Andrews, P. B. Resolution in type theory. Journal of Symbolic Logic 36 (1971) 414 -- 432.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....Uses of Higher Order Logic in Computational Linguistics Dale A. Miller and Gopalan Nadathur Computer and Information Science University of Pennsylvania Philadelphia, PA 19104 3897 Abstract Consideration of the question of meaning in the framework of linguistics often requires an allusion to sets and other higher order notions. The traditional approach to representing and reasoning about meaning in a computa tional setting has been to use knowledge ....

....Uses of Higher Order Logic in Computational Linguistics Dale A. Miller and Gopalan Nadathur Computer and Information Science University of Pennsylvania Philadelphia, PA 19104 3897 Abstract Consideration of the question of meaning in the framework of linguistics often requires an allusion to sets and other higher order notions. The traditional approach to representing and reasoning about meaning in a computa tional setting has been to use knowledge ....

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Peter B. Andrews, "Resolution in Type Theory," Journal of Symbolic Logic 36 (1971), 414 - 432.

....1 ; t m are called its arguments, A t 1 : t m ) is called its body and the term is said to be rigid if A is a constant or an element of fx 1 ; x n g, and exible otherwise. Every term in our typed language can be transformed into such a form modulo the lambda conversion rules [4]. Moreover, the results of applying a substitution to a term and to any one of its head normal forms are equal under these rules. Thus, we may restrict our attention to terms in such a form as we henceforth do. Huet s uni cation procedure consists of the repetitive use of two phases for ....

Peter B. Andrews. Resolution in type theory. Journal of Symbolic Logic, 36:414-432, 1971.

....metaprogramming. These and many other gains are discussed extensively in [14] Semantics: Declarative and Operational Over the past decade a series of papers have provided semantics for certain classical or intuitionistic fragments of HOHH (see [22, 3] based on work by Henkin [13] and Andrews [1] on the semantics of Church s classical Type theory [6] Nadathur s dissertation [20] provides a notion of term model for the HOHH fragment. Miller de nes a Kripke like bottom up semantics for a rst order fragment [15] in which the syntax of programs is built into the notion of model. In A ....

Peter Andrews. Resolution in type theory. Journal of Symbolic Logic, 36:414-432, 1971.

....metaprogramming. These and many other gains are discussed extensively in [7] Semantics: Declarative and Operational Over the past decade a series of papers have provided semantics for certain classical or intuitionistic fragments of HOHH (see [16, 3] based on work by Henkin [6] and Andrews [1] on the semantics of Church s classical Type theory [4] Nadathur s dissertation [12] provides a notion of term model for the HOHH fragment and lays ground work for a bottom up semantics. Miller defined a Kripke like bottom up semantics for a first order fragment [8] in which the syntax of ....

....is built into the notion of model. In A semantics for Prolog , D.A. Wolfram [16] provided semantics for the higher order Horn clause fragment of Prolog, a result which was independently established in Bai and Blair [3] and was implicit in Nadathur [13] These authors used Andrews V complexes [1] to provide a classical model theory for the higher order Horn clause fragment of Prolog. However, the larger hereditarily Harrop fragment requires constructive logic and a constructive model theory. Operationality: bottom up semantics In [8] Miller produced a variant of Kripke style models for ....

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Peter Andrews. Resolution in type theory. Journal of Symbolic Logic, 36:414--432, 1971.

....a stronger notion of a variable and a term, but at the same time encompasses only the most primitive logical notions that are relevant in this context; only these notions appear to be of consequence from the perspective of computational applications. Our logic is closely related to that of [1], the only real differences being the inclusion of j conversion as a rule of inference and the incorporation of a larger number of propositional connectives and quantifiers as primitives. We describe this logic below, simultaneously introducing the logical vocabulary used in the rest of the ....

....( normal) form of G. From the Church Rosser Theorem, described in, e.g. 3] for a calculus without type symbols but applicable to the language under consideration as well, it follows that a fi normal ( normal) form of a formula is unique up to a renaming of bound variables. Further, it is known [1, 8] that a fi normal form exists for every formula in the typed calculus; this may be obtained by repeatedly replacing subformulas of the form [x:A] B by S x B A, preceded, perhaps, by some ff conversion steps. Such a formula may be converted into a normal form by replacing each subformula of ....

