35 research outputs found
Witnessed Symmetric Choice and Interpretations in Fixed-Point Logic with Counting
At the core of the quest for a logic for Ptime is a mismatch between algorithms making arbitrary choices and isomorphism-invariant logics. One approach to tackle this problem is witnessed symmetric choice. It allows for choices from definable orbits certified by definable witnessing automorphisms.
We consider the extension of fixed-point logic with counting (IFPC) with witnessed symmetric choice (IFPC+WSC) and a further extension with an interpretation operator (IFPC+WSC+I). The latter operator evaluates a subformula in the structure defined by an interpretation. When similarly extending pure fixed-point logic (IFP), IFP+WSC+I simulates counting which IFP+WSC fails to do. For IFPC+WSC, it is unknown whether the interpretation operator increases expressiveness and thus allows studying the relation between WSC and interpretations beyond counting.
In this paper, we separate IFPC+WSC from IFPC+WSC+I by showing that IFPC+WSC is not closed under FO-interpretations. By the same argument, we answer an open question of Dawar and Richerby regarding non-witnessed symmetric choice in IFP. Additionally, we prove that nesting WSC-operators increases the expressiveness using the so-called CFI graphs. We show that if IFPC+WSC+I canonizes a particular class of base graphs, then it also canonizes the corresponding CFI graphs. This differs from various other logics, where CFI graphs provide difficult instances
Witnessed Symmetric Choice and Interpretations in Fixed-Point Logic with Counting
At the core of the quest for a logic for PTime is a mismatch between
algorithms making arbitrary choices and isomorphism-invariant logics. One
approach to overcome this problem is witnessed symmetric choice. It allows for
choices from definable orbits which are certified by definable witnessing
automorphisms.
We consider the extension of fixed-point logic with counting (IFPC) with
witnessed symmetric choice (IFPC+WSC) and a further extension with an
interpretation operator (IFPC+WSC+I). The latter operator evaluates a
subformula in the structure defined by an interpretation. This structure
possibly has other automorphisms exploitable by the WSC-operator. For similar
extensions of pure fixed-point logic (IFP) it is known that IFP+WSCI simulates
counting which IFP+WSC fails to do. For IFPC+WSC it is unknown whether the
interpretation operator increases expressiveness and thus allows studying the
relation between WSC and interpretations beyond counting.
We separate IFPC+WSC from IFPC+WSCI by showing that IFPC+WSC is not closed
under FO-interpretations. By the same argument, we answer an open question of
Dawar and Richerby regarding non-witnessed symmetric choice in IFP.
Additionally, we prove that nesting WSC-operators increases the expressiveness
using the so-called CFI graphs. We show that if IFPC+WSC+I canonizes a
particular class of base graphs, then it also canonizes the corresponding CFI
graphs. This differs from various other logics, where CFI graphs provide
difficult instances.Comment: 46 pages, 5 figures, [v2] and [v3] Corrected minor mistakes and added
figure
Choiceless Polynomial Time with Witnessed Symmetric Choice
We extend Choiceless Polynomial Time (CPT), the currently only remaining
promising candidate in the quest for a logic capturing PTime, so that this
extended logic has the following property: for every class of structures for
which isomorphism is definable, the logic automatically captures PTime.
For the construction of this logic we extend CPT by a witnessed symmetric
choice operator. This operator allows for choices from definable orbits. But,
to ensure polynomial time evaluation, automorphisms have to be provided to
certify that the choice set is indeed an orbit.
We argue that, in this logic, definable isomorphism implies definable
canonization. Thereby, our construction removes the non-trivial step of
extending isomorphism definability results to canonization. This step was a
part of proofs that show that CPT or other logics capture PTime on a particular
class of structures. The step typically required substantial extra effort.Comment: 65 pages. Full version of a paper to appear at LICS 22. v2: corrected
typos and small mistake
Canonization for Bounded and Dihedral Color Classes in Choiceless Polynomial Time
In the quest for a logic capturing Ptime the next natural classes of structures to consider are those with bounded color class size. We present a canonization procedure for graphs with dihedral color classes of bounded size in the logic of Choiceless Polynomial Time (CPT), which then captures Ptime on this class of structures. This is the first result of this form for non-abelian color classes.
The first step proposes a normal form which comprises a "rigid assemblage". This roughly means that the local automorphism groups form 2-injective 3-factor subdirect products. Structures with color classes of bounded size can be reduced canonization preservingly to normal form in CPT.
In the second step, we show that for graphs in normal form with dihedral color classes of bounded size, the canonization problem can be solved in CPT. We also show the same statement for general ternary structures in normal form if the dihedral groups are defined over odd domains
The Iteration Number of the Weisfeiler-Leman Algorithm
We prove new upper and lower bounds on the number of iterations the
-dimensional Weisfeiler-Leman algorithm (-WL) requires until
stabilization. For , we show that -WL stabilizes after at most
iterations (where denotes the number of vertices of the
input structures), obtaining the first improvement over the trivial upper bound
of and extending a previous upper bound of for
[Lichter et al., LICS 2019].
We complement our upper bounds by constructing -ary relational structures
on which -WL requires at least iterations to stabilize. This
improves over a previous lower bound of [Berkholz,
Nordstr\"{o}m, LICS 2016].
We also investigate tradeoffs between the dimension and the iteration number
of WL, and show that -WL, where , can
simulate the -WL algorithm using only many iterations, but still requires at least
iterations for any (that is sufficiently smaller than ).
The number of iterations required by -WL to distinguish two structures
corresponds to the quantifier rank of a sentence distinguishing them in the -variable fragment of first-order logic with counting
quantifiers. Hence, our results also imply new upper and lower bounds on the
quantifier rank required in the logic , as well as tradeoffs between
variable number and quantifier rank.Comment: 30 pages, 1 figure, full version of a paper accepted at LICS 2023;
second version improves the presentation of the result
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 \'O 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 Lov\'asz-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.Comment: Minor corrections in Section
Labor Supply Shocks, Native Wages, and the Adjustment of Local Employment
By exploiting a commuting policy that led to a sharp and unexpected inflow of Czech workers to areas along the German-Czech border, we examine the impact of an exogenous immigration-induced labor supply shock on local wages and employment of natives. On average, the supply shock leads to a moderate decline in local native wages and a sharp decline in local native employment. These average effects mask considerable heterogeneity across groups: while younger natives experience larger wage effects, employment responses are particularly pronounced for older natives. This pattern is inconsistent with standard models of immigration but can be accounted for by a model that allows for a larger labor supply elasticity or a higher degree of wage rigidity for older than for young workers. We further show that the employment response is almost entirely driven by diminished inflows of natives into work rather than outflows into other areas or nonemployment, suggesting that "outsiders" shield "insiders" from the increased competition.Christian Dustmann acknowledges funding through the ERC Advanced
Grant 323992-DMEA and by the DFG (DU1024/1-1). Jan Stuhler acknowledges
funding from the German National Academic Foundation, the Spanish Ministry of
Economy and Competitiveness (MDM2014-0431 and ECO2014-55858-P), and the
Comunidad de Madrid (MadEco-CM S2015/HUM-3444)