Separating Rank Logic from Polynomial Time

Lichter, Moritz

doi:10.1109/lics52264.2021.9470598

Cited by 9 publications

(14 citation statements)

References 21 publications

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“…We next present a high-level exposition of the results in [6] and [13] on invertible-map equivalences of CFI-structures, and their consequences for graph isomorphism and descriptive complexity. We refer to the full versions of these papers, published on ArXiv [5,14].…”

Section: Invertible-map Equivalences For Generalised Cfi-structuresmentioning

confidence: 99%

“…More recently, a somewhat different CFI-construction over the rings Z 2 i has been used by Lichter [14] to separate rank logic from PTIME. His construction of CFI-structures CFI [G, Z 2 i , λ] is not based on 3-regular graphs, but on highly connected regular graphs of large degree and girth.…”

Section: Invertible-map Equivalences For Generalised Cfi-structuresmentioning

confidence: 99%

“…Further, but this is a minor point, the last component is not a function on vertices, but a function λ : E → Z The analysis of these CFI-structures is done in terms of the game-theoretic description of the invertible-map equivalences, the so-called invertible-map game introduced in [8], using combinatorial objects called blurers. The main technical result of [14] shows that these CFI-structures cannot be told apart by invertible-map equivalences for the prime 2.…”

Section: Invertible-map Equivalences For Generalised Cfi-structuresmentioning

confidence: 99%

“…To combine the results of [5] and [14] we want to show that the CFI-structures CFI [G, Z 2 i , λ] are not just ≡ IM k,{2} -equivalent but in fact ≡ IM k,P -equivalent for the set of all primes P. For this, we have to show that the differences in the two CFI-constructions do not really matter.…”

Section: Combining the Constructionsmentioning

confidence: 99%

“…For d-regular graphs, the inner vertices can be replaced by d-ary relations, which is done in [5]. Alternatively, [14] leaves out the outer vertices and directly connects the inner vertices, which is important to yield structures of the same signature for different degrees of the base graph. When using only one sort of vertices (so either only inner or only outer ones) fewer case distinctions are needed.…”

Section: Combining the Constructionsmentioning

confidence: 99%

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Limitations of the Invertible-Map Equivalences

Dawar¹,

Grädel²,

Lichter³

2021

Preprint

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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.

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Section: Invertible-map Equivalences For Generalised Cfi-structuresmentioning

confidence: 99%

Section: Invertible-map Equivalences For Generalised Cfi-structuresmentioning

confidence: 99%

Section: Invertible-map Equivalences For Generalised Cfi-structuresmentioning

confidence: 99%

Section: Combining the Constructionsmentioning

confidence: 99%

Section: Combining the Constructionsmentioning

confidence: 99%

See 3 more Smart Citations

Limitations of the Invertible-Map Equivalences

Dawar¹,

Grädel²,

Lichter³

2021

Preprint

Self Cite

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Limitations of the invertible-map equivalences

Dawar

Grädel

Lichter

2022

Journal of Logic and Computation

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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 ${\equiv ^{\text {IM}}_{k, Q}}$ 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.

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The expressive power of finitely many generalized quantifiers

Dawar¹,

Hella

Proceedings Ninth Annual IEEE Symposium on Logic in Computer Science

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We study Lindström quantifiers that satisfy certain closure properties which are motivated by the study of polymorphisms in the context of constraint satisfaction problems (CSP). When the algebra of polymorphisms of a finite structure B satisfies certain equations, this gives rise to a natural closure condition on the class of structures that map homomorphically to B. The collection of quantifiers that satisfy closure conditions arising from a fixed set of equations are rather more general than those arising as CSP. For any such conditions P, we define a pebble game that delimits the distinguishing power of the infinitary logic with all quantifiers that are P-closed. We use the pebble game to show that the problem of deciding whether a system of linear equations is solvable in Z /2 Z is not expressible in the infinitary logic with all quantifiers closed under a near-unanimity condition.

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Separating Rank Logic from Polynomial Time

Cited by 9 publications

References 21 publications

Limitations of the Invertible-Map Equivalences

Limitations of the Invertible-Map Equivalences

Limitations of the invertible-map equivalences

The expressive power of finitely many generalized quantifiers

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