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Two-Restricted One Context Unification is in Polynomial Time

Adrià Gascón, Manfred Schmidt-Schauß, Ashish Tiwari, Marc Herbstritt
<span title="2015-09-04">2015</span> <i > <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/kjskj3qctjdkbigtx4fkcqiyii" style="color: black;">Annual Conference for Computer Science Logic</a> </i> &nbsp;
We show that the case of 1CU where the context variable occurs at most twice in the input (1CU2r) is solvable in polynomial time.  ...  One Context Unification (1CU) extends first-order unification by introducing a single context variable.  ...  is not the empty context, then unification can be performed in polynomial time.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.4230/lipics.csl.2015.405">doi:10.4230/lipics.csl.2015.405</a> <a target="_blank" rel="external noopener" href="https://dblp.org/rec/conf/csl/GasconST15.html">dblp:conf/csl/GasconST15</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/lukahafjincjzirdf5bnor4dli">fatcat:lukahafjincjzirdf5bnor4dli</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20200710102217/https://www.research.ed.ac.uk/portal/files/24376057/25.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/1d/86/1d86464dc6a1bbd5ceb7c905e63c98a79b8bdd44.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.4230/lipics.csl.2015.405"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> Publisher / doi.org </button> </a>

Page 8525 of Mathematical Reviews Vol. , Issue 2002K [page]

<span title="">2002</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
Then, a given context equation with two context variables is (eventually) translated into a finite set of systems of word equations with linear constant restrictions.  ...  of finite systems of context equations with two context variables is decidable.  ... 
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One Context Unification Problems Solvable in Polynomial Time

Adria Gascon, Ashish Tiwari, Manfred Schmidt Schaus
<span title="">2015</span> <i title="IEEE"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/cnybbxuptncftdgxtodn5edz7m" style="color: black;">2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science</a> </i> &nbsp;
One context unification is known to be in NP, but it is not known to be solvable in polynomial time.  ...  In this paper, we present a polynomial time algorithm for certain interesting classes of the one context unification problem.  ...  On the other hand, context unification is a restricted form of second-order unification, which is undecidable [9] .  ... 
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Context unification with one context variable

Adrià Gascón, Guillem Godoy, Manfred Schmidt-Schauß, Ashish Tiwari
<span title="">2010</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/ezljl2d3lzga5efenbxdvvfcpa" style="color: black;">Journal of symbolic computation</a> </i> &nbsp;
We also exhibit particular cases where one context unification is polynomial. multi-equations.  ...  In this paper we analyze the special case of context unification where the use of at most one context variable is allowed and show that it is in NP.  ...  In special situations, the first phase can be done in polynomial time too, and hence one context unification is polynomial in these cases.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.jsc.2008.10.005">doi:10.1016/j.jsc.2008.10.005</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/wv6ggga5pvcpdmewrgi67pmu5e">fatcat:wv6ggga5pvcpdmewrgi67pmu5e</a> </span>
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Well-Nested Context Unification [chapter]

Jordi Levy, Joachim Niehren, Mateu Villaret
<span title="">2005</span> <i title="Springer Berlin Heidelberg"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/2w3awgokqne6te4nvlofavy5a4" style="color: black;">Lecture Notes in Computer Science</a> </i> &nbsp;
We distinguish a new decidable variant of CU well-nested CU -and present a new unification algorithm that solves well-nested context equations in non-deterministic polynomial time.  ...  Context unification (CU) is the open problem of solving context equations for trees.  ...  , we can check if it is a unifier in polynomial time on |e| + |σ|.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/11532231_11">doi:10.1007/11532231_11</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/kh23avzugrgrfmnm544ij4tkum">fatcat:kh23avzugrgrfmnm544ij4tkum</a> </span>
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On the complexity of Bounded Second-Order Unification and Stratified Context Unification

J. Levy, M. Schmidt-Schauss, M. Villaret
<span title="2010-05-05">2010</span> <i title="Oxford University Press (OUP)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/mwzmwwmscfe4vnzrjlt5itd2ju" style="color: black;">Logic Journal of the IGPL</a> </i> &nbsp;
This paper is a join of two separate previous, preliminary papers on NP-completeness of Bounded Second-Order Unification and Stratified Context Unification.  ...  Stratified Context Unification is a decidable restriction of Context Unification, whose decidability is a long-standing open problem.  ...  In [BLM05, SS05] and [Lif06, Lif07] it is proved that, given a singleton tree grammar, we can decide in polynomial time (O(n 3 )) on the size n of the grammar whether two non-terminals define the same  ... 
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Nominal Unification with Atom and Context Variables

Manfred Schmidt-Schauß, David Sabel, Michael Wagner
<span title="2018-06-29">2018</span> <i > <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/wkjtc7dggzfktmoov66odshrra" style="color: black;">International Conference on Rewriting Techniques and Applications</a> </i> &nbsp;
For terms without context-variables and atom-variables, NomUnifyASD runs in polynomial time, is unitary, and extends the classical problem by permitting distinct-variable constraints.  ...  We show that NomUnifyASD is sound and complete for this problem class, and outputs a set of unifiers with constraints in nondeterministic polynomial time if the final constraints are satisfiable.  ...  The nominal unification problem in NL aS where freshness-and dvcconstraints are permitted inis solvable in polynomial time.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.4230/lipics.fscd.2018.28">doi:10.4230/lipics.fscd.2018.28</a> <a target="_blank" rel="external noopener" href="https://dblp.org/rec/conf/rta/Schmidt-Schauss18.html">dblp:conf/rta/Schmidt-Schauss18</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/dldzfeytmrhxhh2mr3cgeyoeem">fatcat:dldzfeytmrhxhh2mr3cgeyoeem</a> </span>
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Page 10281 of Mathematical Reviews Vol. , Issue 2004m [page]

