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Parameter reduction in grammarcompressed trees
 In 12th FoSSaCS, volume 5504 of LNCS
, 2009
"... Abstract. Trees can be conveniently compressed with linear straightline contextfree tree grammars. Such grammars generalize straightline contextfree string grammars which are widely used in the development of algorithms that execute directly on compressed structures (without prior decompression) ..."
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Abstract. Trees can be conveniently compressed with linear straightline contextfree tree grammars. Such grammars generalize straightline contextfree string grammars which are widely used in the development of algorithms that execute directly on compressed structures (without prior decompression). It is shown that every linear straightline contextfree tree grammar can be transformed in polynomial time into a monadic (and linear) one. A tree grammar is monadic if each nonterminal uses at most one context parameter. Based on this result, a polynomial time algorithm is presented for testing whether a given nondeterministic tree automaton with sibling constraints accepts a tree given by a linear straightline contextfree tree grammar. It is shown that if tree grammars are nondeterministic or nonlinear, then reducing their numbers of parameters cannot be done without an exponential blowup in grammar size. 1
Algorithmics on SLPcompressed strings: a survey,
 Groups Complex. Cryptol.
, 2012
"... Abstract Results on algorithmic problems on strings that are given in a compressed form via straightline programs are surveyed. A straightline program is a contextfree grammar that generates exactly one string. In this way, exponential compression rates can be achieved. Among others, we study pat ..."
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Abstract Results on algorithmic problems on strings that are given in a compressed form via straightline programs are surveyed. A straightline program is a contextfree grammar that generates exactly one string. In this way, exponential compression rates can be achieved. Among others, we study pattern matching for compressed strings, membership problems for compressed strings in various kinds of formal languages, and the problem of querying compressed strings. Applications in combinatorial group theory and computational topology and to the solution of word equations are discussed as well. Finally, extensions to compressed trees and pictures are considered.
On the complexity of Bounded SecondOrder Unification and Stratified Context Unification
, 2010
"... Bounded SecondOrder Unification is a decidable variant of undecidable SecondOrder Unification. Stratified Context Unification is a decidable restriction of Context Unification, whose decidability is a longstanding open problem. This paper is a join of two separate previous, preliminary papers on ..."
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Bounded SecondOrder Unification is a decidable variant of undecidable SecondOrder Unification. Stratified Context Unification is a decidable restriction of Context Unification, whose decidability is a longstanding open problem. This paper is a join of two separate previous, preliminary papers on NPcompleteness of Bounded SecondOrder Unification and Stratified Context Unification. It clarifies some omissions in these papers, joins the algorithmic parts that construct a minimal solution, and gives a clear account of a method of using singleton tree grammars for compression that may have potential usage for other algorithmic questions in related areas.
Onecontext Unification with STGCompressed Terms is in NP
, 2012
"... Onecontext unification is an extension of firstorder term unification in which a variable of arity one standing for a context may occur in the input terms. This problem arises in areas like program analysis, term rewriting and XML processing and is known to be solvable in nondeterministic polynomi ..."
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Onecontext unification is an extension of firstorder term unification in which a variable of arity one standing for a context may occur in the input terms. This problem arises in areas like program analysis, term rewriting and XML processing and is known to be solvable in nondeterministic polynomial time. We prove that this problem can be solved in nondeterministic polynomial time also when the input is compressed using Singleton Tree Grammars (STG’s). STG’s are a grammarbased compression method for terms that generalizes the directed acyclic graph representation. They have been recently considered as an efficient inmemory representation for large terms, since several operations on terms can be performed efficiently on their STG representation without a prior decompression.
Pattern Matching of Compressed Terms and Contexts and Polynomial Rewriting
, 2011
"... A generalization of the compressed string pattern match that applies to terms with variables is investigated: Given terms s and t compressed by singleton tree grammars, the task is to find an instance of s that occurs as a subterm in t. We show that this problem is in NP and that the task can be pe ..."
