Dowek, G., Hardin, T., and Kirchner, C. 1995. Higher-order unification via explicit substitutions. In Symposium on Logic in Computer Science, LICS'95, D. Kozen, Ed. IEEE Computer Society Press, San Diego, California, 366--374.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....about capture of variables. In the previous example the instantiation of Y with x results in the term #x:A.x, while the substitution of x for Y in #x:A.Y results in #z:A.x. Notice that unless A and B represent the same type, the resulting terms in both cases may be ill typed. As pointed out in [28, 15], open terms in the # calculus reveal new challenges. Assume, for example, that an open term is involved in a # redex. The # rule can create substitutions applied to meta variables that cannot be e#ective while the meta variables are not instantiated. In this case, a notation for suspended ....

....the index i 1 to the term mapped by the substitution S on the index i. 2. A Framework to Represent Incomplete Proof Terms. The important elements of our framework are: explicit substitutions, open terms, and dependent types. A simply typed version of ## on open terms has been studied in [15]. In [31, 33] we propose the ##L calculus which is a dependent typed version of a variant of ##. The ##L calculus is confluent and weakly normalizing on well typed terms. As usual in explicit substitution calculi, expressions of ##L are structured in terms and substitutions. The ##L calculus ....

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G. Dowek, T. Hardin, and C. Kirchner, Higher-order unification via explicit substitutions (extended abstract), in Proceedings of the Tenth Annual IEEE Symposium on Logic in Computer Science, San Diego, California, 26--29 June 1995, IEEE Computer Society Press, pp. 366--374. 26

....In consequence, technical nuisances due to higher order aspects of the # calculus, for example # conversion, can be minimized or eliminated in explicit substitution calculi. For instance, higher order unification problems have been reformulated in a first order setting via some variants of ## [8, 9, 25, 5]. However, explicit substitutions are not free of di#culties. Typed versions of these calculi lead to unexpected problems. It is well known now that ## does not preserve strong normalization [30] that is, well typed terms may not terminate in ##. Furthermore, as a rewrite system, ## is not ....

....are place holders in a proof term, and an explicit substitution notation is necessary to delay the application of substitutions to meta variables waiting to be instantiated. Meta variables have also been used as unification variables in the higher order unification methods presented in [8, 9, 25]. In order to apply explicit substitution techniques in a dependent type framework, we develop a # calculus of explicit substitutions, called ##L , with dependent types and support for meta variables. The rest of this section gives an overview of the dependent type theory in which we are ....

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G. Dowek, T. Hardin, and C. Kirchner, Higher-order unification via explicit substitutions (extended abstract), in Proceedings of the Tenth Annual IEEE Symposium on Logic in Computer Science, San Diego, California, 26--29 June 1995, IEEE Computer Society Press, pp. 366--374.

....interest in studying the calculi on open terms is that they alow, for instance, the representation of incomplete proofs where the place holder stands for the still unknown part of the proof. Calculi on open terms have also provided the tools to prune the search space in unification algorithms (cf. [DHK95]) 3. Simulation of fi reduction: If a evaluates in the calculus (using only fi reduction) to b, does a evaluate to b in the subst calculus (using the fi rule and the substitutions rules) 4. Preservation of Termination (PSN) If a terminates in the calculus, does it terminate in the ....

G. Dowek, T. Hardin, and C. Kirchner. Higher order unification via explicit substitutions (extended abstract). In Proceedings, Tenth Annual IEEE Symposium on Logic in Computer Science, San Diego, pages 366-- 374, 1995.

....the technical details can be encapsulated within the instance relation. We would like to understand our notion of unification as a particular case of second order unification. One step in this direction would be to consider a modular constraint based presentation of secondorder unification such as [5]. Flexible bounds might partly capture, within principal types, what constraint based algorithms capture as partially unresolved multi sets of unification constraints. Another example of restricted unification within second order terms is unification under a mixed prefix [18] However, our notion ....

