T. Hardin. Confluence Results for the Pure Strong Categorical Logic CCL : -calculi as Subsystems of CCL. Theoretical Computer Science, 65(2):291--342, 1989.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....meta operators (see lemmas 1 6 in [KR95] In section 3 we introduce the s e calculus as the s calculus extended by the addition of these rules and prove its local confluence by studying the critical pairs. In order to prove (global) confluence using the interpretation technique (cf. [Har89] and [CHL91] we are led to study the corresponding calculus of substitutions s e (the calculus obtained by deleting the rule which starts fi reduction) We prove the weak normalisation of s e by showing that innermost strategies always terminate. The strong normalisation is, as far as we know, ....

T. Hardin. Confluence Results for the Pure Strong Categorical Logic CCL : -calculi as Subsystems of CCL. Theoretical Computer Science, 65(2):291--342, 1989.

....of terms. We shall prove strong normalisation of the corresponding calculus of substitution by translating it into the oe calculus [ACCL91] and therefore the relation between both calculi will be made explicit. The confluence of the s calculus is obtained by the interpretation method ([Har89], CHL92] The proof of the preservation of normalisation follows the lines of an analogous result for the AE calculus (cf. BBLRD95] The relation between s and AE is also studied. 1 Introduction Most literature on the calculus considers substitution as an implicit operation. It means ....

T. Hardin. Confluence Results for the Pure Strong Categorical Logic CCL : -calculi as Subsystems of CCL. Theoretical Computer Science, 65(2):291--342, 1989.

....must be added. The calculus thus obtained, s e has been shown confluent (cf. KR97] The combination of s e with generalized reduction has not yet been studied. Theorem 3 (Confluence of s and gs) The s and gs calculi are confluent on s. Proof of Theorem 3 We use the interpretation method (cf. [Har89, CHL96]) To prove confluence of the s calculus, remove each (g) from the proof below. For the confluence of the gs calculus, leave each (g) but remove the parentheses that embrace the gs. The proof goes as follows: We interpret the (g)s calculus into the (g) calculus via s normalization: Gamma ....

T. Hardin. Confluence results for the pure strong categorical logic CCL: -calculi as subsystems of CCL. Theoretical Computer Science, 65:291--342, 1989.

....within abstractions, the t caculus can be considered as a calculus written in the s style which works with the updating mechanism of the oe calculi and therefore as a calculus that links both s and oe styles. In this paper we introduce t, we prove its confluence using the interpretation method ([Har89], CHL92] we make explicit the relationship between t and exp, which happens to be a sort of inmersion, and we use this inmersion to prove the PSN for t using the PSN of exp. We compare t with oe by providing an inmersion of the former into the latter and argue about the impossibility of such an ....

T. Hardin. Confluence Results for the Pure Strong Categorical Logic CCL : -calculi as Subsystems of CCL. Theoretical Computer Science, 65(2):291--342, 1989.

....we will have uniqueness of normal forms. All attempts to find a direct proof for the confluence of B failed. In particular, parallelization does not seem to work in order to show confluence of p oe. Instead we will make use of an interpretation technique due to Hardin, which was identified in [12]. This method has subsequently been used several times in order to show confluence for systems of explicit substitutions. A first step towards the proof of the Church Rosser property for B is to show that B oe is Church Rosser. In order to apply an old result by Newman [25] we establish that B ....

Hardin, T. Confluence results for the pure strong categorical logic CCL: --calculi as subsystems of CCL. Theoretical Computer Science 65 (1989), 291-- 342.

....confluent on Q , so by Newman s Lemma, OE is confluent. ut Corollary 2 Let x be in Q , then x has exactly one OE normal form. Proof. By Corollary 1 and Theorem 2. ut 5 OE Confluence An useful technique to prove confluence in explicit substitutions calculi is the interpretation method ([Har89]) Based on this method, Kes96] proposes a general scheme of abstract properties which are sufficient to guarantee ground confluence on explicit substitutions calculi. Although the interpretation method can be used to prove confluence on terms with meta variables ( R io93] we use a technique ....

