We show that the 2-dimensional Weisfeiler-Leman algorithm stabilizes n-vertex graphs after at most O(n log n) iterations. This implies that if such graphs are distinguishable in 3-variable first order logic with counting, then they can also be distinguished in this logic by a formula of quantifier depth at most O(n log n).For this we exploit a new refinement based on counting walks and argue that its iteration number differs from the classic Weisfeiler-Leman refinement by at most a logarithmic factor. We then prove matching linear upper and lower bounds on the number of iterations of the walk refinement. This is achieved with an algebraic approach by exploiting properties of semisimple matrix algebras. We also define a walk logic and a bijective walk pebble game that precisely correspond to the new walk refinement.Via the above-mentioned correspondence between the Weisfeiler-Leman (WL) algorithm and the logic [7] we obtain the following corollary.Corollary 2. If two n-vertex graphs can be distinguished by a sentence in 3-variable first order logic with counting C 3 , then there is also a C 3 sentence of quantifier depth at most O(n log n) that distinguishes the two graphs.To prove Theorem 1, we take an algebraic point of view and use a one to one correspondence between coherent configurations and coherent algebras [11]. The WL algorithm produces the former as output, whereas the latter are semisimple matrix algebras closed with respect to the Hadamard multiplication.

Build systems are used in all but the smallest software projects to invoke the right build tools on the right files in the right order. A build system must be sound (after a build, generated files consistently reflect the latest source files) and efficient (recheck and rebuild as few build units as possible). Contemporary build systems provide limited efficiency because they lack support for expressing finegrained file dependencies. We present a build system called pluto that supports the definition of reusable, parameterized, interconnected builders. When run, a builder notifies the build system about dynamically required and produced files as well as about other builders whose results are needed. To support fine-grained file dependencies, we generalize the traditional notion of time stamps to allow builders to declare their actual requirements on a file's content. pluto collects the requirements and products of a builder with their stamps in a build summary. This enables pluto to provides provably sound and optimal incremental rebuilding. To support dynamic dependencies, our rebuild algorithm interleaves dependency analysis and builder execution and enforces invariants on the dependency graph through a dynamic analysis. We have developed pluto as a Java API and used it to implement more than 25 builders. We describe our experience with migrating a larger Ant build script to pluto and compare the respective build times.

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This note draws conclusions that arise by combining two recent papers, by Anuj Dawar, Erich Grädel, and Wied Pakusa, published at ICALP 2019 and by Moritz Lichter, published at LICS 2021. In both papers, the main technical results rely on the combinatorial and algebraic analysis of the invertible-map equivalences ≡ IM k,Q on certain variants of Cai-Fürer-Immerman structures (CFI-structures for short). These ≡ IM k,Q -equivalences, for a natural number k and a set of primes Q, refine the well-known Weisfeiler-Leman equivalences used in algorithms for graph isomorphism. The intuition is that two graphs G ≡ IM k,Q H cannot be distinguished by iterative refinements of equivalences on k-tuples defined via linear operators on vector spaces over fields of characteristic p ∈ Q.In the first paper it has been shown, using considerable algebraic machinery, that for a prime q / ∈ Q, the ≡ IM k,Q equivalences are not strong enough to distinguish between non-isomorphic CFI-structures over the field Fq. In the second paper, a similar but not identical construction for CFI-structures over the rings Z 2 i has, again by rather involved combinatorial and algebraic arguments, been shown to be indistinguishable with respect to ≡ IM k,{2} . Together with earlier work on rank logic, this second result suffices to separate rank logic from polynomial time.We show here that the two approaches can be unified to prove that CFI-structures over the rings Z 2 i are in fact indistinguishable with respect to ≡ IM k,P , for the set P of all primes. In particular, this implies the following two results.There is no fixed k such that the invertible-map equivalence ≡ IM k,P coincides with isomorphism on all finite graphs. No extension of fixed-point logic by linear-algebraic operators over fields can capture polynomial time.

This note draws conclusions that arise by combining two recent papers, by Anuj Dawar, Erich Grädel and Wied Pakusa, published at ICALP 2019, and by Moritz Lichter, published at LICS 2021. In both papers, the main technical results rely on the combinatorial and algebraic analysis of the invertible-map equivalences ${\equiv ^{\text {IM}}_{k, Q}}$ on certain variants of Cai–Fürer–Immerman structures (CFI-structures for short). These ${\equiv ^{\text {IM}}_{k, Q}}$-equivalences, for a natural number $k$ and a set of primes $Q$, refine the well-known Weisfeiler–Leman equivalences used in algorithms for graph isomorphism. The intuition is that two graphs $G{\equiv ^{\text {IM}}_{k, Q}}H$ cannot be distinguished by iterative refinements of equivalences on $k$-tuples defined via linear operators on vector spaces over fields of characteristic $p \in Q$. In the first paper it has been shown, using considerable algebraic machinery, that for a prime $q \notin Q$, the ${\equiv ^{\text {IM}}_{k, Q}}$ equivalences are not strong enough to distinguish between non-isomorphic CFI-structures over the field $\mathbb {F}_q$. In the second paper, a similar but not identical construction for CFI-structures over the rings $\mathbb {Z}_{2^i}$ has, again by rather involved combinatorial and algebraic arguments, been shown to be indistinguishable with respect to ${\equiv ^{\text {IM}}_{k, \{2\}}}$. Together with an earlier work on rank logic, this second result suffices to separate rank logic from polynomial time. We show here that the two approaches can be unified to prove that CFI-structures over the rings $\mathbb {Z}_{2^i}$ are in fact indistinguishable with respect to ${\equiv ^{\text {IM}}_{k, {\mathbb {P}}}}$, for the set ${\mathbb {P}}$ of all primes. In particular, this implies the following two results. First, there is no fixed $k$ such that the invertible-map equivalence ${\equiv ^{\text {IM}}_{k, {\mathbb {P}}}}$ coincides with isomorphism on all finite graphs. Second, no extension of fixed-point logic by linear-algebraic operators over fields can capture polynomial time.