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Andrews, P. B. Resolution in type theory. Journal of Symbolic Logic 36 (1971) 414 -- 432.

....propositions is then avoided, but instead of looking for syntactically equal instances of two terms, unification has to search for convertible instances of those terms. In other words, the conversion steps of the early proof searching methods are integrated within the unification process [2]: this is the so called higher order unification, or unification in simply typed calculus [26] A similar approach has been applied successfully to many theories of interest, by removing equational axioms like associativity and commutativity from the theory and integrating them within the ....

P. B. Andrews, Resolution in type theory, Journal of Symbolic Logic, 36 (1971), pp. 414--432.

....of equivalent propositions is then avoided, but instead of looking for syntactically equal instances of two terms, unification has to search for convertible instances of those terms. In other words, the conversion steps of the early proof searching methods are mixed with the unification process [And71]: this is the so called higher order unification, or unification in simply typed calculus [Hue75] A similar transformation has been applied successfully to a lot of theories: equational axioms are removed from the theory but mixed with the unification process, leading to equational unification ....

P. Andrews. Resolution in type theory. Journal of Symbolic Logic, 36:414--432, 1971.

....and developed by Robinson has been extended in two ways: equational unication and higher order unication. The same idea is at the root of both extensions. It consists in incorporating in the unication process some knowledge about the underlying theory, in the higher order case, fij conversion [2] and in the case of a rst order theory, some equational axioms, such as associativity [12] Nevertheless although the goal is the same, the unication processes designed so far are fairly different. In this paper, we show that higher order uni cation can be reduced to rst order equational uni ....

P. Andrews. Resolution in type theory. Journal of Symbolic Logic, 36:414432, 1971.

....of a given programming language for certain applications. Although less work has been done on theorem proving in higher order logic than in first order logic as claimed in the last point, the nature of proofs in higherorder logic is far from mysterious. For example, higherorder resolution [1] and unification [8] has been developed, and based on these principles, several theorem provers for various higher order logics (see [2] and its references) have been built and tested. The experience with such systems shows that theorem proving in such a logic is difficult. It is not clear, ....

Peter B. Andrews, "Resolution in Type Theory," Journal of Symbolic Logic 36 (1971), 414 -- 432.

....higher order logic is almost as old as the field of deduction systems itself. The first successful attempts to mechanize and implement higherorder logic were those of Huet [Hue73] and Jensen and Pietrzykowski [JP76] They combine the resolution principle for higher order logic (first studied in [And71] with higher order unification. The unification problem in typed calculi is much more complex than that for first order terms, since it has to take the theory of fffij equality into account. In particular the higher order unification problem is undecidable and sets of solutions need not to have ....

Peter B. Andrews. Resolution in type theory. Journal of Symbolic Logic, 36(3):414--432, 1971.

....its pure term rewriting character. On the other hand examples demonstrate that the extensionality rules harmonise quite well with the difference reducing HO RUE resolution idea. 1 Introduction Higher Order (HO) Theorem Proving based on the resolution method has been first examined by Andrews [And71] and Huet [Hue72] Whereas the former avoids unification the latter generally delays the computation of unifiers and instead adds unification constraints to the clauses in order to tackle the undecidability problem of HO unification. More recent papers concentrate on the adaption of sorts [Koh94] ....

....8Qo (P = Q) P , Q) are valid (see [Ben99,BK97] Satisfiability and validity (M j= F or M j= Phi) of a formula F or set of formulas Phi in a model M is defined as usual. The completeness proofs employ the abstract consistency method of [BK97] Ben99] which extends Andrews HO adaptation [And71] of Smullyan s approach [Smu63] to Henkin semantics. Here we only mention the two main aspects: Definition 1 (Acc for Henkin Models) Let Sigma be a signature and Gamma Sigma a class of sets of Sigma sentences. If the following conditions (all but r e ) hold for all A;B 2 cwff o , F; G 2 ....

P. B. Andrews. Resolution in type theory. JSL, 36(3):414--432, 1971.

....explicitly in the proof. Indeed, these axioms are now redundant: as the terms 0 y and y are congruent, the axiom 8y 0 y = y is congruent to the equality axiom 8y y = y. Hence, it can be dropped. Using the terminology introduced by Plotkin, these axioms have been builtin (Plotkin, 1972; Andrews, 1971; Peterson and Stickel, 1981; Stickel, 1985; Jouannaud and Kirchner, 1986; March e, 1994; Viry, 1995; Viry, 1998) In the example above, the congruence is just the congruent closure of the relation induced on terms by the term rewriting system. In many situations, it is also natural to consider ....

