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Third Order Matching is Decidable
 Annals of Pure and Applied Logic
, 1999
"... The higher order matching problem is the problem of determining whether a term is an instance of another in the simply typed calculus, i.e. to solve the equation a = b where a and b are simply typed terms and b is ground. The decidability of this problem is still open. We prove the decidability of ..."
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The higher order matching problem is the problem of determining whether a term is an instance of another in the simply typed calculus, i.e. to solve the equation a = b where a and b are simply typed terms and b is ground. The decidability of this problem is still open. We prove the decidability of the particular case in which the variables occurring in the problem are at most third order. Introduction The higher order matching problem is the problem of determining whether a term is an instance of another in the simply typed calculus i.e. to solve the equation a = b where a and b are simply typed terms and b is ground. Pattern matching algorithms are used to check if a proposition can be deduced from another by elimination of universal quantifiers or by introduction of existential quantifiers. In automated theorem proving, elimination of universal quantifiers and introduction of existential quantifiers are mixed and full unification is required, but in proofchecking and semiaut...
A New Characterization of Lambda Definability
, 1993
"... . We give a new characterization of lambda definability in Henkin models using logical relations defined over ordered sets with varying arity. The advantage of this over earlier approaches by Plotkin and Statman is its simplicity and universality. Yet, decidability of lambda definability for heredit ..."
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. We give a new characterization of lambda definability in Henkin models using logical relations defined over ordered sets with varying arity. The advantage of this over earlier approaches by Plotkin and Statman is its simplicity and universality. Yet, decidability of lambda definability for hereditarily finite Henkin models remains an open problem. But if the variable set allowed in terms is also restricted to be finite then our techniques lead to a decision procedure. 1 Introduction An applicative structure consists of a family (A oe ) oe2T of sets, one for each type oe, together with a family (app oe;ø ) oe;ø 2T of application functions, where app oe;ø maps A oe!ø \Theta A oe into A ø . For an applicative structure to be a model of the simply typed lambda calculus (in which case we call it a Henkin model, following [4]), one requires two more conditions to hold. It must be extensional which means that the elements of A oe!ø are uniquely determined by their behavior under app oe;ø...
A simple proof of a theorem of Statman
 Theoretical Computer Science
, 1992
"... In this note, we reprove a theorem of Statman that deciding the fijequality of two firstorder typable terms is not elementary recursive [Sta79]. The basic idea of our proof, like that of Statman's, is the Henkin quantifier elimination procedure [Hen63]. However, our coding is much simpler, and eas ..."
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In this note, we reprove a theorem of Statman that deciding the fijequality of two firstorder typable terms is not elementary recursive [Sta79]. The basic idea of our proof, like that of Statman's, is the Henkin quantifier elimination procedure [Hen63]. However, our coding is much simpler, and easy to understand. 1 Introduction A well known theorem of Richard Statman states that if we have two terms that are firstorder typable, deciding whether the terms reduce to the same normal form is not Kalmar elementary: namely, it cannot be decided in f k (n) steps for any fixed integer k 0, where n is the length of the two terms, and f 0 (n) = n, f t+1 (n) = 2 f t (n) . The theorem is often cited, but in contrast, its proof is not well understood. In this note, we give a simple proof of the theorem. The key idea that vastly simplifies the technical details of the proof is to use list iteration as a quantifier elimination procedure. 2 Preliminaries 2.1 Deciding truth of formulas in high...
Hereditarily Sequential Functionals: A GameTheoretic Approach to Sequentiality
, 1996
"... The aim of this thesis is to give a new understanding of sequential computations in higher types. We present a new computation model for higher types based on a game describing the interaction between a functional and its arguments. The functionals which may be described in this way are called hered ..."
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The aim of this thesis is to give a new understanding of sequential computations in higher types. We present a new computation model for higher types based on a game describing the interaction between a functional and its arguments. The functionals which may be described in this way are called hereditarily sequential. We show that this computation model captures exactly the notion of computability in higher types introduced by Kleene in his pioneering work starting 1959. We study the order structure of the hereditarily sequential functionals and discuss the occurring difficulties. These functionals form a fully abstract model for PCF and we discuss which problems remain still open for a satisfactory solution to the full abstraction problem of PCF. Zusammenfassung Ziel dieser Arbeit ist es, eine neue Beschreibung sequentieller Berechnungen in hoheren Typen zu geben. Wir stellen dazu ein neues Berechnungsmodell fur hohere Typen vor, in dem die Interaktion zwischen einem Funktional und se...
Models of Lambda Calculi and Linear Logic: Structural, Equational and ProofTheoretic Characterisations
, 1994
"... Models of Lambda Calculi and Linear Logic: Structural, Equational and ProofTheoretic Characterisations Ralph Loader, of St. Hugh's College, Oxford. Thesis submitted for the Degree of D.Phil. Michaelmas term, 1994. T his thesis is an investigation into models of typed calculi and of linear logic. ..."
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Models of Lambda Calculi and Linear Logic: Structural, Equational and ProofTheoretic Characterisations Ralph Loader, of St. Hugh's College, Oxford. Thesis submitted for the Degree of D.Phil. Michaelmas term, 1994. T his thesis is an investigation into models of typed calculi and of linear logic. The models we investigate are denotational in nature; we construct various categories, in which types (or formulae) are interpreted by objects, and terms (proofs) by morphisms. The results we investigate compare particular properties of the syntax and the semantics of a calculus, by trying to use syntax to characterise features of a model, or vice versa. There are four chapters in the thesis, one each on linear logic and the simply typed calculus, and two on inductive datatypes. In chapter one, we look at some models of linear logic, and prove a full completeness result for multiplicative linear logic. We form a model, the linear logical predicates , by abstracting a little the structure ...
