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40
On the Undecidability of SecondOrder Unification
 INFORMATION AND COMPUTATION
, 2000
"... ... this paper, and it is the starting point for proving some novel results about the undecidability of secondorder unification presented in the rest of the paper. We prove that secondorder unification is undecidable in the following three cases: (1) each secondorder variable occurs at most t ..."
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Cited by 33 (16 self)
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... this paper, and it is the starting point for proving some novel results about the undecidability of secondorder unification presented in the rest of the paper. We prove that secondorder unification is undecidable in the following three cases: (1) each secondorder variable occurs at most twice and there are only two secondorder variables; (2) there is only one secondorder variable and it is unary; (3) the following conditions (i)#(iv) hold for some fixed integer n: (i) the arguments of all secondorder variables are ground terms of size <n, (ii) the arity of all secondorder variables is <n, (iii) the number of occurrences of secondorder variables is #5, (iv) there is either a single secondorder variable or there are two secondorder variables and no firstorder variables.
Matching Power
 Proceedings of RTA’2001, Lecture Notes in Computer Science, Utrecht (The Netherlands
, 2001
"... www.loria.fr/{~cirstea,~ckirchne,~lliquori} Abstract. In this paper we give a simple and uniform presentation of the rewriting calculus, also called Rho Calculus. In addition to its simplicity, this formulation explicitly allows us to encode complex structures such as lists, sets, and objects. We pr ..."
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Cited by 31 (20 self)
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www.loria.fr/{~cirstea,~ckirchne,~lliquori} Abstract. In this paper we give a simple and uniform presentation of the rewriting calculus, also called Rho Calculus. In addition to its simplicity, this formulation explicitly allows us to encode complex structures such as lists, sets, and objects. We provide extensive examples of the calculus, and we focus on its ability to represent some object oriented calculi, namely the Lambda Calculus of Objects of Fisher, Honsell, and Mitchell, and the Object Calculus of Abadi and Cardelli. Furthermore, the calculus allows us to get object oriented constructions unreachable in other calculi. In summa, we intend to show that because of its matching ability, the Rho Calculus represents a lingua franca to naturally encode many paradigms of computations. This enlightens the capabilities of the rewriting calculus based language ELAN to be used as a logical as well as powerful semantical framework. 1
HigherOrder Coloured Unification and Natural Language Semantics
, 1996
"... In this paper, we show that HigherOrder Coloured Unification  a form of unification developed for automated theorem proving  provides a general theory for modeling the interface between the interpretation process and other sources of linguistic, non semantic information. In particular, it prov ..."
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Cited by 24 (14 self)
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In this paper, we show that HigherOrder Coloured Unification  a form of unification developed for automated theorem proving  provides a general theory for modeling the interface between the interpretation process and other sources of linguistic, non semantic information. In particular, it provides the general theory for the Primary Occurrence Restriction which (Dalrymple et al., 1991)'s analysis called for. 1 Introduction It is well known that HigherOrder Unification (HOU) can be used to construct the semantics of Natural Language: (Dalrymple et al., 1991)  henceforth, DSP  show that it allows a treatment of VP Ellipsis which successfully captures the interaction of VPE with quantification and nominal anaphora; (Pulman, 1995; Gardent and Kohlhase, 1996) use HOU to model the interpretation of focus and its interaction with focus sensitive operators, adverbial quantifiers and second occurrence expressions; (Gardent et al., 1996) shows that HOU yields a simple but precise ...
HigherOrder Matching and Tree Automata
 Proc. Conf. on Computer Science Logic
, 1997
"... ions x 1 : : : xn are assumed to have arity one. For instance, x 1 x 2 :c(x 3 :x 3 ; x 2 (x 1 )) (assumed in normal form) has the following representation as a tree: x 1 x 2 c \Gamma \Gamma @ @ x 3 x 2 x 3 x 1 In what follows, we assume that F is finite. This is not a restriction as, for countab ..."
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Cited by 21 (0 self)
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ions x 1 : : : xn are assumed to have arity one. For instance, x 1 x 2 :c(x 3 :x 3 ; x 2 (x 1 )) (assumed in normal form) has the following representation as a tree: x 1 x 2 c \Gamma \Gamma @ @ x 3 x 2 x 3 x 1 In what follows, we assume that F is finite. This is not a restriction as, for countably infinite alphabets, there is always another alphabet F 0 , which is finite, and an injective tree homomorphism h from T (F) into T (F) 0 such that h(T (F)) is recognizable by a finite tree automaton and the size of h(t) is linear with respect to the size of t. 1 However, for sake of clarity, we will keep the standard notations instead of using the encodings of F . 3.2 2automata We will use a slight modification of tree automata. The main difference with the definitions of [13, 4] is the presence of special symbols 2 ø which should be interpreted as any term of type ø . This slight modification is necessary because, for instance, the set of all closed terms is not recognizable by ...
Decidable higherorder unification problems
 AUTOMATED DEDUCTION  CADE12. SPRINGER LNAI 814
, 1994
"... Secondorder unification is undecidable in general. Miller showed that unification of socalled higherorder patterns is decidable and unitary. Weshow that the unification of a linear higherorder pattern s with an arbitrary secondorder term that shares no variables with s is decidable and finitar ..."