G. Dowek, T. Hardin, C. Kirchner, and F. Pfenning. Higher-order unification via explicit substitutions: the case of higher-order patterns. In M. Maher, editor, Joint international conference and symposium on logic programming, pages 259--273, 1996.

....can be encapsulated within the instance relation and its properties. We would like to understand our unification algorithm as a particular case of second order unification. One step in this direction would be to consider a modular constraint based presentation of second order unification such as [DHKP96] Flexible bounds might capture, within principal types, what constraint based algorithms capture as partially unresolved multi sets of unification constraints. Another example of restricted unification within second order terms is unification under a mixed prefix [Mil92] However, the notion of ....

Gilles Dowek, Thrse Hardin, Claude Kirchner, and Frank Pfenning. Higher-order unification via explicit substitutions: the case of higher-order patterns. In M. Maher, editor, Joint international conference and symposium on logic programming, pages 259--273, 1996.

....higher order unification is technically complicated without being completely satisfactory from a pragmatic point of view. The reason lies in the difference between substitution for first order terms and for # terms. The former is a simple operation of textual replacement (sometimes called grafting [6], or context substitution [11, Sect. 2.1] whereas the latter also involves renamings to avoid capture. Capture avoidance ensures that substitution respects # equivalence, but it complicates higher order unification algorithms. Furthermore it is the simple textual form of substitution rather than ....

....application, capture avoiding substitution and ## equivalence. Does it have to be so No For one thing, several authors have already noted that one can make sense of possibly capturing substitution modulo # equivalence by using explicit substitutions in the term representation language: see [6, 12, 14, 27]. Compared with those works, we make a number of simplifications. First, we find that we do not need to use function variables, application or ## equivalence in our representation language leaving just binders and # equivalence. Secondly, instead of using explicit substitutions of names for ....

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G. Dowek, T. Hardin, and C. Kirchner. Higher-order unification via explicit substitutions. In Tenth Annual Symposium on Logic in Computer Science, pages 366--374. IEEE Computer Society Press, Washington, 1995.

....in the introduction, all these calculi are a priori good candidates for efficient implementations of a functional language for instance. But as far as we know, no work has been done concerning this practical aspect ( Bor95] implements oe for a higher order unification algorithm described in [DHK95], but he does not deal with the efficiency of the system) So we want to compare some calculi with regard to their time efficiency when considered as reduction machineries. Our aim is then to add new columns to the previous table instead of new lines. 19 Chapter 3 The implemented systems It is ....

....operational aspect. Also note that we have done strong reduction, but there exists other kinds of reduction (and of normal forms) for instance weak reduction is used in functionnal languages like LISP, and head reduction is used by higher order theorem proving for higher order unification like in [DHK95]. We also have considered closed terms, considering metavariables could be interesting but, since not all systems are confluent on open or semi open terms, that would eliminate many systems which are not confluent with them. With the amount of time available, it would have been very difficult to ....

Gilles Dowek, Th'er`ese Hardin, and Claude Kirchner. Higher-order unification via explicit substitutions (extended abstract). In Proceedings of the 10 th Annual IEEE Symposium on Logic in Computer Science, pages 366--374, San Diego (California, USA), june 1995. IEEE Computer Society Press. 58

....as sets of first order rewrite rules which internalize the substitutions by the use of closures : a[s] where a is a term and s a substitution, belonging to the grammar of terms. As a side effect, higher order unification ( 7] is replaced by standard techniques of first order unification ([5]) The first rule of all the calculi is the translation of the fi reduction which introduces substitutions on terms. Other rules permit to reduce substituted terms and substitutions. Many systems have already been proposed, using similar representation for terms, but introducing different ....

G. Dowek, T. Hardin, and C. Kirchner. Higher order unification via explicit substitutions. In Proceedings, Tenth Annual IEEE Symposium on Logic in Computer Science, pages 366--374, june 1995. 11

....and explicit substitutions is the formulation of different first order calculi for the calculus [1, 4, 17, 15, 6, 22] which is the paradigmatic example of a higher order (term) rewriting system. Other examples are the translations of higher order unification to first order unification modulo [11], higher order logic to first order logic modulo [12] higher order theorem proving to first order theorem proving modulo [9] etc. All these translations have a theoretical interest because the expressive power of higher and first order formalisms is put in evidence, but another practical issue ....

