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On the Complexity of the Tiden-Arnborg Algorithm for Unification modulo One-Sided Distributivity

Paliath Narendran, Andrew Marshall, Bibhu Mahapatra
2010 Electronic Proceedings in Theoretical Computer Science  
We prove that the Tiden and Arnborg algorithm for equational unification modulo one-sided distributivity is not polynomial time bounded as previously thought.  ...  A set of counterexamples is developed that demonstrates that the algorithm goes through exponentially many steps.  ...  They present an algorithm for equational unification modulo a one-sided distributivity axiom: x × (y + z) = x × y + x × z This unification problem has recently been of interest in cryptographic protocol  ... 
doi:10.4204/eptcs.42.5 fatcat:6ottk2aarrhmdgaa7ryuzw7fy4

On Unification Modulo One-Sided Distributivity: Algorithms, Variants and Asymmetry

Andrew Marshall, Catherine Meadows, Paliath Narendran, Franz Baader
2015 Logical Methods in Computer Science  
An algorithm for unification modulo one-sided distributivity is an early result by Tid\'en and Arnborg.  ...  We give the first asymmetric unification algorithm for one-sided distributivity.  ...  In particular, the algorithm for unification modulo a one-sided distributivity axiom X × (Y + Z) = X × Y + X × Z is an early result by Tidén and Arnborg [22] .  ... 
doi:10.2168/lmcs-11(2:11)2015 fatcat:tcjmrfmvnzghbbbzsgqpsnidla

Page 467 of Mathematical Reviews Vol. , Issue 89A [page]

1989 Mathematical Reviews  
We also present some original results on LR automata.” 89a:68141 68Q50 03B35 03C05 03D35 68Q20 68TI5 Tidén, Erik; Arnborg, Stefan (S-RIT-C) Unification problems with one-sided distributivity.  ...  Then the complexity of some problems involving the one-sided distributivity axiom D,, the two-sided distributivity axiom D, and the axiom U (x +1 =1+*x = x) is investigated.  ... 

Decidability of Unification in the Theory of One-Sided Distributivity and a Multiplicative Unit

Manfred Schmidt-Schauß
1996 Journal of symbolic computation  
Unification in the equational theory of one-sided distributivity and a multiplicative unit is shown to be decidable.  ...  Schulz extending the decision algorithm for string unification by Makanin.  ...  Acknowledgements The author wishes to thank the referees for carefully reading drafts of the paper.  ... 
doi:10.1006/jsco.1996.0054 fatcat:yum36hpqrzcjzdlc7xxcnq6b7a

Bounded ACh Unification [article]

Ajay Kumar Eeralla, Christopher Lynch
2019 arXiv   pre-print
Since the unification modulo ACh theory is undecidable, we define a variant of the problem called bounded ACh unification.  ...  We consider the problem of the unification modulo an equational theory ACh, which consists of a function symbol h that is homomorphic over an associative-commutative operator +.  ...  In their paper (Tiden and Arnborg 1987), Tiden and Arnborg presented an algorithm for unification modulo one-sided distributivity: x × (y + z) = (x × y) + (x × z), and also it has been shown that it is  ... 
arXiv:1811.05602v2 fatcat:eajtulodjzftnloisosohkswm4

Theories of Homomorphic Encryption, Unification, and the Finite Variant Property

Fan Yang, Santiago Escobar, Catherine Meadows, José Meseguer, Paliath Narendran
2014 Proceedings of the 16th International Symposium on Principles and Practice of Declarative Programming - PPDP '14  
This new verification procedure also allows us to construct a rough metric of the complexity of a theory with respect to variant unification, or variant complexity.  ...  Recent advances in the automated analysis of cryptographic protocols have aroused new interest in the practical application of unification modulo theories, especially theories that describe the algebraic  ...  The earliest work on homomorphic theories is by Tiden and Arnborg [42] who gave a unification algorithm modulo the theory of one-sided distributivity: 1 x * (y + z) = (x * y) + (x * z).  ... 
doi:10.1145/2643135.2643154 dblp:conf/ppdp/YangEMMN14 fatcat:ss5awwz6wffnnlp6i67rg5svcu

More problems in rewriting [chapter]

Nachum Dershowitz, Jean-Pierre Jouannaud, Jan Willem Klop
1993 Lecture Notes in Computer Science  
Is satisfiability of equations in the theory of distributivity (unification modulo a distributivity axiom) decidable? The question should read "modulo one right-and one left-distributivity axiom".  ...  (With just one of these, the problem had already been solved in [Tiden and Arnborg, 1987] . ) A partial positive solution is given in [Contejean, 1993] , based on a striking result on the structure of  ... 
doi:10.1007/3-540-56868-9_39 fatcat:cibthhywyrgmvcsv73tryp3v2e

An algorithm for distributive unification [chapter]

Manfred Schmidt-Schauß
1996 Lecture Notes in Computer Science  
We consider unification of terms under the equational theory of two-sided distributivity D with the axioms x*(y+z) = x*y + x*z and (x+y)*z = x*z + y*z.  ...  The notion of a greatest common divisor for a free AC1-algebra is defined in the obvious way. We can use this notion also for two terms that have a unique factorization.  ...  We have provided a decision algorithm for unifiability of equations under the theory of twosided distributivity, answering a question that was open for some years.  ... 
doi:10.1007/3-540-61464-8_60 fatcat:webxvrl4nzb6xfpnnsflrbakea

Solving *-problems modulo Distributivity by a Reduction to AC1-unification

Evelyne Contejean
1993 Journal of symbolic computation  
We show that uni cation modulo both-sided distributivity of the symbol on + can be reduced to AC1-uni cation for all uni cation problems which do not involve the + operator.  ...  Moreover, in this case, we can describe \almost all" solutions in a nite way, although there are in general in nitely many minimal solutions for such problems.  ...  Arnborg & Tiden (1987) have shown that one-sided (left or right) distributivity has a decidable uni cation, whereas adding a unit element for and associativity of + makes uni cation undecidable.  ... 
doi:10.1006/jsco.1993.1060 fatcat:zdc4ovequnelraq4fg5w26ppsy

Unification of Higher-order Patterns modulo Simple Syntactic Equational Theories

Alexandre Boudet
2000 Discrete Mathematics and Theoretical Computer Science   unpublished
We present an algorithm for unification of higher-order patterns modulo simple syntactic equational theories as defined by Kirchner [14].  ...  If the syntactic algorithm for a theory E terminates in the first-order case, then our algorithm will also terminate for pattern E-unification.  ...  The assumption that E is a simple theory is essentially technical, and could be dropped as it has been done in the first-order case for unification in combinations of equational theories.  ... 
fatcat:p6c4m6xydba2ndjwxsxfokticu