T. Hardin. Confluence results for the Pure Strong Categorical Logic CCC: -calculi as subsystems of CCL. Theoretical Computer Science, 65(2):291--342, 1989.

....terms. However, we believe that s e still deserves attention because, while offering a new style to work with explicit substitutions, it is able to simulate one step of classical fi reduction, whereas i is not. To prove confluence we introduce a generalization of the interpretation method (cf. [Har89] and [CHL92] to a technique which uses weak normal forms (instead of strong ones) This technique is general enough to apply to many reduction systems and we consider it as a powerful tool to obtain confluence. Strong normalisation of the corresponding calculus of substitutions s e , is left as a ....

....one step of clasical fi reduction as shown in [MH95] it simulates only a big step beta reduction. Furthermore, this lack of the simulation property is an uncommon feature among calculi of explicit substitutions. As the strong normalisation of s e remains open, the interpretation method (cf. [Har89], CHL92] which is usually used to prove the confluence of a calculus with explicit substitutions is not applicable to s e . In section 1 we propose a generalization of the interpretation method which enables us to prove the confluence of s e with just weak normal forms. The method is general ....

[Article contains additional citation context not shown here]

T. Hardin. Confluence Results for the Pure Strong Categorical Logic CCL : -calculi as Subsystems of CCL. Theoretical Computer Science, 65(2):291--342, 1989.

....rules must be added. The calculus thus obtained, s e has been shown confluent (cf. 26] The combination of s e with generalised reduction has not yet been studied. Theorem 3.13 (Confluence of s and gs) The s and gs calculi are confluent on s. Proof. We use the interpretation method (cf. [16, 10]) To prove confluence of the s calculus, remove all the (g) s from the proof below. For the confluence of the gs calculus, leave the (g) s but remove the ( s that embrace the g s. The proof goes as follows: We interpret the (g)s calculus into the (g) calculus via s normalisation: b s(b) a s(a) ....

T. Hardin. Confluence results for the pure strong categorical logic CCL: -calculi as subsystems of CCL. Theor. Comp. Sc., 65(2):291--342, 1989.

....meta operators (see lemmas 1 6 in [KR95] In section 3 we introduce the s e calculus as the s calculus extended by the addition of these rules and prove its local confluence by studying the critical pairs. In order to prove (global) confluence using the interpretation technique (cf. [Har89] and [CHL91] we are led to study the corresponding calculus of substitutions s e (the calculus obtained by deleting the rule which starts fi reduction) We prove the weak normalisation of s e by showing that innermost strategies always terminate. The strong normalisation is, as far as we know, ....

T. Hardin. Confluence Results for the Pure Strong Categorical Logic CCL : -calculi as Subsystems of CCL. Theoretical Computer Science, 65(2):291--342, 1989.

....the sg calculus is powerful enough to simulate gfi reduction. Lemma 6 (Simulation of gfi reduction) Let a; b 2 , if a gfi b then a sg b. Proof: Induction on a using Lemma 4. 2 Theorem 3 (Confluence of sg) The sg calculus is confluent on s. Proof: We use the interpretation method (cf. [14, 9]) We interpret the sg calculus into the g calculus via s normalisation: a Gamma Gamma Gamma Gamma Gamma Gamma Gamma Gamma sg R R sg b c s s s s(a) s(b) s(c) Gamma Gamma Gamma Gamma Gamma Gamma Gamma Gamma gfi R R gfi R ....

T. Hardin. Confluence Results for the Pure Strong Categorical Logic CCL : -calculi as Subsystems of CCL. Theoretical Computer Science, 65(2):291--342, 1989.