<span title="">2004</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
However, we show how to compute under restriction to acyclic Horn theories polynomially many explanations in input-polynomial time and all explanations in polynomial total time, respectively.  ...  in the two given graphs is merged together in the new graph.  ... 
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Complex Analogies: Remarks on the Complexity of HDTP [chapter]

Robert Robere, Tarek Richard Besold
<span title="">2012</span> <i title="Springer Berlin Heidelberg"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/2w3awgokqne6te4nvlofavy5a4" style="color: black;">Lecture Notes in Computer Science</a> </i> &nbsp;
Amongst others, we show that restricted higher-order antiunification as currently used in HDTP is W [1]-hard (and thus NP-hard) already for quite simple cases.  ...  of the key mechanisms underlying HDTP, together with a short discussion of and reflection on the obtained results.  ...  F Anti-Unification is solvable in polynomial time. 2. FP Anti-Unification is NP-complete and W[1]-hard w.r.t. parameter set {h, p}. 3.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/978-3-642-35101-3_45">doi:10.1007/978-3-642-35101-3_45</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/4t6ksaoprrgidcft6cde5rrepm">fatcat:4t6ksaoprrgidcft6cde5rrepm</a> </span>
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Context Matching for Compressed Terms

Adria Gascón, Guillem Godoy, Manfred Schmidt-Schauss
<span title="">2008</span> <i title="IEEE"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/zw5e5duxvbdsthglbdpw3xoqae" style="color: black;">Logic in Computer Science</a> </i> &nbsp;
The main result is a polynomial time algorithm for context matching with dags, when the number of different context variables is fixed for the problem.  ...  NP-completeness is obtained when the terms are represented using singleton tree grammars. The special cases of first-order matching and also unification with STGs are shown to be decidable in PTIME.  ...  This kind of restriction has already been considered for context unification restricted to two context variables [19] , and also proved useful in the context of program verification with procedure calls  ... 
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<a target="_blank" rel="noopener" href="https://web.archive.org/web/20170809185647/http://www.cs.upc.edu/~ggodoy/papers/dagcontextmatchingSTG-LICS.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/af/64/af6475131c1fd969db6bb8878a166be8298a0b7e.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1109/lics.2008.17"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> ieee.com </button> </a>

A decision algorithm for distributive unification

Manfred Schmidt-Schauβ
<span title="">1998</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/elaf5sq7lfdxfdejhkqbtz6qoq" style="color: black;">Theoretical Computer Science</a> </i> &nbsp;
One spin-off is an algorithm that decides the word-problem w.r.t. D in polynomial time. This is the basis for an I .P-algorithm for D-matching, hence D-matching is .I 'Y-complete.  ...  A further (future) spin-off is a decision algorithm for stratified context unification problems.  ...  Acknowledgements I would like to thank Hubert Comon for a discussion on context unification, Evelyne Contejean for discussions on D-unification.  ... 
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Stratified Context Unification Is NP-Complete [chapter]

Jordi Levy, Manfred Schmidt-Schauß, Mateu Villaret
<span title="">2006</span> <i title="Springer Berlin Heidelberg"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/2w3awgokqne6te4nvlofavy5a4" style="color: black;">Lecture Notes in Computer Science</a> </i> &nbsp;
Stratified Context Unification is a specialization where the nesting of second-order variables in E is restricted.  ...  polynomial algorithm that compares compacted terms in polynomial time.  ...  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  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/11814771_8">doi:10.1007/11814771_8</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/zbhcjiuxjrewxajajwmjsmygki">fatcat:zbhcjiuxjrewxajajwmjsmygki</a> </span>
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Page 1703 of Mathematical Reviews Vol. , Issue 86d [page]

<span title="">1986</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
On the sequential nature of unification. J. Logic Programming 1 (1984), no. 1, 35-50. The unification problem is considered in the context of parallel computation.  ...  However, as the authors show, one special case of unification can be solved in O(log? n) par- allel time using polynomial many processors.  ... 
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Page 7111 of Mathematical Reviews Vol. , Issue 97K [page]

<span title="">1997</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
Apart from the reduced search space due to the inherited ordering and unification restrictions of the inferences, symbolic constraints turn out to be especially useful in the context of built- in equational  ...  In every inference, instead of producing as many conclusions as minimal E-unifiers of two terms s and 1, one single conclusion is generated with an additional F-unification problem s = ¢ kept in its constraint  ... 
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Page 7895 of Mathematical Reviews Vol. , Issue 99k [page]

<span title="">1999</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
The abstraction on both data and control makes our system suitable in any applicative context where unification is required.” 99k:68088 68Q42 03B35 08A70 68-02 68Q40 Baader, Franz (D-AACH; Aachen); Nipkow  ...  Summary: “In this paper we define a two-rule reduction system for the lazy computation of unification on first-order terms.  ... 
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