorija categorii i teoria dokazatelstv.I. Aktualnye problemy logiki i metodologii nauky, pages 252--278, 1979.

.... ) This relation is confluent, but when taken together with j contractions confluence is lost as equation 2 demonstrates for a detailed discussion the reader should consult [3] These deficiencies in j contractions have recently led several authors [1, 2, 6, 12] to reconsider the old proposal [11, 13, 14] that j conversion be interpreted as an expansion f ) x T :fx if f : T T 0 and x 6 2FV(f) and the resulting rewrite relation has been shown confluent. In these works infinite reduction sequences such as f ) x T :fx ) x T : y T :fy)x ) are avoided by imposing syntactic ....

G. E. Mints. Teorija categorii i teoria dokazatelstv.i. Aktualnye problemy logiki i metodologii nauky, pages 252--278, 1979.

.... x: x:fx ) f Similar problems arise if we try to enrich the calculus with extra rewrite rules which may be confluent by themselves, but which when taken in conjunction with j contraction fail to be confluent [16] Recently several researchers [2,15,20,19,22, 49] have adopted older proposals [41,62,68] that j conversion be interpreted as an expansion: t ) x:tx if t : A B and the resulting rewrite relation has been shown confluent. In these works infinite reduction sequences such as: f ) x:fx ) x: y:fy)x ) 1.2) are prohibited by imposing syntactic restrictions to limit the possibilities ....

....and then contracting any remaining fi redexes. Historically the use of j expansions, as opposed to j contractions, can be traced back to Prawitz [68] and Huet [41] The formulation of the restrictions on expansion required to recover strong normalisation were originally proposed by Mints [62] although it is only recently that several researchers [2,15,19,22,49] using different proof strategies, have proved confluence and strong normalisation. The Linear Calculus, I; Omega ; Although substantial agreement exists on the nature of rewriting for final type constructors such as those ....

[Article contains additional citation context not shown here]

G. E. Mints. Teorija categorii i teoria dokazatelstv.i. Aktualnye problemy logiki i metodologii nauky, pages 252--278, 1979.

....can show that: no equality is lost . normalization is preserved Roberto Di Cosmo Glasgow, September 96 6 Chronology I 1970s: the first expansion 1971 Prawitz suggests to reverse # [Pra71] 1976 Huet uses ## long normal forms for higher order unification [Hue76] 1979 Mints reverses # and SP [Min79] 197 Many people suggest expansions: Martin Lof, Meyer, Statman, etc. 1980s: the contraction 1980 Klop s counterexample to CR for # SP [Klo80] 1981 Pottinger shows CR for typed ### SP [Pot81] 1986 Lambek Scott, Obtulowicz: typed ### SP T is not CR [LS86] 1987 Poigne Voss try completion for ....

Gregory Mints. Teorija categorii i teoria dokazatelstv.I. Aktualnye problemy logiki i metodologii nauky, pages 252--278, 1979.

....fl# an exhaustive fl reduction path. Remark 4.4 If Q j F NaN F NaN Q 0 , then Q[A] j F NaN F NaN Q 0 [A] Proof. It is an easy induction on the structure of Q. 2 Remark 4.5 The reductions fi 2 and j alone are confluent and strongly normalizing. Proof. Folklore for fi 2 , see [Kes93, Cub92, DCK94a, Min79] for j. 2 Lemma 4.6 (Commutation of fi 2 wrt j ) fi 2 commutes (in one step) with j . Proof. We consider all possible critical pairs between j and fi 2 : oe:M ) A] y : B: oe:M ) A] y M [A=oe] y : B: M [A=oe] y j F NaN F NaN fi 2 # F NaN F NaN fi 2 F NaN F NaN fflffl ....

Gregory Mints. Teorija categorii i teoria dokazatelstv.I. Aktualnye problemy logiki i metodologii nauky, pages 252--278, 1979.

....(a[shif t] 1) Gamma c Eta ( a[shif t] 1) shif t] 1) However, our restriction is sufficient to keep the strong normalization property: Lemma 4.4 (Strong normalization of d Eta) The reduction Gamma c Eta is strongly normalizing. Proof. Using a decreasing measure as defined in [Min79], we refer the reader to [Kes97b] for details. It is also worth noticing that even if a term a is not a Lambda abstraction according to definition 4.1, its W normal form may be one: for example 1[cons(1) is not a Lambdaabstraction but W (1[cons(1) 1 is. The same happens with the notion of ....

Grigori Mints. Teorija categorii i teoria dokazatelstv.I. Aktualnye problemy logiki i metodologii nauky, pages 252--278, 1979.

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G. Mints. Teorija categorii i teoria dokazatelstv.I. Aktualnye problemy logiki i metodologii nauky, pages 252--278, 1979.

No context found.

G. Mints. Teorija categorii i teoria dokazatelstv.I. Aktualnye problemy logiki i metodologii nauky, pages 252--278, 1979.

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G. Mints. Teorija categorii i teoria dokazatelstv.I. Aktualnye problemy logiki i metodologii nauky, pages 252--278, 1979.

No context found.

G. Mints. Teorija categorii i teoria dokazatelstv.I. Aktualnye problemy logiki i metodologii nauky, pages 252--278, 1979.

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G. Mints. Teorija categorii i teoria dokazatelstv.I. Aktualnye problemy logiki i metodologii nauky, pages 252--278, 1979.

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