14 research outputs found
Choiceless Computation and Symmetry: Limitations of Definability
The search for a logic capturing PTIME is a long standing open problem in
finite model theory. One of the most promising candidate logics for this is
Choiceless Polynomial Time with counting (CPT). Abstractly speaking, CPT is an
isomorphism-invariant computation model working with hereditarily finite sets
as data structures. While it is easy to check that the evaluation of
CPT-sentences is possible in polynomial time, the converse has been open for
more than 20 years: Can every PTIME-decidable property of finite structures be
expressed in CPT? We attempt to make progress towards a negative answer and
show that Choiceless Polynomial Time cannot compute a preorder with colour
classes of logarithmic size in every hypercube. The reason is that such
preorders have super-polynomially many automorphic images, which makes it
impossible for CPT to define them. While the computation of such a preorder is
not a decision problem that would immediately separate P and CPT, it is
significant for the following reason: The so-called Cai-F\"urer-Immerman (CFI)
problem is one of the standard benchmarks for logics and maybe best known for
separating fixed-point logic with counting (FPC) from P. Hence, it is natural
to consider this also a potential candidate for the separation of CPT and P.
The strongest known positive result in this regard says that CPT is able to
solve CFI if a preorder with logarithmically sized colour classes is present in
the input structure. Our result implies that this approach cannot be
generalised to unordered inputs. In other words, CFI on unordered hypercubes is
a PTIME-problem which provably cannot be tackled with the state-of-the-art
choiceless algorithmic techniques.Comment: Appeared at CSL 202
Subspace-Invariant AC Formulas
We consider the action of a linear subspace of on the set of
AC formulas with inputs labeled by literals in the set , where an element acts on formulas by
transposing the th pair of literals for all such that . A
formula is {\em -invariant} if it is fixed by this action. For example,
there is a well-known recursive construction of depth formulas of size
computing the -variable PARITY function; these
formulas are easily seen to be -invariant where is the subspace of
even-weight elements of . In this paper we establish a nearly
matching lower bound on the -invariant depth
formula size of PARITY. Quantitatively this improves the best known
lower bound for {\em unrestricted} depth
formulas, while avoiding the use of the switching lemma. More generally,
for any linear subspaces , we show that if a Boolean function is
-invariant and non-constant over , then its -invariant depth
formula size is at least where is the minimum Hamming
weight of a vector in
Finite Model Theory and Proof Complexity Revisited: Distinguishing Graphs in Choiceless Polynomial Time and the Extended Polynomial Calculus
This paper extends prior work on the connections between logics from finite model theory and propositional/algebraic proof systems. We show that if all non-isomorphic graphs in a given graph class can be distinguished in the logic Choiceless Polynomial Time with counting (CPT), then they can also be distinguished in the bounded-degree extended polynomial calculus (EPC), and the refutations have roughly the same size as the resource consumption of the CPT-sentence. This allows to transfer lower bounds for EPC to CPT and thus constitutes a new potential approach towards better understanding the limits of CPT. A super-polynomial EPC lower bound for a Ptime-instance of the graph isomorphism problem would separate CPT from Ptime and thus solve a major open question in finite model theory. Further, using our result, we provide a model theoretic proof for the separation of bounded-degree polynomial calculus and bounded-degree extended polynomial calculus
The umbilical cord of finite model theory
Model theory was born and developed as a part of mathematical logic. It has
various application domains but is not beholden to any of them. A priori, the
research area known as finite model theory would be just a part of model theory
but didn't turn out that way. There is one application domain -- relational
database management -- that finite model theory had been beholden to during a
substantial early period when databases provided the motivation and were the
main application target for finite model theory.
Arguably, finite model theory was motivated even more by complexity theory.
But the subject of this paper is how relational database theory influenced
finite model theory.
This is NOT a scholarly history of the subject with proper credits to all
participants. My original intent was to cover just the developments that I
witnessed or participated in. The need to make the story coherent forced me to
cover some additional developments.Comment: To be published in the Logic in Computer Science column of the
February 2023 issue of the Bulletin of the European Association for
Theoretical Computer Scienc
Subspace-Invariant AC^0 Formulas
The n-variable PARITY function is computable (by a well-known recursive construction) by AC^0 formulas of depth d+1 and leaf size n2^{dn^{1/d}}. These formulas are seen to possess a certain symmetry: they are syntactically invariant under the subspace P of even-weight elements in {0,1}^n, which acts (as a group) on formulas by toggling negations on input literals. In this paper, we prove a 2^{d(n^{1/d}-1)} lower bound on the size of syntactically P-invariant depth d+1 formulas for PARITY. Quantitatively, this beats the best 2^{Omega(d(n^{1/d}-1))} lower bound in the non-invariant setting
Choiceless Logarithmic Space
One of the most important open problems in finite model theory is the question whether there is a logic characterising efficient computation. While this question usually concerns Ptime, it can also be applied to other complexity classes, and in particular to Logspace which can be seen as a formalisation of efficient computation for big data. One of the strongest candidates for a logic capturing Ptime is Choiceless Polynomial Time (CPT). It is based on the idea of choiceless algorithms, a general model of symmetric computation over abstract structures (rather than their encodings by finite strings). However, there is currently neither a comparably strong candidate for a logic for Logspace, nor a logic transferring the idea of choiceless computation to Logspace.
We propose here a notion of Choiceless Logarithmic Space which overcomes some of the obstacles posed by Logspace as a less robust complexity class. The resulting logic is contained in both Logspace and CPT, and is strictly more expressive than all logics for Logspace that have been known so far. Further, we address the question whether this logic can define all Logspace-queries, and prove that this is not the case
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