arXiv subjects

Benedikt Pago

Publications and source records attributed to Benedikt Pago.

13 recordsLinked to original sources

arXiv2026

Preservation Theorems in Semiring Semantics

We study the status of preservation theorems such as the {\L}o\'s-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 {\L}ukasiewicz 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 {\L}o\'s-Tarski theorem holds in general, but not in the finite.

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arXiv2026

Optimal Lower Bounds for Symmetric Modular Circuits

A notorious open question in circuit complexity is whether Boolean operations of arbitrary arity can efficiently be expressed using modular counting gates only. H{\aa}stad'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, Kawa{\l}ek, 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.

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arXiv2026

Lower Bounds in Algebraic Complexity via Symmetry and Homomorphism Polynomials

Valiant's conjecture asserts that the circuit complexity classes VP and VNP are distinct, meaning that the permanent does not admit polynomial-size algebraic circuits. As it is the case in many branches of complexity theory, the unconditional separation of these complexity classes seems elusive. In stark contrast, the symmetric analogue of Valiant's conjecture has been proven by Dawar and Wilsenach (2020): the permanent does not admit symmetric algebraic circuits of polynomial size, while the determinant does. Symmetric algebraic circuits are both a powerful computational model and amenable to proving unconditional lower bounds. In this paper, we develop a symmetric algebraic complexity theory by introducing symmetric analogues of the complexity classes VP, VBP, and VF called symVP, symVBP, and symVF. They comprise polynomials that admit symmetric algebraic circuits, skew circuits, and formulas, respectively, of polynomial orbit size. Having defined these classes, we show unconditionally that $\mathsf{symVF} \subsetneq \mathsf{symVBP} \subsetneq \mathsf{symVP}$. To that end, we characterise the polynomials in symVF and symVBP as those that can be written as linear combinations of homomorphism polynomials for patterns of bounded treedepth and pathwidth, respectively. This extends a previous characterisation by Dawar, Pago, and Seppelt (2026) of symVP. Finally, we show that symVBP and symVP contain homomorphism polynomials which are VBP- and VP-complete, respectively. We give general graph-theoretic criteria for homomorphism polynomials and their linear combinations to be VBP-, VP-, or VNP-complete. These conditional lower bounds drastically enlarge the realm of natural polynomials known to be complete for VNP, VP, or VBP. Under the assumption VFPT $\neq$ VW[1], we precisely identify the homomorphism polynomials that lie in VP as those whose patterns have bounded treewidth.

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arXiv2026

Limitations of Affine Integer Relaxations for Solving Constraint Satisfaction Problems

We show that various recent algorithms for finite-domain constraint satisfaction problems (CSP), which are based on solving their affine integer relaxations, do not solve all tractable and not even all Maltsev CSPs. This rules them out as candidates for a universal polynomial-time CSP algorithm. The algorithms are $\mathbb{Z}$-affine $k$-consistency, BLP+AIP, BA$^{k}$, and CLAP. We thereby answer a question by Brakensiek, Guruswami, Wrochna, and Živný whether BLP+AIP solves all tractable CSPs in the negative. We also refute a conjecture by Dalmau and Opršal (LICS 2024) that every CSP is either solved by $\mathbb{Z}$-affine $k$-consistency or admits a Datalog reduction from 3-colorability. For the cohomological $k$-consistency algorithm, that is also based on affine relaxations, we show that it correctly solves our counterexample but fails on an NP-complete template.

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arXiv2025

Arity hierarchies for quantifiers closed under partial polymorphisms

We investigate the expressive power of generalized quantifiers closed under partial polymorphism conditions motivated by the study of constraint satisfaction problems. We answer a number of questions arising from the work of Dawar and Hella (CSL 2024) where such quantifiers were introduced. For quantifiers closed under partial near-unanimity polymorphisms, we establish hierarchy results clarifying the interplay between the arity of the polymorphisms and of the quantifiers: The expressive power of $(\ell+1)$-ary quantifiers closed under $\ell$-ary partial near-unanimity polymorphisms is strictly between the class of all quantifiers of arity $\ell-1$ and $\ell$. We also establish an infinite hierarchy based on the arity of quantifiers with a fixed arity of partial near-unanimity polymorphisms. Finally, we prove inexpressiveness results for quantifiers with a partial Maltsev polymorphism. The separation results are proved using novel algebraic constructions in the style of Cai-Fürer-Immerman and the quantifier pebble games of Dawar and Hella (2024).

