Generic Expression Hardness Results for Primitive Positive Formula Comparison

We study the expression complexity of two basic problems involving the comparison of primitive positive formulas: equivalence and containment. In particular, we study the complexity of these problems relative to finite relational structures. We present two generic hardness results for the studied problems, and discuss evidence that they are optimal and yield, for each of the problems, a complexity trichotomy

A counterexample to the reconstruction of ω\omega-categorical structures from their endomorphism monoids

We present an example of two countable $\omega$-categorical structures, one of which has a finite relational language, whose endomorphism monoids are isomorphic as abstract monoids, but not as topological monoids -- in other words, no isomorphism between these monoids is a homeomorphism. For the same two structures, the automorphism groups and polymorphism clones are isomorphic, but not topologically isomorphic. In particular, there exists a countable $\omega$-categorical structure in a finite relational language which can neither be reconstructed up to first-order bi-interpretations from its automorphism group, nor up to existential positive bi-interpretations from its endomorphism monoid, nor up to primitive positive bi-interpretations from its polymorphism clone.Comment: 17 page

On the Complexity of Existential Positive Queries

We systematically investigate the complexity of model checking the existential positive fragment of first-order logic. In particular, for a set of existential positive sentences, we consider model checking where the sentence is restricted to fall into the set; a natural question is then to classify which sentence sets are tractable and which are intractable. With respect to fixed-parameter tractability, we give a general theorem that reduces this classification question to the corresponding question for primitive positive logic, for a variety of representations of structures. This general theorem allows us to deduce that an existential positive sentence set having bounded arity is fixed-parameter tractable if and only if each sentence is equivalent to one in bounded-variable logic. We then use the lens of classical complexity to study these fixed-parameter tractable sentence sets. We show that such a set can be NP-complete, and consider the length needed by a translation from sentences in such a set to bounded-variable logic; we prove superpolynomial lower bounds on this length using the theory of compilability, obtaining an interesting type of formula size lower bound. Overall, the tools, concepts, and results of this article set the stage for the future consideration of the complexity of model checking on more expressive logics

On the Scope of the Universal-Algebraic Approach to Constraint Satisfaction

The universal-algebraic approach has proved a powerful tool in the study of the complexity of CSPs. This approach has previously been applied to the study of CSPs with finite or (infinite) omega-categorical templates, and relies on two facts. The first is that in finite or omega-categorical structures A, a relation is primitive positive definable if and only if it is preserved by the polymorphisms of A. The second is that every finite or omega-categorical structure is homomorphically equivalent to a core structure. In this paper, we present generalizations of these facts to infinite structures that are not necessarily omega-categorical. (This abstract has been severely curtailed by the space constraints of arXiv -- please read the full abstract in the article.) Finally, we present applications of our general results to the description and analysis of the complexity of CSPs. In particular, we give general hardness criteria based on the absence of polymorphisms that depend on more than one argument, and we present a polymorphism-based description of those CSPs that are first-order definable (and therefore can be solved in polynomial time).Comment: Extended abstract appeared at 25th Symposium on Logic in Computer Science (LICS 2010). This version will appear in the LMCS special issue associated with LICS 201

Quasivarieties of Wajsberg hoops

In this paper we deal with quasivarieties of residuated structures which form the equivalent algebraic semantics of a positive frag- ment of some substructural logic. Our focus is mainly on varieties and quasivarieties of Wajsberg hoops, which are the equivalent algebraic semantics of the positive fragment of Łukasiewicz many-valued logic. In particular we study the lattice of subquasivari- eties of Wajsberg hoops and we describe completely all the subvarieties of Wajsberg hoops that are primitive. Though the treatment is mostly algebraic in nature, there are obvious connections with the underlying logic