G. Dowek, T. Hardin and C. Kirchner. Higher-order unification via explicit substitutions. Information and Computation, 157:183--235, 2000.

.... is then to translate the pre metaterm ff: b fi) by (S( b fi) in such a way that there is no capture of variables since ( S( b fi) is exactly (2) The solution adopted here for translating pre free o metavariables into the de Bruijn formalism is in some sense what is called pre cooking in [10]: the pre cooking function takes a oe term with t metavariables and suffixes them with as many explicit shift operators as the number of binders present in its parameter path. This avoids variable capture when the higher order unification procedure finds solutions for the t metavariables. We use ....

....of a fixed metavariable coincide modulo their corresponding context. Dealing with such notion of coherence of substitutions in a de Bruijn formalism is also present in other formalisms but in a more restricted form. Thus for example, as mentioned before, a pre cooking function is used in [10] in order to avoid variable capture in the higher order unification procedure. In XRS [17] the notions of binding arity and pseudo binding arity are introduced in order to take into account the parameter path of the different occurrences of t metavariables appearing in a rewrite rule. Then it is ....

[Article contains additional citation context not shown here]

Gilles Dowek, Th'er`ese Hardin, and Claude Kirchner. Higher-order unification via explicit substitutions. In Dexter Kozen, editor, Proceedings of Tenth Annual IEEE Symposium on Logic in Computer Science (LICS), San Diego, USA, 1995.

.... condition on variable assignments in Definition 10) Our solution is then to translate the pre metaterm ff: b fi) by (S( b fi) in such a way that there is no capture of variables since ( S( b fi) is exactly (2) The solution adopted here is in some sense what is called pre cooking in [9]. We use MVar(A) resp. MVar i (A) and MVar t (A) to denote the set of all metavariables (resp. i and t metavariables) of the de Bruijn pre metaterm A. As in the SERS formalism, we also need here a notion of well formed pre metaterm. The first motivation is to guarantee that labels of ....

....of a fixed metavariable coincide modulo their corresponding context. Dealing with such notion of coherence of substitutions in a de Bruijn formalism is also present in other formalisms but in a more restricted form. Thus for example, as mentioned before, a pre cooking function is used in [9] in order to avoid variable capture in the higher order unification procedure. In XRS [16] the notions of binding arity and pseudo binding arity are introduced in order to take into account the parameter path of the different occurrences of t metavariables appearing in a rewrite rule. Our notion ....

[Article contains additional citation context not shown here]

G. Dowek, T. Hardin, and C. Kirchner. Higher-order unification via explicit substitutions. In LICS, 1995.

....extensions are now under consideration, including a linear type system with use once channel types and a library supporting a physically distributed channel abstraction. 3.3.2 Perspectives, work in progress Action structures. The refinement of the presentation of action calculi is in [1]; the paper presents several examples and uses a new notion, action graphs, in doing so. The analysis of closed action calculi appears in [2] and [unpublished report 1] The model theory of action calculi, so called control structures, appears in [3] and [4] Future work will be in classifying ....

....distributed channel abstraction. 3.3.2 Perspectives, work in progress Action structures. The refinement of the presentation of action calculi is in [1] the paper presents several examples and uses a new notion, action graphs, in doing so. The analysis of closed action calculi appears in [2] and [unpublished report 1]. The model theory of action calculi, so called control structures, appears in [3] and [4] Future work will be in classifying action calculi according to their dynamics; it appears that the model theory will help considerably here. Work is also in progress to find ways of defining a bisimilarity ....

