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Bounded Second-Order Unification Is NP-Complete [chapter]

Jordi Levy, Manfred Schmidt-Schauß, Mateu Villaret
2006 Lecture Notes in Computer Science  
It is already known that Bounded Second-Order Unification is decidable and NP-hard, whereas general Second-Order Unification is undecidable.  ...  We prove that Bounded Second-Order Unification is NP-complete, provided that m is given in unary encoding, by proving that a size-minimal solution can be represented in polynomial space, and then applying  ...  Bounded Second-Order Unification is NP-complete. Proof: For any equation E, and any size-minimal solution σ, there exists a STG of polynomial size in |E| that generates σ(X), for every X ∈ V ar(E).  ... 
doi:10.1007/11805618_30 fatcat:3vv2y2svezgjdaddzknhvqjr34

Stratified Context Unification Is NP-Complete [chapter]

Jordi Levy, Manfred Schmidt-Schauß, Mateu Villaret
2006 Lecture Notes in Computer Science  
A corollary of our result is that solvability of rewrite constraints is NP-complete.  ...  Context Unification is the problem to decide for a given set of second-order equations E where all second-order variables are unary, whether there exists a unifier, such that for every second-order variable  ...  Conclusion and Further Research We have shown that stratified context unification is NP-complete by exploiting compaction of terms and polynomial comparison of the compactions using singleton tree grammars  ... 
doi:10.1007/11814771_8 fatcat:zbhcjiuxjrewxajajwmjsmygki

Simplifying the signature in second-order unification

Jordi Levy, Mateu Villaret
2009 Applicable Algebra in Engineering, Communication and Computing  
Our work simplifies the study of Second-Order Unification and some of its variants, like Context Unification and Bounded Second-Order Unification.  ...  If all symbols are monadic, then the problem is NP-complete, whereas it is enough to have just one binary constant to lose decidability.  ...  Theorem 3 Decidability of Bounded Second-Order Unification can be NP-reduced to decidability of Bounded Second-Order Unification with just one binary function symbol, and constants.  ... 
doi:10.1007/s00200-009-0106-4 fatcat:ltrscory3nb7bdrfjy5jjoof2q

Page 60 of Journal of Research and Practice in Information Technology Vol. 15, Issue 2 [page]

1983 Journal of Research and Practice in Information Technology  
Therefore, the complete value of N1 is described by a pair consisting of a pointer to the term np(N) and another poin- ter to the frame in which N is bound.  ...  Similarly, when the two terms corresponding to the second arguments of terms (1) and (2) are matched, V will become bound to ‘vp(v(likes), np(mary))’.  ... 

Complexity of the Higher Order Matching [chapter]

ToMasz Wierzbicki
1999 Lecture Notes in Computer Science  
We obtain this way already known NP-hardness bounds for the order two and three and a new result that the fourth order matching is NEXPTIME-hard. , 82-96. † Supported by KBN grant 8 T 11C 029 13. 1 Since  ...  the term "pattern" has multiple meanings in the theory of higher order unification, e.g., the one introduced by Dale Miller [15] , we would rather avoid it when speaking about unification of an arbitrary  ...  Thus: Corollary 1 Second order pure unification (in particular matching) is in NP. Second order pure unification (in particular matching) with bounded number of variables is in PTIME.  ... 
doi:10.1007/3-540-48660-7_6 fatcat:w4ykjjdclbak5c72c5toaz46ci

Restricted Unification in the DL EL

Franz Baader, Maryam Rostamigiv
2021 International Workshop on Description Logics  
The complexity of deciding unifiability is ExpTime-complete for FL0 and NP-complete for EL.  ...  the decision problem stays NP-complete.  ...  Unification in EL is NP-complete.  ... 
dblp:conf/dlog/BaaderR21 fatcat:rwzldipnive5bpahbu4d3x3g5m

Tractable and Intractable Second-Order Matching Problems [chapter]

Kouichi Hirata, Keizo Yamada, Masateru Harao
1999 Lecture Notes in Computer Science  
It is well-known that the second-order matching problem is NP-complete.  ...  Then, we show that the second-order matching problem is NP-complete for a unary predicate, a unary ground, a ternary function-free predicate, a binary function-free ground, and an 1-fv predicate matching  ...  It is also well-known that the second-order matching problem is NP-complete [2] . Huet and Lang [14] have designed a complete and nonredundant second-order matching algorithm.  ... 
doi:10.1007/3-540-48686-0_43 fatcat:kif2eqafhvcnvnbrfqbawgsgmm

