We study the problem of checking whether an existential sentence (that is, a first-order sentence in prefix form built using existential quantifiers and all Boolean connectives) is true in a finite partially ordered set (in short, a poset). A poset is a reflexive, antisymmetric, and transitive digraph. The problem encompasses the fundamental embedding problem of finding an isomorphic copy of a poset as an induced substructure of another poset.Model checking existential logic is already NP-hard on a fixed poset; thus we investigate structural properties of posets yielding conditions for fixed-parameter tractability when the problem is parameterized by the sentence. We identify width as a central structural property (the width of a poset is the maximum size of a subset of pairwise incomparable elements); our main algorithmic result is that model checking existential logic on classes of finite posets of bounded width is fixedparameter tractable. We observe a similar phenomenon in classical complexity, where we prove that the isomorphism problem is polynomial-time tractable on classes of posets of bounded width; this settles an open problem in order theory.We surround our main algorithmic result with complexity results on less restricted, natural neighboring classes of finite posets, establishing its tightness in this sense. We also relate our work with (and demonstrate its independence of) fundamental fixed-parameter tractability results for model checking on digraphs of bounded degree and bounded clique-width.

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Abstract. We study the complexity of equivalence and isomorphism on primitive positive formulas with respect to a given structure. We study these problems for various fixed structures; we present generic hardness and complexity class containment results, and give classification theorems for the case of two-element (boolean) structures.

a b s t r a c tCommutative, integral and bounded GBL-algebras form a subvariety of residuated lattices which provides the algebraic semantics of an interesting common fragment of intuitionistic logic and of several fuzzy logics.It is known that both the equational theory and the quasiequational theory of commutative GBL-algebras are decidable (in contrast to the noncommutative case), but their complexity has not been studied yet. In this paper, we prove that both theories are in PSPACE, and that the quasiequational theory is PSPACE-hard.

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.set of relations that are definable by a primitive positive formula forms a robust algebraic object known as a relational clone; a known Galois correspondence associates, in a bijective manner, each such relational clone with a clone, a set of operations with certain closure properties. This correspondence provides a way to pass from a relational structure B to an algebra A B whose set of operations is the mentioned clone, in such a way that two structures having the same algebra have the same complexity (for each of the mentioned problems). In a previous paper by the present authors [6], we developed this correspondence and presented some basic complexity results for the problems at hand, including a classification of the complexity of the problems on all two-element structures.Our hardness results. Our first hardness result (Section 3) yields that for any structure B whose associated algebra A B gives rise to a variety V(A B ) that admits the unary type, both the equivalence and containment problems are Π p 2 -hard. Note that this is the maximal complexity possible for these problems, as the problems are contained in the class Π p 2 . The condition of admitting the unary type originates from tame congruence theory, a theory developed to understand the structure of finite algebras [13]. We observe that this result implies a dichotomy in the complexity of the studied problems under the G-set conjecture for the constraint satisfaction problem (CSP), a conjecture put forth by Bulatov, Jeavons, and Krokhin [7] which predicts exactly where the tractability/intractability dichotomy lies for the CSP. (Recall that the CSP can be formulated as the problem of deciding, given a structure B and a primitive positive sentence φ, whether or not φ holds on B.) In particular, under the G-set conjecture, the structures not obeying the described condition have equivalence and containment problems in coNP. The resolution of the G-set conjecture, on which there has been focused and steady progress over the past decade [9,14,10,3], would thus, in combination with our hardness result, yield a coNP/Π p 2 -complete dichotomy for the equivalence and containment problems. In fact, our hardness result already unconditionally implies dichotomies for our problems for all classes of structures where the G-set conjecture has already been established, including the class of three-element structures [9], and the class of conservative structures [8].One formulation of the G-set conjecture is that, for a structure B whose associated algebra A B is idempotent, the absence of the unary type in the variety generated by A B implies that the CSP o...

We introduce a proof system for Hájek's logic BL based on a relational hypersequents framework. We prove that the rules of our logical calculus, called RHBL, are sound and invertible with respect to any valuation of BL into a suitable algebra, called (ω)[0, 1]. Refining the notion of reduction tree that arises naturally from RHBL, we obtain a decision algorithm for BL provability whose running time upper bound is 2 O(n) , where n is the number of connectives of the input formula. Moreover, if a formula is unprovable, we exploit the constructiveness of a polynomial time algorithm for leaves validity for providing a procedure to build countermodels in (ω)[0, 1]. Finally, since the size of the reduction tree branches is O(n 3 ), we can describe a polynomial time verification algorithm for BL unprovability.

We provide a complete classification of solvable instances of the equational unification problem over De Morgan and Kleene algebras with respect to unification type. The key tool is a combinatorial characterization of finitely generated projective De Morgan and Kleene algebras

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