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24
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 36 (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.
On the Complexity of Linear and Stratified Context Matching Problems
, 2001
"... We investigate the complexity landscape of context matching with respect to the number of occurrences of variables (i.e. linearity vs. varity 2) and various restrictions of stratification. We show that stratified context matching (SCM) and varity 2 context matching are NPcomplete, but that stratifi ..."
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Cited by 18 (1 self)
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We investigate the complexity landscape of context matching with respect to the number of occurrences of variables (i.e. linearity vs. varity 2) and various restrictions of stratification. We show that stratified context matching (SCM) and varity 2 context matching are NPcomplete, but that stratified simultaneous monadic context matching (SSMCM) is in P. SSMCM is equivalent to stratified simultaneous word matching (SSWM). We also show that the linear and the Comonrestricted case are in P and of time complexity O(n&sup3;). We give an algorithm for context matching and discuss how the performance of the general case can be improved through the use of information derived from polynomial approximations of the problem.
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 17 (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 ...
DECIDABILITY OF HIGHERORDER MATCHING
"... Abstract. We show that the higherorder matching problem is decidable using a gametheoretic argument. ..."
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Abstract. We show that the higherorder matching problem is decidable using a gametheoretic argument.
Recognizability in the Simply Typed LambdaCalculus, in "16th Workshop on Logic, Language, Information and Computation
 Lecture Notes in Artificial Intelligence, vol. 5514, Springer, 2009, p. 48–60, http://hal.inria.fr/inria00412654/en/. Scientific Books (or Scientific Book chapters
"... Abstract. We define a notion of recognizable sets of simply typed λterms that extends the notion of recognizable sets of strings or trees. This definition is based on finite models. Using intersection types, we generalize the notions of automata for strings and trees so as to grasp recognizability ..."
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Abstract. We define a notion of recognizable sets of simply typed λterms that extends the notion of recognizable sets of strings or trees. This definition is based on finite models. Using intersection types, we generalize the notions of automata for strings and trees so as to grasp recognizability for λterms. We then expose the closure properties of this notion and present some of its applications. 1
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 8 (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.
HigherOrder Positive Set Constraints
 In 16th Int. Workshop Computer Science Logic (CSL
, 2002
"... We introduce a natural notion of positive set constraints on simplytyped terms. We show that satisfiability of these socalled positive higherorder set constraints is decidable in 2NEXPTIME. We explore a number of subcases solvable in 2DEXPTIME, among which higherorder definite set constrai ..."
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Cited by 7 (2 self)
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We introduce a natural notion of positive set constraints on simplytyped terms. We show that satisfiability of these socalled positive higherorder set constraints is decidable in 2NEXPTIME. We explore a number of subcases solvable in 2DEXPTIME, among which higherorder definite set constraints, a.k.a., emptiness of higherorder pushdown processes. This uses a firstorder clause format on socalled shallow higherorder patterns, and automated deduction techniques based on ordered resolution with splitting. This technique is then applied to the task of approximating success sets for a restricted subset of  Prolog, a la Fr uhwirth et al.
Higherorder beta matching with solutions in long betaeta normal form
 Nordic Journal of Computing
, 2006
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Higherorder matching, games and automata
 Proceedings of 22nd Annual IEEE Symposium on Logic in Computer Science, (LICS
, 2007
"... Higherorder matching is the problem given t = u where t, u are terms of simply typed λcalculus and u is closed, is there a substitution θ such that tθ and u have the same normal form with respect to βηequality: can t be pattern matched to u? This paper considers the question: can we characterize ..."
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Cited by 4 (4 self)
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Higherorder matching is the problem given t = u where t, u are terms of simply typed λcalculus and u is closed, is there a substitution θ such that tθ and u have the same normal form with respect to βηequality: can t be pattern matched to u? This paper considers the question: can we characterize the set of all solution terms to a matching problem? We provide an automatatheoretic account that is relative to resource: given a matching problem and a finite set of variables and constants, the (possibly infinite) set of terms that are built from those components and that solve the problem is regular. The characterization uses standard bottomup tree automata. 1.
On the Expressive Power of Schemes
"... We present a calculus, called the schemecalculus, that permits to express natural deduction proofs in various theories. Unlike λcalculus, the syntax of this calculus sticks closely to the syntax of proofs, in particular, no names are introduced for hypotheses. We show that despite its nondetermini ..."
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We present a calculus, called the schemecalculus, that permits to express natural deduction proofs in various theories. Unlike λcalculus, the syntax of this calculus sticks closely to the syntax of proofs, in particular, no names are introduced for hypotheses. We show that despite its nondeterminism, this schemecalculus has the same expressivity as the corresponding typed λcalculus. or, in a simpler way, as λxλy (2 × x × x × y) Other solutions have been investigated. A solution related to Bourbaki’s, is to express the link with a directed edge from each ✷ to the corresponding λ λλ 2 × × × 1.