Tractable and intractable second-order matching problems

Kouichi Hirata, Keizo Yamada, Masateru Harao
2004 Journal of symbolic computation  
It is well-known that the second-order matching problem is NP-complete.  ...  Then, we show that the second-order matching problem is NP-complete for a unary predicate, a unary ground, a ternary function-free predicate, a binary function-free ground, and an 1-fv predicate matching  ...  It is also well-known that the second-order matching problem is NP-complete [2] . Huet and Lang [14] have designed a complete and nonredundant second-order matching algorithm.  ... 
doi:10.1016/j.jsc.2003.09.002 fatcat:c3mvlslzrzdbrexciaqbxd6uau

Page 2056 of Mathematical Reviews Vol. , Issue 2003C [page]

2003 Mathematical Reviews  
2003c:68091 error probabilistic ordered branching programs is NP-complete.  ...  Summary: “Context unification is a variant of second order uni- fication and also a generalization of string unification. Currently it is not known whether context unification is decidable.  ... 

Linear Higher-Order Matching Is NP-Complete [chapter]

Philippe de Groote
2000 Lecture Notes in Computer Science  
We prove that this problem is decidable by showing that it belongs to NP. Then we prove that this problem is in fact NP-complete. Finally, we discuss some heuristics for a practical algorithm.  ...  We consider the problem of higher-order matching restricted to the set of linear λ-terms (i.e., λ-terms where each abstraction λx. M is such that there is exactly one free occurrence of x in M ).  ...  Whether linear higher-order unification is decidable is an open problem related, in the second-order case, to context unification [11] .  ... 
doi:10.1007/10721975_9 fatcat:a6y3c5ebj5exjapfphk4wuhxci

Matching in the Description Logic FL0 with respect to General TBoxes (Extended abstract)

Franz Baader, Oliver Fernández Gil, Pavlos Marantidis
2019 International Workshop on Description Logics  
Whereas in EL both matching [6] and unification [7] are NP-complete problems, the complexity of these problems differs significantly for FL 0 : matching is polynomial, but unification is ExpTime-complete  ...  TBoxes is Exp-Time-complete [3] , ExpTime-hardness of this problem is clear. The first main contribution of this paper is thus to show the ExpTime upper bound.  ... 
dblp:conf/dlog/BaaderGM19 fatcat:lk6q57liybemncvd2ydpwo4xry

Assessing Complexity Results in Feature Theories [article]

Marten Trautwein
1995 arXiv   pre-print
On the one hand, the complexity of feature theories does not provide an upper bound for the complexity of such unification grammars.  ...  On the other hand, the complexity of feature theories need not provide a lower bound. Therefore, we argue for formalisms that describe actual unification grammars instead of feature theories.  ...  An additional NP upper bound is proven for an arbitrary string and grammar, which results in an NP-complete recognition problem.  ... 
arXiv:cmp-lg/9503022v1 fatcat:5xx6mc6jx5hfdh7ok2cqfqijjm

Monadic Second-Order Unification Is NP-Complete [chapter]

Jordi Levy, Manfred Schmidt-Schauß, Mateu Villaret
2004 Lecture Notes in Computer Science  
It is already known that Bounded Second-Order Unification is decidable and NP-hard, whereas general Second-Order Unification is undecidable.  ...  We prove that Bounded Second-Order Unification is NP-complete, provided that m is given in unary encoding, by proving that a size-minimal solution can be represented in polynomial space, and then applying  ...  Bounded Second-Order Unification is NP-complete. Proof: For any equation E, and any size-minimal solution σ, there exists a STG of polynomial size in |E| that generates σ(X), for every X ∈ V ar(E).  ... 
doi:10.1007/978-3-540-25979-4_4 fatcat:cygu7illbvap7kxyhgkn6rtvbe

Scope and Situation Binding in LTAG Using Semantic Unification

Laura Kallmeyer, Maribel Romero
2008 Research on Language and Computation  
Then, within this framework, an analysis of scope is proposed that accounts for the different scopal properties of quantifiers, adverbs, raising verbs and attitude verbs.  ...  In general, the complexity of disambiguating expressions with scope constraints of the form x ≤ y is NP-complete.  ...  Furthermore, the second situation within sometimes must be locally bound by the ∀s-quantification of think, yielding ∃s [s s]; if it could be distantly bound and represented as ∃s [s s 0 ], the sentence  ... 
doi:10.1007/s11168-008-9046-6 fatcat:ok7gp4jounbchfmukd7dk4qhj4

Page 7594 of Mathematical Reviews Vol. , Issue 2004j [page]

2004 Mathematical Reviews  
best known lower bound complexity is known to be NP-hard.  ...  The context unification problem (CUP) can be seen as a variant of second-order unification or as a generalization of the string unification problem.  ... 
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