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A generalization of the compressed string pattern match that applies to terms with variables is investigated: Given terms s and t compressed by singleton tree grammars, the task is to find an instance of s that occurs as a subterm in t. We show that this problem is in NP and that the task can be performed in time O(n cVar(s) ), including the construction of the compressed substitution, and a representation of all occurrences. We show that the special case where s is uncompressed can be performed in polynomial time. As a nice application we show that for an equational deduction of t to t ′ by an equality axiom l = r (a rewrite) a single step can be performed in polynomial time in the size of compression of t and l, r if the number of variables is fixed in l. We also show that n rewriting steps can be performed in polynomial time, if the equational axioms are compressed and assumed to be constant for the rewriting sequence. Another potential application are querying mechanisms on compressed XMLdata bases.
Matching of Compressed Patterns with CharacterVariables
, 2012
"... We consider the problem of finding an instance of a stringpattern s in a given string under compression by straight line programs (SLP). The variables of the string pattern can be instantiated by single characters. This is a generalisation of the fully compressed pattern match, which is the task of ..."
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We consider the problem of finding an instance of a stringpattern s in a given string under compression by straight line programs (SLP). The variables of the string pattern can be instantiated by single characters. This is a generalisation of the fully compressed pattern match, which is the task of finding a compressed string in another compressed string, which is known to have a polynomial time algorithm. We mainly investigate patterns s that are linear in the variables, i.e. variables occur at most once in s, also known as partial words. We show that fully compressed pattern matching with linear patterns can be performed in polynomial time. A polynomialsized representation of all matches and all substitutions is also computed. Also, a related algorithm is given that computes all periods of a compressed linear pattern in polynomial time. A technical key result on the structure of partial words shows that an overlap of h + 2 copies of a partial word w with at most h holes implies that w is strongly periodic.
FirstOrder Unification on Compressed Terms
, 2011
"... Singleton Tree Grammars (STGs) have recently drawn considerable attention. They generalize the sharing of subtrees known from DAGs to sharing of connected subgraphs. This allows to obtain smaller inmemory representations of trees than with DAGs. In the past years some important tree algorithms were ..."
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Singleton Tree Grammars (STGs) have recently drawn considerable attention. They generalize the sharing of subtrees known from DAGs to sharing of connected subgraphs. This allows to obtain smaller inmemory representations of trees than with DAGs. In the past years some important tree algorithms were proved to perform efficiently (without decompression) over STGs; e.g., type checking, equivalence checking, and unification. We present a tool that implements an extension of the unification algorithm for STGs. This algorithm makes extensive use of equivalence checking. For the latter we implemented two variants, the classical exact one and a recent randomized one. Our experiments show that the randomized algorithm performs better. The running times are also compared to those of unification over uncompressed trees.
Fast Equality Test for StraightLine Compressed Strings
, 2011
"... The paper describes a simple and fast randomized test for equality of grammarcompressed strings. The thorough running time analysis is done by applying a logarithmic cost measure. ..."
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The paper describes a simple and fast randomized test for equality of grammarcompressed strings. The thorough running time analysis is done by applying a logarithmic cost measure.
Congruence Closure of Compressed Terms in Polynomial Time
, 2011
"... The wordproblem for a finite set of equational axioms between ground terms is the question whether for terms s, t the equation s = t is a consequence. We consider this problem under grammar based compression of terms, in particular compression with singleton tree grammars (STGs) and with directed ..."
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The wordproblem for a finite set of equational axioms between ground terms is the question whether for terms s, t the equation s = t is a consequence. We consider this problem under grammar based compression of terms, in particular compression with singleton tree grammars (STGs) and with directed acyclic graphs (DAGs) as a special case. We show that given a DAGcompressed ground and reduced term rewriting system T, the Tnormal form of an STGcompressed term s can be computed in polynomial time, and hence the Tword problem can be solved in polynomial time. This implies that the word problem of STGcompressed terms w.r.t. a set of DAGcompressed ground equations can be decided in polynomial time. If the ground term rewriting system (gTRS) T is STGcompressed, we show NPhardness of Tnormalform computation. For compressed, reduced gTRSs we show a PSPACE upper bound on the complexity of the normal form computation of STGcompressed terms. Also special cases are considered and a prototypical implementation is presented.