9 research outputs found
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
Choiceless Polynomial Time, Symmetric Circuits and Cai-F\"urer-Immerman Graphs
Choiceless Polynomial Time (CPT) is currently the only candidate logic for
capturing PTIME (that is, it is contained in PTIME and has not been separated
from it). A prominent example of a decision problem in PTIME that is not known
to be CPT-definable is the isomorphism problem on unordered
Cai-F\"urer-Immerman graphs (the CFI-query). We study the expressive power of
CPT with respect to this problem and develop a partial characterisation of
solvable instances in terms of properties of symmetric XOR-circuits over the
CFI-graphs: The CFI-query is CPT-definable on a given class of graphs only if:
For each graph , there exists an XOR-circuit , whose input gates are
labelled with edges of , such that is sufficiently symmetric with
respect to the automorphisms of and satisfies certain other circuit
properties. We also give a sufficient condition for CFI being solvable in CPT
and develop a new CPT-algorithm for the CFI-query. It takes as input structures
which contain, along with the CFI-graph, an XOR-circuit with suitable
properties. The strongest known CPT-algorithm for this problem can solve
instances equipped with a preorder with colour classes of logarithmic size. Our
result implicitly extends this to preorders with colour classes of
polylogarithmic size (plus some unordered additional structure). Finally, our
work provides new insights regarding a much more general problem: The existence
of a solution to an unordered linear equation system over a
finite field is CPT-definable if the matrix has at most logarithmic rank
(with respect to the size of the structure that encodes the equation system).
This is another example that separates CPT from fixed-point logic with
counting
The Model-Theoretic Expressiveness of Propositional Proof Systems
We establish new, and surprisingly tight, connections between propositional proof complexity and finite model theory.
Specifically, we show that the power of several propositional proof systems, such as Horn resolution, bounded width resolution, and the polynomial calculus of bounded degree, can be characterised in a precise sense by variants of fixed-point logics that are of fundamental importance in descriptive complexity theory.
Our main results are that Horn resolution has the same expressive power as least fixed-point logic, that bounded width resolution captures existential least fixed-point logic, and that the (monomial restriction of the) polynomial calculus of bounded degree solves precisely the problems definable in fixed-point logic with counting
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
A Finite-Model-Theoretic View on Propositional Proof Complexity
We establish new, and surprisingly tight, connections between propositional
proof complexity and finite model theory. Specifically, we show that the power
of several propositional proof systems, such as Horn resolution, bounded-width
resolution, and the polynomial calculus of bounded degree, can be characterised
in a precise sense by variants of fixed-point logics that are of fundamental
importance in descriptive complexity theory. Our main results are that Horn
resolution has the same expressive power as least fixed-point logic, that
bounded-width resolution captures existential least fixed-point logic, and that
the polynomial calculus with bounded degree over the rationals solves precisely
the problems definable in fixed-point logic with counting. By exploring these
connections further, we establish finite-model-theoretic tools for proving
lower bounds for the polynomial calculus over the rationals and over finite
fields
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