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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.
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.
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.