Results 1 
7 of
7
Stratified context unification is npcomplete
 In Proc. of the 3rd International Joint Conference on Automated Reasoning, IJCAR’06
, 2006
"... Abstract. Context Unification is the problem to decide for a given set of secondorder equations E where all secondorder variables are unary, whether there exists a unifier, such that for every secondorder variable X, theabstractionλx.r instantiated for X has exactly one occurrence of the bound va ..."
Abstract

Cited by 7 (2 self)
 Add to MetaCart
Abstract. Context Unification is the problem to decide for a given set of secondorder equations E where all secondorder variables are unary, whether there exists a unifier, such that for every secondorder variable X, theabstractionλx.r instantiated for X has exactly one occurrence of the bound variable x in r. Stratified Context Unification is a specialization where the nesting of secondorder variables in E is restricted. It is already known that Stratified Context Unification is decidable, NPhard, and in PSPACE, whereas the decidability and the complexity of Context Unification is unknown. We prove that Stratified Context Unification is in NP by proving that a sizeminimal solution can be represented in a singleton tree grammar of polynomial size, and then applying a generalization of Plandowski’s polynomial algorithm that compares compacted terms in polynomial time. This also demonstrates the high potential of singleton tree grammars for optimizing programs maintaining large terms. A corollary of our result is that solvability of rewrite constraints is NPcomplete. 1
Polynomial equality testing for terms with shared substructures. Frank report 21, Institut für Informatik
 FB Informatik und Mathematik. J. W. GoetheUniversität Frankfurt am Main
, 2005
"... Abstract. Sharing of substructures like subterms and subcontexts in terms is a common method for spaceefficient representation of terms, which allows for example to represent exponentially large terms in polynomial space, or to represent terms with iterated substructures in a compact form. We prese ..."
Abstract

Cited by 7 (3 self)
 Add to MetaCart
Abstract. Sharing of substructures like subterms and subcontexts in terms is a common method for spaceefficient representation of terms, which allows for example to represent exponentially large terms in polynomial space, or to represent terms with iterated substructures in a compact form. We present singleton tree grammars as a general formalism for the treatment of sharing in terms. Singleton tree grammars (STG) are recursionfree contextfree tree grammars without alternatives for nonterminals and at most unary secondorder nonterminals. STGs generalize Plandowski’s singleton context free grammars to terms (trees). We show that the test, whether two different nonterminals in an STG generate the same term can be done in polynomial time, which implies that the equality test of terms with shared terms and contexts, where composition of contexts is permitted, can be done in polynomial time in the size of the representation. This will allow polynomialtime algorithms for terms exploiting sharing. We hope that this technique will lead to improved upper complexity bounds for variants of second order unification algorithms, in particular for variants of context unification and bounded second order unification.
THE COMPLEXITY OF MONADIC SECONDORDER UNIFICATION ∗
, 1113
"... Abstract. Monadic secondorder unification is secondorder unification where all function constants occurring in the equations are unary. Here we prove that the problem of deciding whether a set of monadic equations has a unifier is NPcomplete, where we use the technique of compressing solutions us ..."
Abstract

Cited by 4 (1 self)
 Add to MetaCart
Abstract. Monadic secondorder unification is secondorder unification where all function constants occurring in the equations are unary. Here we prove that the problem of deciding whether a set of monadic equations has a unifier is NPcomplete, where we use the technique of compressing solutions using singleton contextfree grammars. We prove that monadic secondorder matching is also NPcomplete.
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 ..."
Abstract

Cited by 3 (1 self)
 Add to MetaCart
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.
On the complexity of Bounded SecondOrder Unification and Stratified Context Unification
"... 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 ..."
Abstract
 Add to MetaCart
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.
Simplifying the signature in secondorder unification
, 2009
"... SecondOrder Unification is undecidable even for very specialized fragments. The signature plays a crucial role in these fragments. If all symbols are monadic, then the problem is NPcomplete, whereas it is enough to have just one binary constant to lose decidability. In this work we reduce SecondO ..."
Abstract
 Add to MetaCart
SecondOrder Unification is undecidable even for very specialized fragments. The signature plays a crucial role in these fragments. If all symbols are monadic, then the problem is NPcomplete, whereas it is enough to have just one binary constant to lose decidability. In this work we reduce SecondOrder Unification to SecondOrder Unification with a signature that contains just one binary function symbol and constants. The reduction is based on partially currying the equations by using the binary function symbol for explicit application @. Our work simplifies the study of SecondOrder Unification and some of its variants, like Context Unification and Bounded SecondOrder Unification.