Frank Pfenning. Unification and anti-unification in the Calculus of Constructions. In Sixth Annual IEEE Symposium on Logic in Computer Science, pages 74--85, Amsterdam, The Netherlands, July 1991.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....is undecidable in general. As discovered by Miller [8] there exists a decidable fragment, called higher order patterns. For this fragment, unification and computing the most specific generalization is decidable even in rich type theories with dependent types and polymorphism as shown by Pfenning [12]. However, these algorithms may not be efficient in practice [16] and hence it is not obvious that they are suitable for higher order term indexing techniques. In this paper, we present substitution tree indexing for higher order terms based on linear higher order patterns [16] Linear ....

....first order indexing techniques rely on a post processing step to carry out the consistency checks. To design a higher order indexing technique, we will extend this notion of linearization and standardization. We will require that terms are linear higher order patterns. Higher order patterns [8, 12] are terms where every existential variable must be applied to some distinct bound variables. Linear higher order patterns [16] impose some further restrictions on the structure of terms: First, all existential variables must occur only once. This allows us to delay any expensive consistency ....

Frank Pfenning. Unification and anti-unification in the Calculus of Constructions. In Sixth Annual IEEE Symposium on Logic in Computer Science, pages 74-85, Amsterdam, The Netherlands, July 1991.

....building applications: a logical variable can only 28 be applied to distinct variables and distinct universal variables that are quantified in its scope. A similar restriction applies to all systems of Barendregt s cube [3, 2] and makes corresponding unification problems unitary and decidable [52]. The L calculus lies between a first order theory and the calculus. It is enough for many applications, but cannot be used for coding (as in Section 4.1.1) T g [ B 1 ; B 2 ] T g [ B 1 ] T g [ B 2 ] as t g (B1 and B2) k(D1 (D2 k) t g B1 D1 t g B2 D2 because the underlined ....

F. Pfenning. Unification and anti-unification in the calculus of constructions. In Symp. Logic in Computer Science, pages 74--85, 1991.

....eligible since it enjoys the strong normalization property. Another way of making the unification problem still less difficult is discussed in Section 5.2. There are other more sophisticated higher order domains that also have the strong normalization property and a practical unification problem [13, 54]. The unification problem for simply typed terms modulo axioms ff and fi is semi decidable and infinitary. The latter means that there can be infinitely many most general unifiers. For instance, the problem u(N v(v) u) fffi w(w) has the following infinitely many minimal unifiers: 0 = ....

F. Pfenning. Unification and anti-unification in the calculus of constructions. In Symp. Logic in Computer Science, pages 74--85, 1991.

....Bn:A 1 where Gamma is a context x 1 : B 1 ; xn : Bn and M 0 Delta Gamma as an abbreviation for M 0 Delta x 1 ; xn . Moreover, there is a one to one correspondence between these two solutions. Following Miller s terminology, we say M 0 is the result of raising M . Pfenning [31] investigated this notion in the setting of the calculus of construction which includes LF. We will rewrite the f forall rule to reflect this view to: Gamma AE [M 0 Delta Gamma =x]A 2 f Gamma S : a Delta u Gamma M 0 : Pi Gamma:A 1 f forall Gamma AE Pix : A 1 :A 2 f Gamma ....

....We view the answer substitution as a collection of constraints to the existential variables in a goal A. In general, unification for higher order terms is undecidable, however Pfenning showed that unification of higher order patterns in the context of LF type theory is decidable and unitary[31]. The constraints in essentially describe the solution to a higher order pattern unification problem. However, note that we can extend the notion of constraints to the general case. Substitution : Deltaj ; XA = M We require that all free (existential) variables X defined by a substitution ....

Frank Pfenning. Unification and anti-unification in the Calculus of Constructions. In Sixth Annual IEEE Symposium on Logic in Computer Science, pages 74--85, Amsterdam, The Netherlands, July 1991.

....at CMU [11] Felty and Gunter at Bell Labs, and the authors at Inria. Other works are done in a similar framework for integrating linear logic and logic programming (Pareschi and Andreoli [4] Hodas and Miller [17] or higher order type systems and logic programming (Elliot [10] Pfenning [36, 37]) We present in this paper the broad lines of our implementation of Prolog: Prolog Mali. We have implemented Prolog for its own merits, and as a demonstration that memory management issues are a good guide for implementing logic programming systems. Speed was always our second concern. We ....