Andrews, P. (1971). Resolution in type theory. Journal of Symbolic Logic, 36:414-432.

....by P R Q if and only if there is a conversion step from P to Q. Proposition 2.7 If P R Q and T P then T Q. RR n Sigma3383 14 Gilles Dowek From proposition 2.3, R is an equivalence relation. Thus, we can consider the quotient P=R of classes of propositions modulo R (Plotkin Andrews quotient [1, 11]) We dene the same deduction rules on classes of P=R as on propositions of P . Remark Proof checking is decidable if R is decidable and provided we indicate the substituted term in the elimination rule of the universal quantier and the introduction rule of the existential quantier. We have the ....

P.B. Andrews, Resolution in type theory, The Journal of Symbolic Logic, 36, 3 (1971) pp. 414-432.

....of equivalent propositions is then avoided, but instead of looking for syntactically equal instances of two terms, unication has to search for convertible instances of those terms. In other words, the conversion steps of the early proof searching methods are integrated within the unication process [And71]: this is the so called higher order unication, or unication in simply typed calculus [Hue75] A similar approach has been applied successfully to many theories of interest, by removing equational axioms like associativity and commutativity from the theory and integrating them within the ....

P. Andrews. Resolution in type theory. Journal of Symbolic Logic, 36:414432, 1971.

....since it does not admit complete calculi [G od31] However, there is a more general notion of semantics (the so called Henkinmodels [Hen50] that allows complete calculi and therefore sets the standard for deductive power of calculi. Peter Andrews Unifying Principle for Type Theory [And71] provides a method of higher order abstract consistency that has become the standard tool for completeness proofs in higher order logic, even though it can only be used to show completeness relative to a certain Hilbert style calculus T. A calculus C is called complete relative to a calculus T, ....

....proves all theorems of T. Since T is not necessarily complete with respect to Henkin models, the notion of completeness that can be established by this method is a strictly weaker notion than Henkin completeness. As a consequence, the calculi developed for higher order automated theorem proving [And71, Hue72, Hue73, JP72, Mil83, Koh94b, Koh95] and the corresponding theorem proving systems such as Tps [ABI 96] or earlier versions of the authors Leo 1 are not or cannot be proven complete with respect to Henkin models. Moreover, they are not even sound with respect to T, since all of them ....

[Article contains additional citation context not shown here]

Peter B. Andrews. Resolution in type theory. Journal of Symbolic Logic, 36(3):414{ 432, 1971.

....To instantiate this scheme, we apply it to a term of type o; for example, M(x: x x) is definitionally equal to oe (8 (x: xx) 9 (x: xx) which is the encoding of (8x:x = x) oe (9x:x = x) 3. 2 Higher Order Logic There are many ways to present higher order logic (see, for example, [6, 1, 51, 54]) For our version we follow Church in using the simply typed calculus [35] to form expressions of the logic. The language of simple functional types oe, is given by the following abstract syntax: oe : j o j oe where and o are basic types (of individuals and propositions) The language ....

Andrews, P. B. Resolution in type theory. Journal of Symbolic Logic 36 (1971), 414--432.

....Given t 2 , we know T (t) 2 by Theorem 10. Hence T (t) is weakly normalisable. By Theorem 8, t is strongly normalisable. It can be readily shown that all are weakly normalisable by a well known method originally due to Turing according to Gandy (1980) which can also be found in Andrews (1971) and Girard (1989) Therefore, the simply typed calculus enjoys strong normalisation theorem. Theorem 12 If Gamma t : ff is derivable in F, then T ( Gamma) T (t) jffj is also derivable in F. Proof The proof corresponds exactly to Kolmogorov s double negation embedding for second order ....

Andrews, P.B., (1971), Resolution in type theory, J. Symbolic Logic 36, pp. 414-432.

....in simply typed lambda calculus is strongly normalising. A proof of a weak normalisation theorem for simply typed calculus can be given by a method originating from Turing s work [Gan80] independently invented by Prawitz [Pra65] in proof theory. A detailed account of it can also be found in [And71]. Several authors have invented between techniques to infer from this result strong normalisation of simply typed calculus and related systems, see [Ned73] Klo80] deGr93] and [KW94] Another proof, using a different characterisation of strongly normalising terms is given in [RS95] ....

