Limitations of Algebraic Approaches to Graph Isomorphism Testing

We investigate the power of graph isomorphism algorithms based on algebraic reasoning techniques like Gr\"obner basis computation. The idea of these algorithms is to encode two graphs into a system of equations that are satisfiable if and only if if the graphs are isomorphic, and then to (try to) decide satisfiability of the system using, for example, the Gr\"obner basis algorithm. In some cases this can be done in polynomial time, in particular, if the equations admit a bounded degree refutation in an algebraic proof systems such as Nullstellensatz or polynomial calculus. We prove linear lower bounds on the polynomial calculus degree over all fields of characteristic different from 2 and also linear lower bounds for the degree of Positivstellensatz calculus derivations. We compare this approach to recently studied linear and semidefinite programming approaches to isomorphism testing, which are known to be related to the combinatorial Weisfeiler-Lehman algorithm. We exactly characterise the power of the Weisfeiler-Lehman algorithm in terms of an algebraic proof system that lies between degree-k Nullstellensatz and degree-k polynomial calculus

Generalized Quantifiers and Logical Reducibilities

We consider extensions of first order logic (FO) and least fixed point logic (LFP) with generalized quantifiers in the sense of Lindström [Lin66]. We show that adding a finite set of such quantifiers to LFP fails to capture all polynomial time properties of structures, even over a fixed signature. We show that this strengthens results in [Hel92] and [KV92a]. We also consider certain regular infinite sets of Lindström quantifiers, which correspond to a natural notion of logical reducibility. We show that if there is any recursively enumerable set of quantifiers that can be added to FO (or LFP) to capture P, then there is one with strong uniformity conditions. This is established through a general result, linking the existence of complete problems for complexity classes with respect to the first order translations of [Imm87] or the elementary reductions of [LG77] with the existence of recursive index sets for these classes

Pebble Games and Linear Equations

We give a new, simplified and detailed account of the correspondence between levels of the Sherali-Adams relaxation of graph isomorphism and levels of pebble-game equivalence with counting (higher-dimensional Weisfeiler-Lehman colour refinement). The correspondence between basic colour refinement and fractional isomorphism, due to Ramana, Scheinerman and Ullman, is re-interpreted as the base level of Sherali-Adams and generalised to higher levels in this sense by Atserias and Maneva, who prove that the two resulting hierarchies interleave. In carrying this analysis further, we here give (a) a precise characterisation of the level-k Sherali-Adams relaxation in terms of a modified counting pebble game; (b) a variant of the Sherali-Adams levels that precisely match the k-pebble counting game; (c) a proof that the interleaving between these two hierarchies is strict. We also investigate the variation based on boolean arithmetic instead of real/rational arithmetic and obtain analogous correspondences and separations for plain k-pebble equivalence (without counting). Our results are driven by considerably simplified accounts of the underlying combinatorics and linear algebra

Logarithmic Weisfeiler--Leman and Treewidth

In this paper, we show that the $(3k+4)$-dimensional Weisfeiler--Leman algorithm can identify graphs of treewidth $k$ in $O(\log n)$ rounds. This improves the result of Grohe & Verbitsky (ICALP 2006), who previously established the analogous result for $(4k+3)$-dimensional Weisfeiler--Leman. In light of the equivalence between Weisfeiler--Leman and the logic $\textsf{FO} + \textsf{C}$ (Cai, F\"urer, & Immerman, Combinatorica 1992), we obtain an improvement in the descriptive complexity for graphs of treewidth $k$. Precisely, if $G$ is a graph of treewidth $k$, then there exists a $(3k+5)$-variable formula $\varphi$ in $\textsf{FO} + \textsf{C}$ with quantifier depth $O(\log n)$ that identifies $G$ up to isomorphism

Enhancing Fixed Point Logic with Cardinality Quantifiers

Let Q IPP be any quantifier such that FO(QIFP), first-order logic enhanced with Q IPP and its vectorizations, equals inductive fixed point logic, IFP in expressive power. It is known that for certain quantifiers Q, the equivalence FO(QIFP) ≡ IFP is no longer true if Q is added on both sides. Rather, we have FO (QIFP, Q) < IFP(Q) in such cases. We extend these results to a great variety of quantifiers, namely all unbounded simple cardinality quantifiers. Our argument also applies to partial fixed point logic, PFP. In order to establish an analogous result for least fixed point logic, LFP, we exhibit a general method to pass from arbitrary quantifiers to monotone quantifiers. Our proof shows that the three isomorphism problem is not definable in, infinitary logic extended with all monadic quantifiers and their vectorizations, where a finite bound is imposed to the number of variables as well as to the number of nested quantifiers in Q1. This strengthens a result of Etessami and Immerman by which tree isomorphism is not definable in TC + COUNTIN

Relating structure and power: Comonadic semantics for computational resources

Combinatorial games are widely used in finite model theory, constraint satisfaction, modal logic and concurrency theory to characterize logical equivalences between structures. In particular, Ehrenfeucht–Fraïssé games, pebble games and bisimulation games play a central role. We show how each of these types of games can be described in terms of an indexed family of comonads on the category of relational structures and homomorphisms. The index k is a resource parameter that bounds the degree of access to the underlying structure. The coKleisli categories for these comonads can be used to give syntax-free characterizations of a wide range of important logical equivalences. Moreover, the coalgebras for these indexed comonads can be used to characterize key combinatorial parameters: tree depth for the Ehrenfeucht–Fraïssé comonad, tree width for the pebbling comonad and synchronization tree depth for the modal unfolding comonad. These results pave the way for systematic connections between two major branches of the field of logic in computer science, which hitherto have been almost disjoint: categorical semantics and finite and algorithmic model theory