Categorical Completeness Results for the SimplyTyped LambdaCalculus
 Proceedings of TLCA '95, Springer LNCS 902
, 1995
"... . We investigate, in a categorical setting, some completeness properties of betaeta conversion between closed terms of the simplytyped lambda calculus. A cartesianclosed category is said to be complete if, for any two unconvertible terms, there is some interpretation of the calculus in the catego ..."
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. We investigate, in a categorical setting, some completeness properties of betaeta conversion between closed terms of the simplytyped lambda calculus. A cartesianclosed category is said to be complete if, for any two unconvertible terms, there is some interpretation of the calculus in the category that distinguishes them. It is said to have a complete interpretation if there is some interpretation that equates only interconvertible terms. We give simple necessary and sufficient conditions on the category for each of the two forms of completeness to hold. The classic completeness results of, e.g., Friedman and Plotkin are immediate consequences. As another application, we derive a syntactic theorem of Statman characterizing betaeta conversion as a maximum consistent congruence relation satisfying a property known as typical ambiguity. 1 Introduction In 1970 Friedman proved that betaeta conversion is complete for deriving all equalities between the (simplytyped) lambdadefinable...
The Semantics of Types in Programming Languages
 Handbook of Logic in Computer Science
, 1994
"... ion: [[H B x : u: M 0 : u ! v]]ae is the function from [[u]] to [[v]] given by d 7! [[H; x : u B M 0 : v]](ae[x 7! d]), that is, the function f defined by f(d) = [[H; x : u B M 0 : v]](ae[x 7! d]): 22 Carl A. Gunter ffl Application: [[H B L(N ) : t]]ae is the value obtained by applying the ..."
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ion: [[H B x : u: M 0 : u ! v]]ae is the function from [[u]] to [[v]] given by d 7! [[H; x : u B M 0 : v]](ae[x 7! d]), that is, the function f defined by f(d) = [[H; x : u B M 0 : v]](ae[x 7! d]): 22 Carl A. Gunter ffl Application: [[H B L(N ) : t]]ae is the value obtained by applying the function [[H B L : s ! t]]ae to argument [[H B N : s]]ae where s is the unique type such that H ` L : s ! t and H ` N : s. It will save us quite a bit of ink to drop the parentheses that appear as part of expressions such as [[H; x : u B M 0 : v]](ae[x 7! d]) and simply write [[H; x : u B M 0 : v]]ae[x 7! d]. Doing so appears to violate the convention of associating applications to the left, but there is little chance of confusion in the case of expressions such as these. Hence, we will adopt the convention that the postfix update operator binds more tightly than general application. It can be shown that this assignment of meanings respects our equational rules. This is the soundness ...
On statman's finite completeness theorem
, 1992
"... Abstract We give a complete selfcontained proof of Statman's finite completeness theorem and of a corollary of this theorem stating that the *definability conjecture implies the higherorder matching conjecture. ..."
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Abstract We give a complete selfcontained proof of Statman's finite completeness theorem and of a corollary of this theorem stating that the *definability conjecture implies the higherorder matching conjecture.
Complexity of the Higher Order Matching
 Automated Deduction. Volume 1632 of LNCS
, 2000
"... We use the standard encoding of Boolean values in simply typed lambda calculus to develop a method of translating SAT problems for various logics into higher order matching. We obtain this way already known NPhardness bounds for the order two and three and a new result that the fourth order matchin ..."
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We use the standard encoding of Boolean values in simply typed lambda calculus to develop a method of translating SAT problems for various logics into higher order matching. We obtain this way already known NPhardness bounds for the order two and three and a new result that the fourth order matching is NEXPTIMEhard. 1 Introduction Consider two normalized simply typed lambda terms M and N, where N is closed (does not contain free variables). The higher order matching problem M ? = N (also known as pattern matching, 1 range problem or instantiation problem) is to decide whether there exists a substitution # for free variables in M, such that M# is ##reducible to N. Matching is a special case of unification, where the restriction that N is closed is removed (and a solution of M ? = N is a substitution # such that M# and N# are equal modulo ##conversion). The order of a problem M ? = N is the highest functionality order of free variables occurring in M. At the time of writin...
The Undecidability of lambdaDefinability
"... this article, we shall show that the PlotkinStatman conjecture [4, 7] is false. The conjecture was that, in a model of the simply typed calculus with only finitely many elements at each type, definability (by a closed term of the calculus) is decidable. This conjecture had been shown to imply many ..."
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this article, we shall show that the PlotkinStatman conjecture [4, 7] is false. The conjecture was that, in a model of the simply typed calculus with only finitely many elements at each type, definability (by a closed term of the calculus) is decidable. This conjecture had been shown to imply many things, for example, Statman [7] (see also Wolfram's book [8]) has shown it implies the decidability of pure higher order pattern matching (a problem that remains open at the time of writing) and is equivalent to higher order pattern matching with ffifunctions. The proof of undecidability given here uses encodings of semiThue systems as definability problems. It had been thought that definability might be characterised by invariance under logical relations, which would imply the PlotkinStatman conjecture. We give a relatively simple counterexample to this, using our encoding of word problems.