P.B. Andrews (1971), Resolution in type theory, J. Symbolic Logic 36, pp. 414-432.

....(t) for I terms t if we can find any normalisation sequences for them. In order to get a tighter bound, the key is to find shorter normalisation sequences. We start with a weak normalisation proof due to Turing according to [Gan80] which can also be found in many other literatures such as [And71, GLT89]. Definition 5.5 The rank ae(T) of a simple type T is defined as follows. ae(T) 0 if T is atomic; 1 maxfae(T 0 ) ae(T 1 )g if T = T 0 T 1 . The rank ae(r) of a fi redex r = x U :v V )u U is ae(U V ) and the rank of a term t is ae(t) 8 : h0; 0i if t is in fi normal ....

....in t with ae(r) ki, otherwise. The ranks of terms are lexicographically ordered. Notice that a fi redex has a redex rank, which is a number, and also has a term rank, which is a pair of numbers. Since several (weak) normalisation proofs for in the following style have been given in details [And71, GLT89], we only make some observations which we need shortly. Observations Now let us observe the following. ffl If t Gamma fi t 0 and fi redex r 0 in t 0 is a residual of some fi redex r in t, then ae(r 0 ) ae(r) ffl Given t = C[r] with ae(t) hk; ni, where r = x V :u U )v V is ....

P.B. Andrews (1971), Resolution in type theory, J. Symbolic Logic 36, pp. 414-432.

....free proof then: c (ffA 1 g; fA n g; f:B 1 ; B p gg) 2 and thus, by the proposition 4.3 c (ffA 1 g; fA n g; f:B 1 ; B p gg) 7 7 2[C] where C is a E uni able set of constraints. 2 Theorem Proving Modulo 27 5. A typical example: Higher order logic In (Andrews, 1971) P.B. Andrews proposes to build in conversion axioms of higher order logic. When automating reasonning, uni cation is then replaced by uni cation modulo conversion, usually called higher order uni cation (Huet, 1975) which is the kernel of the constrained resolution de ned by G. Huet (Huet, ....

Andrews, P.: 1971, `Resolution in Type Theory'. Journal of Symbolic Logic 36, 414-432.

....on (t) for I terms t if we can find any normalization sequences for them. In order to get a tighter bound, the key is to find shorter normalization sequences. We start with a weak normalization proof due to Turing according to [Gan80] which can also be found in many other literatures such as [And71] and [GLT89] Definition 23 The rank ae(T ) of a simple type T is defined as follows. ae(T ) 0 if T is atomic; 1 maxfae(T 0 ) ae(T 1 )g if T = T 0 T 1 . The rank ae(r) of a fi redex r = x U :v V )u U is ae(U V ) and the rank of a term t is ae(t) h0; 0i if t is in ....

P.B. Andrews (1971), Resolution in type theory, J. Symbolic Logic 36, pp. 414-432.

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Peter B. Andrews, Resolution in Type Theory, Journal of Symbolic Logic 36 (1971), 414-432.

....procedure which was in principle complete for first order logic, though not for type theory. As a start toward extending the mating method to a complete method for proving theorems of higher order logic, it was shown in [6] that a sentence is a theorem of elementary type theory (the system of [1] and [2] if and only if it has a tautologous development, where a development is the analogue of a Herbrand expansion of a sentence of first order logic. Once one has found a tautologous development for a theorem, one can construct a proof of it in natural deduction style without further search. ....

....theorem appears in [36] where it is noted that the theorem can be used in solving integral equations of the second kind; it justifies showing that a fixed point equation has a solution by showing that the iterated equation has a unique solution. This was posed as a problem for theorem provers in [1], and it is gratifying that we are finally able to prove it automatically. It is a hard theorem for TPS because so many flexible literals are created when the definition of equality is instantiated. The very natural, though overly detailed, proof that TPS finds is shown in Figure 5.1. Let us ....

Andrews, P. B.: Resolution in type theory, J. Symbolic Logic 36 (1971), 414--342.

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Andrews, P. B. Resolution in type theory. Journal of Symbolic Logic 36 (1971) 414 -- 432.

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Peter B. Andrews. Resolution in type theory. Journal of Symbolic Logic, 3(36):414-432, 1971.

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