F. Pfenning. Unification and anti-unification in the calculus of constructions. In Symp. Logic in Computer Science, pages 74--85, 1991.

....of a set of disjoint ( parallel ) redexes. Lemma 7. Sets of disjoint redexes are independent. Proof. Trivial from the restriction to trees. fi 4 Dependence After having sketched the behaviour of independent steps, we briefly recapitulate some basic theory of dependent steps as known from e.g. [Mil91, Pfe, MN94, Pre95]. The basic results ( Mil91, Pre95] we need are: 1. Matching for patterns is decidable. This yields that the rewrite relation of a PRS is decidable. 2. Unification of patterns is decidable as well, and most general unifiers do exist. This yields that one can compute so called critical pairs ....

....of the span, imprecisely) denoted by l t g. The critical unification is the largest substructure on which either of the steps operate. That patterns are closed under all these operations and critical unifications and intersections exist in case of dependent steps follows from results in [Mil91, Pfe, MN94, Pre95]. Lemma 9. 1. A set of redexes is independent if and only if its elements are pairwise independent. 2. Two redexes are independent if and only if they don t form a dependent span. 3. Every dependent span is obtained by putting a critical span in a context. Proof. This is tedious. The first item ....

Frank Pfenning. Unification and anti-unification in the calculus of constructions. pp. 74--85 in LICS'91.

.... [26] One can see the extension of Prolog with simply typed terms merely as the instance CLP( of the CLP scheme [22] Other possible domains in Prolog flavour are the typed calculi of Barendregt s cube [2, 18] Elliot and Pfenning have studied unification in some of these calculi [13, 34]. We describe the computation domain of Prolog, the various notions of variables and substitutions in this domain, and the unification problem. 1.2.1.1 Simply typed terms We give a quick introduction to the typed calculus (see Barendregt or Revesz for more information [3, 35] Types Simple ....

F. Pfenning. Unification and anti-unification in the calculus of constructions. In Symp. Logic in Computer Science, pages 74--85, 1991.

....substitutions allows a finer and granular control on the application of substitutions. In particular, Strecker s definition of fi corresponds to a strategy in Pi L where a (Beta) rule if followed by Pi L normalization. Unification algorithms for dependent types have been studied previously in [22, 12], and for the case of explicit substitutions in [10] In comparison to Agda [8] our calculus erases the distinction between standard term and term with an explicit substitution. Acknowledgments The authors would like to thank C. Braga for his help with Maude and to the anonymous referees for ....

F. Pfenning. Unification and anti-unification in the calculus of constructions. In Proc. of LICS, pages 74--85, 1991.

....at CMU [14] Felty and Gunter at Bell Labs, and the authors at Inria. Other works are done in a similar framework for integrating linear logic and logic programming (Pareschi and Andreoli [3] Hodas and Miller [19] or higher order type systems and logic programming (Elliot [13] Pfenning [38, 39]) We present in this paper the broad lines of our implementation of Prolog: Prolog Mali. We have implemented Prolog for its own merits, and as a demonstration that memory management issues are a good guide for implementing logic programming systems. Speed was always our second concern. We ....

F. Pfenning. Unification and anti-unification in the calculus of constructions. In Symp. Logic in Computer Science, pages 74--85, 1991.

No context found.

Frank Pfenning. Unification and anti-unification in the Calculus of Constructions. In Sixth Annual IEEE Symposium on Logic in Computer Science, pages 74--85, Amsterdam, The Netherlands, July 1991.

No context found.

Frank Pfenning. Unification and anti-unification in the Calculus of Constructions. In Sixth Annual IEEE Symposium on Logic in Computer Science, pages 74--85, Amsterdam, The Netherlands, July 1991.

No context found.

Frank Pfenning. Unification and anti-unification in the Calculus of Constructions. In Sixth Annual IEEE Symposium on Logic in Computer Science, pages 74--85, Amsterdam, The Netherlands, July 1991.

No context found.

Frank Pfenning. Unification and anti-unification in the Calculus of Constructions. In Sixth Annual IEEE Symposium on Logic in Computer Science, pages 74--85, Amsterdam, The Netherlands, July 1991.

No context found.

Frank Pfenning. Unification and anti-unification in the Calculus of Constructions. In Sixth Annual IEEE Symposium on Logic in Computer Science, pages 74--85, Amsterdam, The Netherlands, July 1991.