....that Deltaexp is powerful enough to simulate fi reduction: Lemma 8 (Simulation of fi reduction) For pure terms a; b: if a fi b then a fi oe b. Proof: By induction on a using Lemma 7.2 and .3. 2 Theorem 9 The Deltaexp calculus is confluent. Proof: We use the interpretation method [10, 16]. If a fi oe b 1 and a fi oe b 2 then by Lemma 7.7, oe(a) fi oe(b i ) for i 2 f1; 2g, and by CR of Delta, 9c such that oe(b i ) fi c, and by Lemma 8 oe(b i ) fi oe c. Hence, b i fi oe c. 2 Finally, the following is the converse of the generalised PSN result we are aiming ....

T. Hardin. Confluence Results for the Pure Strong Categorical Logic CCL : -calculi as Subsystems of CCL. Theoretical Computer Science, 65(2):291--342, 1989.

....within abstractions, the t caculus can be considered as a calculus written in the s style which works with the updating mechanism of the oe calculi and therefore as a calculus that links both s and oe styles. In this paper we introduce t, we prove its confluence using the interpretation method ([Har89], CHL92] we make explicit the relationship between t and exp, which happens to be a sort of inmersion, and we use this inmersion to prove the PSN for t using the PSN of exp. We compare t with oe by providing an inmersion of the former into the latter and argue about the impossibility of such an ....

T. Hardin. Confluence Results for the Pure Strong Categorical Logic CCL : -calculi as Subsystems of CCL. Theoretical Computer Science, 65(2):291--342, 1989.

....must be added. The calculus thus obtained, s e has been shown confluent (cf. KR97] The combination of s e with generalized reduction has not yet been studied. Theorem 3 (Confluence of s and gs) The s and gs calculi are confluent on s. Proof of Theorem 3 We use the interpretation method (cf. Har89, CHL96] To prove confluence of the s calculus, remove each (g) from the proof below. For the confluence of the gs calculus, leave each (g) but remove the parentheses that embrace the gs. The proof goes as follows: We interpret the (g)s calculus into the (g) calculus via s normalization: a ....

T. Hardin. Confluence results for the pure strong categorical logic CCL: -calculi as subsystems of CCL. Theoretical Computer Science, 65:291--342, 1989.

....of terms. We shall prove strong normalisation of the corresponding calculus of substitution by translating it into the oe calculus [ACCL91] and therefore the relation between both calculi will be made explicit. The confluence of the s calculus is obtained by the interpretation method ([Har89], CHL92] The proof of the preservation of normalisation follows the lines of an analogous result for the AE calculus (cf. BBLRD95] The relation between s and AE is also studied. 1 Introduction Most literature on the calculus considers substitution as an implicit operation. It means that ....

T. Hardin. Confluence Results for the Pure Strong Categorical Logic CCL : -calculi as Subsystems of CCL. Theoretical Computer Science, 65(2):291--342, 1989.

....with a double occurrence in this rule can be considered as a constant in the set of semi open expressions. In particular, there are not reduction rules for natural numbers. 3 Confluence An useful technique to prove confluence in calculi of explicit substitutions is the interpretation method [11, 17]. Although the interpretation method can be used to prove confluence on terms with meta variables (cf. 32] we prefer to use a technique that was coined in [34] the Yokouchi Hikita s Lemma. This lemma seems to be suitable for left linear calculi of explicit substitutions [3, 31, 25] Lemma 6 ....

T. Hardin. Confluence results for the Pure Strong Categorical Logic CCC: -calculi as subsystems of CCL. Theoretical Computer Science, 65(2):291--342, 1989.

....2, we introduce the Delta calculus and state some of its properties. In Section 3, we extend the Delta calculus with explicit substitutions. In Section 4, we establish the confluence and preservation of strong normalisation (PSN) of the Deltaexp calculus. We use the interpretation method [15] to show confluence and the decency method to establish PSN [7] We also show that the structure preserving method of [9] does not apply to the Deltaexp calculus. In Section 5 we introduce the simply typed version of Deltaexp and show that it has the desirable properties such as subject ....