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Cited by 17 (4 self)
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Secondorder unification is undecidable in general. Miller showed that unification of socalled higherorder patterns is decidable and unitary. Weshow that the unification of a linear higherorder pattern s with an arbitrary secondorder term that shares no variables with s is decidable and finitary. A few extensions of this unification problem are still decidable: unifying two secondorder terms, where one term is linear, is undecidable if the terms contain bound variables but decidable if they don't.
The rewriting calculus  Part I
, 2001
"... The ρcalculus integrates in a uniform and simple setting firstorder rewriting, λcalculus and nondeterministic computations. Its abstraction mechanism is based on the rewrite rule formation and its main evaluation rule is based on matching modulo a theory T. In this first part, the calculus is mot ..."
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Cited by 16 (1 self)
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The ρcalculus integrates in a uniform and simple setting firstorder rewriting, λcalculus and nondeterministic computations. Its abstraction mechanism is based on the rewrite rule formation and its main evaluation rule is based on matching modulo a theory T. In this first part, the calculus is motivated and its syntax and evaluation rules for any theory T are presented. In the syntactic case, i.e. when T is the empty theory, we study its basic properties for the untyped case. We first show how it uniformly encodes λcalculus as well as firstorder rewriting derivations. Then we provide sufficient conditions for ensuring confluence of the calculus.
Higherorder Matching for Program Transformation
, 1999
"... We present a simple, practical algorithm for higher order matching in the context of automatic program transformation. Our algorithm finds more matches than the standard second order matching algorithm of Huet and Lang, but it has an equally simple specification, and it is better suited to the tr ..."
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Cited by 14 (1 self)
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We present a simple, practical algorithm for higher order matching in the context of automatic program transformation. Our algorithm finds more matches than the standard second order matching algorithm of Huet and Lang, but it has an equally simple specification, and it is better suited to the transformation of programs in modern programming languages such as Haskell or ML. The algorithm has been implemented as part of the MAG system for transforming functional programs. 1 Background and motivation 1.1 Program transformation Many program transformations are conveniently expressed as higher order rewrite rules. For example, consider the wellknown transformation that turns a tail recursive function into an imperative loop. The pattern f x = if p x then g x else f (h x ) is rewritten to the term f x = j[ var r ; r := x ; while :(p r) do r := h r ; r := g r ; return r ]j Carefully consider the pattern in this rule: it involves two bound variables, namely f and x , and ...
The "Hardest" Natural Decidable Theory
 12th Annual IEEE Symp. on Logic in Computer Science (LICS'97)', IEEE
, 1997
"... We prove that any decision procedure for a modest fragment of L. Henkin's theory of pure propositional types [7, 12, 15, 11] requires time exceeding a tower of 2's of height exponential in the length of input. Until now the highest known lower bounds for natural decidable theories were at most linea ..."
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Cited by 10 (4 self)
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We prove that any decision procedure for a modest fragment of L. Henkin's theory of pure propositional types [7, 12, 15, 11] requires time exceeding a tower of 2's of height exponential in the length of input. Until now the highest known lower bounds for natural decidable theories were at most linearly high towers of 2's and since midseventies it was an open problem whether natural decidable theories requiring more than that exist [12, 2]. We give the affirmative answer. As an application of this today's strongest lower bound we improve known and settle new lower bounds for several problems in the simply typed lambda calculus. 1. Introduction In his survey paper [12] A. Meyer mentioned (p. 479), as a curious empirical observation, that all known natural decidable nonelementary problems require at most (upper bound) F (1; n) = exp 1 (n) = 2 2 \Delta \Delta \Delta 2 oe n Turing machine steps on inputs of length n to decide 1 . Until now the highest known lower bounds for natu...
Decidability of Bounded HigherOrder Unification
, 2002
"... It is shown that unifiability of terms in the simply typed lambda calculus with beta and eta rules becomes decidable if there is a bound on the number of bound variables and lambdas in a unifier in etaexpanded betanormal form. ..."
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Cited by 7 (0 self)
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It is shown that unifiability of terms in the simply typed lambda calculus with beta and eta rules becomes decidable if there is a bound on the number of bound variables and lambdas in a unifier in etaexpanded betanormal form.
Generation as Deduction on Labelled Proof Nets
 Logical Aspects of Computational Linguistics, LACL’96, volume 1328 of Lecture Notes in Artificial Intelligence
"... . In the framework of labelled proof nets the task of parsing in categorial grammar can be reduced to the problem of firstorder matching under theory. Here we shall show how to use the same method of labelled proof nets to reduce the task of generating to the problem of higherorder matching. 1 Int ..."
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Cited by 5 (1 self)
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. In the framework of labelled proof nets the task of parsing in categorial grammar can be reduced to the problem of firstorder matching under theory. Here we shall show how to use the same method of labelled proof nets to reduce the task of generating to the problem of higherorder matching. 1 Introduction Categorial grammar provides a mechanism for the analysis of linguistic expressions on the basis of lexicalism and the parsing as deduction paradigm ([17]). 3 In accordance with lexicalism each lexical entry of the language encapsulates all the information needed to analyse the lexical item, and the grammar itself only needs to know how to manage these resources. In the particular case of categorial grammar, a lexical categorisation is a formula, or type, constructed over some basic types by logical connectives; and the grammar constitutes the connectives' syntactic behaviour (i.e. the laws governing the connectives). Within the parsing as deduction paradigm the problem of analys...