Benedikt Pago

Anuj Dawar, Benedikt Pago, Tim Seppelt

The central open question of algebraic complexity is whether VP is unequal to VNP, which is saying that the permanent cannot be represented by families of polynomial-size algebraic circuits. For symmetric algebraic circuits, this has been confirmed by Dawar and Wilsenach (2020, 2025) who showed exponential lower bounds on the size of symmetric circuits for the permanent. In this work, we set out to develop a more general symmetric algebraic complexity theory. Our main result is that a family of symmetric polynomials admits small symmetric circuits if and only if they can be written as a linear combination of homomorphism counting polynomials of graphs of bounded treewidth. We also establish a relationship between the symmetric complexity of subgraph counting polynomials and the vertex cover number of the pattern graph. As a concrete example, we examine the symmetric complexity of immanant families (a generalisation of the determinant and permanent) and show that a known conditional dichotomy due to Curticapean (2021) holds unconditionally in the symmetric setting.

Sophie Brinke, Anuj Dawar, Erich Grädel et al.

We study the status of preservation theorems such as the Łoś-Tarski theorem and the homomorphism preservation theorem in the context of semiring semantics. Semiring semantics has its origins in the provenance analysis of database queries. Depending on the underlying semiring, it allows us to track which atomic facts are responsible for the truth of a statement or practical information about the evaluation such as costs or confidence. The systematic development of semiring semantics for first-order logic and other logical systems raises the question to what extent classical model-theoretic results can be generalised to this setting and how such results depend on the underlying semiring. The definitions of semantic properties such as preservation under extensions, substructures, or homomorphisms naturally generalise to the setting of semiring semantics. However, the status of the corresponding preservation theorem strongly depends on the algebraic properties of the particular semirings. We prove that these preservation theorems do indeed hold for all lattice semirings (a quite large class, encompassing practically relevant semirings and in particular all min-max semirings). The proofs combine adaptations of the classical compactness and amalgamation methods with specific reduction methods for logical entailment that have been developed in semiring semantics. On the other side, variants of the existential preservation theorem fail for many other semirings, including the tropical semiring, the Viterbi semiring, the Łukasiewicz semiring, and the natural semiring. Surprisingly, the existential preservation theorem does hold for finite interpretations in a number of semirings, including all lattice semirings. Thus, the situation for these semirings is in sharp contrast to the Boolean case, where the Łoś-Tarski theorem holds in general, but not in the finite.

Benedikt Pago

A notorious open question in circuit complexity is whether Boolean operations of arbitrary arity can efficiently be expressed using modular counting gates only. Håstad's celebrated switching lemma yields exponential lower bounds for the dual problem - realising modular arithmetic with Boolean gates - but, a similar lower bound for modular circuits computing the Boolean AND function has remained elusive for almost 30 years. We solve this problem for the restricted model of symmetric circuits: We consider MOD$_m$-circuits of arbitrary depth, and for an arbitrary modulus $m \in \mathbb{N}$, and obtain subexponential lower bounds for computing the $n$-ary Boolean AND function, under the assumption that the circuits are syntactically symmetric under all permutations of their $n$ input gates. This lower bound is matched precisely by a construction due to (Idziak, Kawałek, Krzaczkowski, LICS'22), leading to the surprising conclusion that the optimal symmetric circuit size is already achieved with depth $2$. Motivated by another construction from (LICS'22), which achieves smaller size at the cost of greater depth, we also prove tight size lower bounds for circuits with a more liberal notion of symmetry characterised by a nested block structure on the input variables.