Colors Make Theories Hard

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9706)


The satisfiability problem for conjunctions of quantifier-free literals in first-order theories \(\mathcal {T}\) of interest–“\(\mathcal {T}\) -solving” for short–has been deeply investigated for more than three decades from both theoretical and practical perspectives, and it is currently a core issue of state-of-the-art SMT solving. Given some theory \(\mathcal {T}\) of interest, a key theoretical problem is to establish the computational (in)tractability of \(\mathcal {T}\)-solving, or to identify intractable fragments of \(\mathcal {T}\) .

In this paper we investigate this problem from a general perspective, and we present a simple and general criterion for establishing the NP-hardness of \(\mathcal {T}\)-solving, which is based on the novel concept of “colorer” for a theory \(\mathcal {T}\).

As a proof of concept, we show the effectiveness and simplicity of this novel criterion by easily producing very simple proofs of the NP-hardness for many theories of interest for SMT, or of some of their fragments.


Domain Size Satisfiability Problem Colorable Theory Closed Term Candidate Problem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.DISIUniversity of TrentoTrentoItaly

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