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arXiv2025

Symmetric Proofs in the Ideal Proof System

We consider the Ideal Proof System (IPS) introduced by Grochow and Pitassi and pose the question of which tautologies admit symmetric proofs, and of what complexity. The symmetry requirement in proofs is inspired by recent work establishing lower bounds in other symmetric models of computation. We link the existence of symmetric IPS proofs to the expressive power of logics such as fixed-point logic with counting and Choiceless Polynomial Time, specifically regarding the graph isomorphism problem. We identify relationships and tradeoffs between the symmetry of proofs and other parameters of IPS proofs such as size, degree and linearity. We study these on a number of standard families of tautologies from proof complexity and finite model theory such as the pigeonhole principle, the subset sum problem and the Cai-Fürer-Immerman graphs, exhibiting non-trivial upper bounds on the size of symmetric IPS proofs.

cs.LO↗

arXiv2025

Symmetric Algebraic Circuits and Homomorphism Polynomials

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.

cs.CC↗

arXiv2024

Choiceless Computation and Symmetry: Limitations of Definability

The search for a logic capturing PTIME is a long standing open problem in finite model theory. One of the most promising candidate logics for this is Choiceless Polynomial Time with counting (CPT). Abstractly speaking, CPT is an isomorphism-invariant computation model working with hereditarily finite sets as data structures. While it is easy to check that the evaluation of CPT-sentences is possible in polynomial time, the converse has been open for more than 20 years: Can every PTIME-decidable property of finite structures be expressed in CPT? We attempt to make progress towards a negative answer and show that Choiceless Polynomial Time cannot compute a preorder with colour classes of logarithmic size in every hypercube. The reason is that such preorders have super-polynomially many automorphic images, which makes it impossible for CPT to define them. While the computation of such a preorder is not a decision problem that would immediately separate P and CPT, it is significant for the following reason: The so-called Cai-Fürer-Immerman (CFI) problem is one of the standard benchmarks for logics and maybe best known for separating fixed-point logic with counting (FPC) from P. Hence, it is natural to consider this also a potential candidate for the separation of CPT and P. The strongest known positive result in this regard says that CPT is able to solve CFI if a preorder with logarithmically sized colour classes is present in the input structure. Our result implies that this approach cannot be generalised to unordered inputs. In other words, CFI on unordered hypercubes is a PTIME-problem which provably cannot be tackled with the state-of-the-art choiceless algorithmic techniques.

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arXiv2023

Limitations of Game Comonads via Homomorphism Indistinguishability

Abramsky, Dawar, and Wang (2017) introduced the pebbling comonad for k-variable counting logic and thereby initiated a line of work that imports category theoretic machinery to finite model theory. Such game comonads have been developed for various logics, yielding characterisations of logical equivalences in terms of isomorphisms in the associated co-Kleisli category. We show a first limitation of this approach by studying linear-algebraic logic, which is strictly more expressive than first-order counting logic and whose k-variable logical equivalence relations are known as invertible-map equivalences (IM). We show that there exists no finite-rank comonad on the category of graphs whose co-Kleisli isomorphisms characterise IM-equivalence, answering a question of Ó Conghaile and Dawar (CSL 2021). We obtain this result by ruling out a characterisation of IM-equivalence in terms of homomorphism indistinguishability and employing the Lovász-type theorems for game comonads established by Dawar, Jakl, and Reggio (2021). Two graphs are homomorphism indistinguishable over a graph class if they admit the same number of homomorphisms from every graph in the class. The IM-equivalences cannot be characterised in this way, neither when counting homomorphisms in the natural numbers, nor in any finite prime field.