....obtained from Deltaexp by leaving out Delta. 4 Confluence and preservation of Strong Normalisation In this section, we show that the Deltaexp calculus enjoys confluence and preservation of strong normalisation. 4. 1 Confluence Confluence is proved as usual, using the interpretation method of [10, 15]. Lemma 5 Let a; b 2 T e . The following holds: 1. oe is SN and CR. Hence, every term c 2 T e has a unique oe normal form, denoted oe(c) 2. oe(ab) oe(a)oe(b) oe(x:a) x:oe(a) oe( Deltax:a) Deltax:oe(a) oe(a[x : b] oe(a) oe(b) x] 3. Projection: If a fi oe b then oe(a) ....

T. Hardin. Confluence Results for the Pure Strong Categorical Logic CCL : -calculi as Subsystems of CCL. Theoretical Computer Science, 65(2):291--342, 1989.

....for first order lambda calculi with expansions and algebraic TRS s, via a translation. Roberto Di Cosmo Glasgow, September 96 12 Related lemmas . An instance of the previous lemma is used in [BTG94] A similar lemma is used in [CDC95] Hardin s interpretation method can be found in [Har89] . Generalized interpretation methods are introduced by Kesner [Kes96] where it is used to show CR for # d , a calculus with explicit substitutions with a weak form of composition that preserves SN, and KamareddineRios [KR95] A related method is due to Curien and Ghelli [CG91] Roberto Di ....

Therese Hardin. Confluence results for the pure strong categorical logic C.C.L.; #-calculi as subsystems of C.C.L. Theoretical Computer Science, 65(2):291--342, 1989.

....j [ N ] ffi ( of type A in the same typing context: we conclude by noticing that fij reduction terminates on typed terms. 5.2. Confluence of SKIn We are finally in position to show that: Theorem 5.4. Confluence) SKIn and Sigma are confluent. Proof. We apply Hardin s interpretation method (Hardin, 1989) several times. This method is as follows. Let R and R 0 be two reduction relations, such that R 0 is normalizing and confluent. Let R 0 (M) denote the unique R 0 normal form of M . R then induces a reduction relation R 0 on R 0 normal forms, defined as the smallest such that M R N ....

Hardin, T. (1989). Confluence results for the pure strong categorical logic CCL. Lambda-calculi as subsystems of CCL. Theoretical Computer Science, 65(3):291--342.

.... properties of Weak and Strong Calculi of Explicit Substitutions Pierre Louis Curien Th er ese Hardin y Jean Jacques L evy z Abstract Categorical combinators [12, 21, 43] and more recently oe calculus [1, 23] have been introduced to provide an explicit treatment of substitutions in the calculus. We reintroduce here the ingredients of these calculi in a self contained and stepwise way, with a special emphasis on confluence properties. The main new results of the ....

....oe calculus [1, 23] have been introduced to provide an explicit treatment of substitutions in the calculus. We reintroduce here the ingredients of these calculi in a self contained and stepwise way, with a special emphasis on confluence properties. The main new results of the paper w.r.t. [12, 21, 1, 23] are the following: 1. We present a confluent weak calculus of substitutions, where no variable clashes can be feared. 2. We solve a conjecture raised in [1] oe calculus is not confluent (it is confluent on ground terms only) This unfortunate result is repaired by presenting a confluent ....

[Article contains additional citation context not shown here]

T. Hardin, Confluence Results for the Pure Strong Categorical Logic CCL. -calculi as subsystems of CCL, Theoretical Computer Sc., 65, pp 291--342, 1989.

No context found.

T. Hardin. Confluence Results for the Pure Strong Categorical Logic CCL : -calculi as Subsystems of CCL. Theoretical Computer Science, 65(2):291--342, 1989.

No context found.

T. Hardin. Confluence Results for the Pure Strong Categorical Logic CCL : #-calculi as Subsystems of CCL. Theoretical Computer Science, 65(2):291--342, 1989.

No context found.

T. Hardin. Confluence Results for the Pure Strong Categorical Logic CCL : -calculi as Subsystems of CCL. Theoretical Computer Science, 65(2):291--342, 1989.

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC