5 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
LIPIcs, Volume 261, ICALP 2023, Complete Volume
LIPIcs, Volume 261, ICALP 2023, Complete Volum
A Fixed-Point Logic with Symmetric Choice
Gire and Hoang introduce a fixed-point logic with a `symmetric' choice operator that makes a nondeterministic choice from a definable set of tuples at each stage in the inductive construction of a relation, as long as the set of tuples is an automorphism class of the structure. We present a clean definition of the syntax and semantics of this logic and investigate its expressive power. We extend the logic of Gire and Hoang with parameterized and nested fixed points and first-order combinations of fixed points. We show that the ability to supply parameters to fixed points strictly increases the power of the logic. Our logic can express the graph isomorphism problem and we show that, on almost all structures, it captures P , the class of problems decidable in polynomial time by a deterministic Turing machine with an oracle for graph isomorphism