# The Computational Structure of Monotone Monadic SNP and Constraint Satisfaction: A Study through Datalog and Group Theory

@article{Feder1998TheCS,
title={The Computational Structure of Monotone Monadic SNP and Constraint Satisfaction: A Study through Datalog and Group Theory},
author={Tom{\'a}s Feder and Moshe Y. Vardi},
journal={SIAM J. Comput.},
year={1998},
volume={28},
pages={57-104}
}
• Published 1 February 1999
• Mathematics, Computer Science
• SIAM J. Comput.
This paper starts with the project of finding a large subclass of NP which exhibits a dichotomy. The approach is to find this subclass via syntactic prescriptions. While the paper does not achieve this goal, it does isolate a class (of problems specified by) "monotone monadic SNP without inequality" which may exhibit this dichotomy. We justify the placing of all these restrictions by showing, essentially using Ladner's theorem, that classes obtained by using only two of the above three…
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#### References

SHOWING 1-10 OF 50 REFERENCES
Inductive pebble games and the expressive power of datalog
• Mathematics, Computer Science
PODS '89
• 1989
This paper sketches a proof that the query “find all pairs of nodes connected by a directed simple path of even length” cannot be expressed in DATALOG, the language of function-free Horn clauses.
The decision problem for the probabilities of higher-order properties
• Mathematics, Computer Science
STOC
• 1987
Logics which on the one hand go beyond fixpoint in terms of expressive power and on the other possess the 0-1 law are investigated, which establishes that the associated decision problem is NEXPTIME-complete and proofs of the decidability and complexity results require certain combinatorial machinery.
The complexity of satisfiability problems
An infinite class of satisfiability problems is considered which contains these two particular problems as special cases, and it is shown that every member of this class is either polynomial-time decidable or NP-complete.
Algorithmic aspects of type inference with subtypes
• Computer Science
POPL '92
• 1992
It is NP-hard to decide whether a lambda term has a type with respect to a fixed subtype hierarchy (involving only atomic type names), and PSPACE upper bounds for deciding polymorphic typability are given.
Easy Problems for Tree-Decomposable Graphs
• Computer Science, Mathematics
J. Algorithms
• 1991
Abstract Using a variation of the interpretability concept we show that all graph properties definable in monadic second-order logic (MS properties) with quantification over vertex and edge sets can
Duality and Polynomial Testing of Tree Homomorphisms
• Mathematics
• 1996
Let H be a fixed digraph. We consider the H-colouring problem, i.e., the problem of deciding which digraphs G admit a homomorphism to H. We are interested in a characterization in terms of the
Generalized first-order spectra, and polynomial. time recognizable sets
The spectrum of a first-order sentence σ is the set of cardinalities of its finite models. Jones and Selman showed that a set C of numbers (written in binary) is a spectrum if and only if C is in the
Optimization, Approximation, and Complexity Classes
• Mathematics, Computer Science
J. Comput. Syst. Sci.
• 1991
It follows that such a complete problem has a polynomial-time approximation scheme iff the whole class does, and that a number of common optimization problems are complete for MAX SNP under a kind of careful transformation that preserves approximability.
The gap between monotone and non-monotone circuit complexity is exponential
The proof is immediate by combining the Alon—Boppana version of another argument of Razborov with results of Grötschel—Lovász—Schrijver on the Lovász — capacity of a graph.
On the Structure of Polynomial Time Reducibility
• Mathematics, Computer Science
JACM
• 1975
The method of showing density ymlds the result that if P ~ NP then there are members of NP -P that are not polynomml complete is shown, which means there is a strictly ascending sequence with a minimal pair of upper bounds to the sequence.