database query optimizers

Abstract: We revisit the complexity classification of constraint satisfaction problems (CSPs) for first-order reducts of $(\mathbb{Z},<)$, established by Bodirsky, Martin, and Mottet in 2018. Following their approach, we give a self-contained proof of this classification, correcting several errors in the argument. We further show that, with the exception of cases arising from finite-domain CSPs, all tractable cases are solved by a well-known uniform algorithm, eliminating the need for the collection of ad-hoc algorithms used in the original paper. A central component of the proof allows the analysis to be restricted to first-order reducts of $(\mathbb{Z},<)$ that contain the successor relation. We substantially simplify this reduction and generalize the underlying construction. This generalization allows us to apply the same technique to first-order reducts of the triangular lattice graph and of infinite regular trees, yielding a complete complexity classification for these classes of structures, in particular a $\mathsf{P}$ versus $\mathsf{NP}$-complete dichotomy.

Abstract: We study definability in fixed-point logic with counting (FPC) for a class of constraint satisfaction problems (CSPs) that simultaneously generalises finite-domain CSPs and temporal CSPs. The templates for the CSPs are first-order expansions of the structure $([k] \times {\mathbb Q};<_1,\dots,<_k,\sim)$ where $<_i \, := \{((i,x),(i,x')) \mid x<x'\}$ and $\sim \, := \{((i,x),(i,x')) \mid i \in [k], x,x' \in {\mathbb Q}\}$, i.e., $k$ copies of $({\mathbb Q};<)$. Such a CSP is definable in FPC precisely when the template pp-constructs neither the ternary temporal relation $X$, which holds on three elements if the minimum is attained exactly twice, nor a nontrivial finite affine structure. Equivalently, the template has global polymorphisms that witness the tractability of the temporal CSPs on the copies, and that provide 3-4 weak near-unanimity operations on the set of copies of ${\mathbb Q}$. In this case, we prove that the CSP is even expressible in the less expressive logic FP$^*$, which is the restriction of FPC without counting terms (i.e., the extension of first-order logic by a fixed point operator and an ordered auxiliary number sort). Our algorithm samples finitely many values in each copy and then performs nested constraint-singleton consistency tests (which is a uniform polynomial-time algorithm for finite-domain CSPs). The correctness proof uses finite-domain CSP theory to combine consistency certificates by a single polymorphism term, fixing each variable to one copy without destroying the certificates needed for the temporal algorithm. We then use a known normal form for algebraic products to control the remaining interactions between different copies.

Abstract: Propositional Dynamic Logic (PDL) is a well-established modal logic of programs. Among its many extensions, the loop operator stands out for capturing the cyclic behaviour of programs: a world satisfies loop(pi) precisely when it can return to itself along a path matching the regular expression pi. While the satisfiability problem for the resulting logic LPDL was settled already in the 1980s, the decidability of its finite satisfiability problem has remained open ever since. The core difficulty is inherent to the finite setting: requiring and forbidding cycles at once is easily reconciled over arbitrary models, but over finite ones it defeats the existing finite-model-theoretic techniques. We make the first substantial progress towards closing this gap. We prove that finite satisfiability is ExpTime-complete for LPDL formulae (also in the presence of the converse operator) in negation normal form, under the restriction that every negated non-atomic loop has the form loop(pi^+) for some arbitrarily complex program pi -- a generalised form of self-reachability; positive occurrences of loops remain unrestricted. At the heart of our approach lies a novel way of representing intended finite models as their infinite, cactus-like unravellings: these are recognisable by two-way tree automata and satisfy a tailored omega-regular foldability condition that lets us finitise the accepted structures.

Abstract: Otto's Theorem characterises the bisimulation-invariant PTIME queries over graphs as exactly those that can be formulated in the polyadic mu-calculus. This extends to a characterisation of the full bisimulation-invariant exponential-time hierarchy, using a higher-order extension of the polyadic mu-calculus, called PHFL, which is naturally stratified by type order. The bisimulation-invariant fragment of k-EXPTIME is captured by the fragment of PHFL whose formulas have type order at most k. This is obtained using a generalisation of the Immerman-Vardi Theorem by Freire and Martins. Here we provide such a characterisation of the exponential-space hierarchy. We show that the bisimulation-invariant fragment of k-EXPSPACE is captured by the fragment of PHFL of type order k+1 whose formulas are tail-recursive. Tail recursion limits the interplay between fixpoints and higher-order functions so that model checking based on fixpoint iteration can be made more space efficient in a way comparable to tail recursion in programming languages. The capturing result makes use of a generalisation of Vardi's characterisation of PSPACE as first-order logic with partial fixpoints, namely that k-EXPSPACE is captured by higher-order logic of order k+1 with partial fixpoints.

Abstract: Equality saturation is a promising technique for program optimization which sidesteps the phase ordering problem. However, current e-graph implementations grow exponentially large, even for simple examples. Many practical applications involve associative and commutative (AC) operators. We present an extension of relational e-matching that handles AC operators natively by storing terms as multisets. Preliminary results show that equality saturation modulo AC uses asymptotically less memory in certain cases.

Abstract: For a graph $H$, $\langle H \rangle$ denotes the class of all subdivisions of $H$ and $tw(H)$ denotes the treewidth of $H$. In this paper, we prove the following. For $k\geq 1, R\geq 1$, let $G,H$ be two graphs such that strong isometric path complexities (Chakraborty et al. [\textsc{Disc. Math., 2026}]) of both $G$ and $\langle H \rangle$ are at most $k$, and $G$ admits an honest, ``nicely rooted'' $R$-bounded $H$-partition. Then, there is a graph $F$ with $tw(F)\leq tw(H)$ such that $G$ admits a $(1,33\cdot R\cdot k^2)$-quasi-isometry to $F$. Using results of Albrechtsen, Distel, and Georgakopoulos (2025), we also obtain that $K_{2,t}$-asymptotic minor-free graphs admit quasi-isometries with additive distortion to $K_{2,t}$-minor-free graphs. This answers an open question raised by the above authors. As part of our proof, we combine the graph-partition based method and the layering partition based method (Chepoi et al. [\textsc{Discrete Comput. Geom.} 2012]) to obtain additive quasi-isometry when both the source and all subdivisions of the target graph have bounded strong isometric path complexity.

Abstract: In this article, we prove that the problem of solving $\operatorname{CSP}(\mathbf{A})$, where $\mathbf{A}$ is a finite relational template which admits a Maltsev polymorphism is in a specific complexity class DET, which is related to the complexity of computing the determinant of a matrix with integer entries. Such a class is intimately related to well-studied MOD-logspace classes in the theory of computational complexity. To prove this fact, we develop a new algorithm for solving syntactically simple binary instances of Maltsev constraint satisfaction problems, rather different from the well-known Bulatov-Dalmau algorithm, which does not require the explicit use or knowledge of a Maltsev polymorphism of the template but, rather, utilizes a graph whose vertices are 2-generated subuniverses of $\mathbb{A}$, where $\mathbb{A}$ is the Maltsev algebra parametrizing $\operatorname{CSP}(\mathbf{A})$. The theoretical importance of this algorithm is reflected in two facts: (1) it places the problem $ \operatorname{CSP}(\mathbf{A})$ in a complexity class related to the deterministic logspace with counting, which, in itself, has a strong connection to a variety of standard algorithmic problems in linear algebra, and (2) it only makes use of the relational structure of the template without the need for the explicit use of a compatible Maltsev polymorphism, depending entirely on the strong ``symmetry" of constraints compatible with such polymorphisms and the knowledge of 2-generated subuniverses of $\mathbb{A}$.