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Dowek, G., Hardin, T., Kirchner, C.: "Higher-Order Unification via Explicit Substitutions ". Proceedings of the Tenth Annual IEEE Symposium on Logic in Computer Science, San Diego, California, June 1995

.... By showing that higher order unification was indeed an instance of first order unification for an appropriate equational axiomatization of the typed lambda eta calculus, Claude Kirchner and his coauthors have provided with a new way of integrating built in theories in functional languages [38]. 6 Final CCL Report 7 2.3 Combination of constraints systems Three site are mainly involved in this important question, CIS, Nancy, and COSYTEC. The question is of course relevant both theoretically and practically, since the goal is to automate the implementation of an algorithm for solving ....

....patterns of P matches t. We have described in [58] an algorithm that solves the problem in time (jtj jP j) log jP j. Higher order unification is equational unification for fij conversion. But it is not first order equational unification, as substitution has to avoid capture. We have shown in [38] that higher order unification can be reduced to first order equational unification in a suitable equational theory: the oe calculus of explicit substitutions. We have described a new algorithm for solving a conjunction of linear diophantine equations, inequations and disequations in natural ....

Gilles Dowek, Th'er`ese Hardin, and Claude Kirchner. Higher-order unification via explicit substitutions, extended abstract. In Dexter Kozen, editor, Proceedings of LICS'95, pages 366--374, San Diego, June 1995.

No context found.

Gilles Dowek, Th'er`ese Hardin, and Claude Kirchner. Higher-order unification via explicit substitutions, extended abstract. In Dexter Kozen, editor, Proceedings of LICS'95, pages 366--374, San Diego, June 1995.

No context found.

Dowek, G., Hardin, T., and Kirchner, C. 1995. Higher-order unification via explicit substitutions. In Symposium on Logic in Computer Science, LICS'95, D. Kozen, Ed. IEEE Computer Society Press, San Diego, California, 366--374.

No context found.

Gilles Dowek, Therese Hardin, and Claude Kirchner. Higher-order unification via explicit substitutions. In D. Kozen, editor, Proceedings of the Tenth Annual Symposium on Logic in Computer Science, pages 366--374, San Diego, California, June 1995. IEEE Computer Society Press.

No context found.

Gilles Dowek, Therese Hardin, and Claude Kirchner. Higher-order unification via explicit substitutions. Information and Computation, 157(1--2):183--235, 2000.

No context found.

Dowek, G., Hardin, T., and Kirchner, C. (1995). Higher-order unification via explicit substitutions. In Symposium on Logic in Computer Science (LICS '95), pages 366--507, Los Alamitos, Ca., USA. IEEE Computer Society Press.

No context found.

G. Dowek, T. Hardin, C. Kirchner, F. Pfenning, Higher-order unification via explicit substitutions: the case of higher-order patterns, in: Proc. of JICSLP, 1996, pp. 259-- 273.

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G. Dowek, T. Hardin, C. Kirchner, Higher-order unification via explicit substitutions, in: 10th Annual Symposium on Logic in Computer Science, IEEE Computer Society Press, Washington, 1995, pp. 366--374.

No context found.

Gilles Dowek, Therese Hardin, and Claude Kirchner. Higher-order unification via explicit substitutions. In D. Kozen, editor, Proceedings of the Tenth Annual Symposium on Logic in Computer Science, pages 366--374, San Diego, California, June 1995. IEEE Computer Society Press.

No context found.

G. Dowek, T. Hardin, C. Kirchner, F. Pfenning, Higher-order unification via explicit substitutions: the case of higher-order patterns, in: Proc. of JICSLP, 1996, pp. 259-- 273.

No context found.

G. Dowek, T. Hardin, C. Kirchner, Higher-order unification via explicit substitutions, in: 10th Annual Symposium on Logic in Computer Science, IEEE Computer Society Press, Washington, 1995, pp. 366--374.

No context found.

Gilles Dowek, Thrse Hardin, Claude Kirchner, and Frank Pfenning. Higher-order unification via explicit substitutions: the case of higher-order patterns. In M. Maher, editor, Joint international conference and symposium on logic programming, pages 259--273, 1996.

No context found.

Gilles Dowek, Th'er`ese Hardin, and Claude Kirchner. Higher order unification via explicit substitutions. Information and Computation, 157(1/2):183--235, 2000.

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