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arXiv2023

Choiceless Polynomial Time, Symmetric Circuits and Cai-Fürer-Immerman Graphs

Choiceless Polynomial Time (CPT) is currently the only candidate logic for capturing PTIME (that is, it is contained in PTIME and has not been separated from it). A prominent example of a decision problem in PTIME that is not known to be CPT-definable is the isomorphism problem on unordered Cai-Fürer-Immerman graphs (the CFI-query). We study the expressive power of CPT with respect to this problem and develop a partial characterisation of solvable instances in terms of properties of symmetric XOR-circuits over the CFI-graphs: The CFI-query is CPT-definable on a given class of graphs only if: For each graph $G$, there exists an XOR-circuit $C$, whose input gates are labelled with edges of $G$, such that $C$ is sufficiently symmetric with respect to the automorphisms of $G$ and satisfies certain other circuit properties. We also give a sufficient condition for CFI being solvable in CPT and develop a new CPT-algorithm for the CFI-query. It takes as input structures which contain, along with the CFI-graph, an XOR-circuit with suitable properties. The strongest known CPT-algorithm for this problem can solve instances equipped with a preorder with colour classes of logarithmic size. Our result implicitly extends this to preorders with colour classes of polylogarithmic size (plus some unordered additional structure). Finally, our work provides new insights regarding a much more general problem: The existence of a solution to an unordered linear equation system $A \cdot x = b$ over a finite field is CPT-definable if the matrix $A$ has at most logarithmic rank (with respect to the size of the structure that encodes the equation system). This is another example that separates CPT from fixed-point logic with counting.

cs.LO↗

arXiv2023

Lower bounds for Choiceless Polynomial Time via Symmetric XOR-circuits

Choiceless Polynomial Time (CPT) is one of the few remaining candidate logics for capturing PTIME. In this paper, we make progress towards separating CPT from polynomial time by firstly establishing a connection between the expressive power of CPT and the existence of certain symmetric circuit families, and secondly, proving lower bounds against these circuits. We focus on the isomorphism problem of unordered Cai-Fürer-Immerman-graphs (the CFI-query) as a potential candidate for separating CPT from P. Results by Dawar, Richerby and Rossman, and subsequently by Pakusa, Schalthöfer and Selman show that the CFI-query is CPT-definable on linearly ordered and preordered base graphs with small colour classes. We define a class of CPT-algorithms, that we call "CFI-symmetric algorithms", which generalises all the known ones, and show that such algorithms can only define the CFI-query on a given class of base graphs if there exists a family of symmetric XOR-circuits with certain properties. These properties include that the circuits have the same symmetries as the base graphs, are of polynomial size, and satisfy certain fan-in restrictions. Then we prove that such circuits with slightly strengthened requirements (i.e. stronger symmetry and fan-in and fan-out restrictions) do not exist for the n-dimensional hypercubes as base graphs. This almost separates the CFI-symmetric algorithms from polynomial time - up to the gap that remains between the circuits whose existence we can currently disprove and the circuits whose existence is necessary for the definability of the CFI-query by a CFI-symmetric algorithm.

cs.CC↗

arXiv2022

Finite Model Theory and Proof Complexity revisited: Distinguishing graphs in Choiceless Polynomial Time and the Extended Polynomial Calculus

This paper extends prior work on the connections between logics from finite model theory and propositional/algebraic proof systems. We show that if all non-isomorphic graphs in a given graph class can be distinguished in the logic Choiceless Polynomial Time with counting (CPT), then they can also be distinguished in the bounded-degree extended polynomial calculus (EPC), and the refutations have roughly the same size as the resource consumption of the CPT-sentence. This allows to transfer lower bounds for EPC to CPT and thus constitutes a new potential approach towards better understanding the limits of CPT. A super-polynomial EPC lower bound for a PTIME-instance of the graph isomorphism problem would separate CPT from PTIME and thus solve a major open question in finite model theory. Further, using our result, we provide a model theoretic proof for the separation of bounded-degree polynomial calculus and bounded-degree extended polynomial calculus.

cs.LO↗

arXiv2022

A Finite-Model-Theoretic View on Propositional Proof Complexity

We establish new, and surprisingly tight, connections between propositional proof complexity and finite model theory. Specifically, we show that the power of several propositional proof systems, such as Horn resolution, bounded-width resolution, and the monomial calculus of bounded degree, can be characterised in a precise sense by variants of fixed-point logics that are of fundamental importance in descriptive complexity theory. Our main results are that Horn resolution has the same expressive power as least fixed-point logic, that bounded-width resolution captures existential least fixed-point logic, and that the polynomial calculus with bounded degree over the rationals solves precisely the problems definable in fixed-point logic with counting. We also study the bounded-degree polynomial calculus. Over the rationals, it captures fixed-point logic with counting if we restrict the bit-complexity of the coefficients. For unrestricted coefficients, we can only say that the bounded-degree polynomial calculus is at most as powerful as bounded variable infinitary counting logic, but a precise logical characterisation of its power remains an open problem. These connections between logics and proof systems allow us to establish finite-model-theoretic tools for proving lower bounds for the polynomial calculus over the rationals and also over finite fields. This is a corrected version of the paper (arXiv:1802.09377) published originally on January 23, 2019.

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