No context found.

Frank Pfenning. Unification and anti-unification in the Calculus of Constructions. In Sixth Annual IEEE Symposium on Logic in Computer Science, pages 74--85, Amsterdam, The Netherlands, July 1991.

No context found.

Frank Pfenning. Unification and anti-unification in the Calculus of Constructions. In Sixth Annual IEEE Symposium on Logic in Computer Science, pages 74--85, Amsterdam, The Netherlands, July 1991.

No context found.

Frank Pfenning. Unification and anti-unification in the Calculus of Constructions. In Sixth Annual IEEE Symposium on Logic in Computer Science, pages 74--85, Amsterdam, The Netherlands, July 1991.

....meta theory of such formalisms. Moreover, calculi featuring a unit type and a unit element do not admit a Church Rosser theorem unless all terms are expanded [JG95] this means that typing information must be stored and maintained in otherwise type free procedures such as pattern unification [Mil91, Pfe91]. The benefits of the spine calculus representation, in conjunction with explicit substitutions, have been exploited in a new implementation of the logical framework LF [HHP93] as the higher order logic programming language Twelf [PS99] LF is based on the type theory , a refinement of ....

Frank Pfenning. Unification and anti-unification in the Calculus of Constructions. In Sixth Annual IEEE Symposium on Logic in Computer Science, pages 74--85, Amsterdam, The Netherlands, July 1991.

....awarded to the first author. In this paper, we discuss the e#cient implementation of higher order pattern unification for the dependently typed lambda calculus. Unlike Huet s general higher order unification algorithm which involves branching and backtracking, higher order pattern unification [6, 13] is deterministic and decidable. An important step toward the e#cient implementation of higher order pattern unification was the development based on explicit substitutions and de Bruijn indices [3] for the simply typed lambda calculus. This allows a clear distinction between bound and existential ....

....we will consider the pattern fragment of the modal lambdacalculus. Higher order patterns are terms where existential variables must be applied to distinct bound variables. This fragment was first identified by Miller [6] for the simply typed lambda calculus, and later extended by Pfenning [13] to the dependently typed and polymorphic case. We enforce that all terms are in normal form, and the type of existential variables has been lowered and is atomic. We call a normal term U an atomic pattern, if all the subterms of the form u[#] are such that # = y 1 x 1 , y k x k where y 1 ....

[Article contains additional citation context not shown here]

Frank Pfenning. Unification and anti-unification in the Calculus of Constructions. In Sixth Annual IEEE Symposium on Logic in Computer Science, pages 74--85, Amsterdam, The Netherlands, July 1991.

....meta theory of such formalisms. Moreover, calculi featuring a unit type and a unit element do not admit a Church Rosser theorem unless all terms are expanded [JG95] this means that typing information must be stored and maintained in otherwise type free procedures such as pattern unification [Mil91, Pfe91]. The benefits of the spine calculus representation, in conjunction with explicit substitutions, have been exploited in a new implementation of the logical framework LF [HHP93] as the higher order logic programming language Twelf [PS99] LF is based on the type theory , a refinement of ....

Frank Pfenning. Unification and anti-unification in the Calculus of Constructions. In Sixth Annual IEEE Symposium on Logic in Computer Science, pages 74--85, Amsterdam, The Netherlands, July 1991.

No context found.

Frank Pfenning. Unification and anti-unification in the calculus of constructions. In Proceedings, Sixth Annual IEEE Symposium on Logic in Computer Science, pages 74--85, Amsterdam, The Netherlands, 15--18 July 1991. IEEE Computer Society Press.

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Frank Pfenning. Unification and anti-unification in the calculus of constructions. In Proceedings, Sixth Annual IEEE Symposium on Logic in Computer Science, pages 74--85, Amsterdam, The Netherlands, 15--18 July 1991. IEEE Computer Society Press.

No context found.

F. Pfenning. Unification and anti-unification in the calculus of constructions. In 6th IEEE Symposium on Logic in Computer Science, pages 74--85. IEEE Computer Society Press, 1991.

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Frank Pfenning. Unification and anti-unification in the calculus of constructions. In Proc. 6th IEEE LICS, pages 74--85, 1991. 50

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Frank Pfenning. Unification and anti-unification in the Calculus of Constructions. In Sixth Annual IEEE Symposium on Logic in Computer Science, pages 74--85, Amsterdam, The Netherlands, July 1991.

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