19 research outputs found
Rank Logic is Dead, Long Live Rank Logic!
Motivated by the search for a logic for polynomial time, we study rank logic (FPR) which extends fixed-point logic with counting (FPC) by operators that determine the rank of matrices over finite fields. While FPR can express most of the known queries that separate FPC from PTIME, nearly nothing was known about the limitations of its expressive power.
In our first main result we show that the extensions of FPC by rank operators over different prime fields are incomparable. This solves an open question posed by Dawar and Holm and also implies that rank logic, in its original definition with a distinct rank operator for every field, fails to capture polynomial time. In particular we show that the variant of rank logic FPR* with an operator that uniformly expresses the matrix rank over finite fields is more expressive than FPR.
One important step in our proof is to consider solvability logic FPS which is the analogous extension of FPC by quantifiers which express the solvability problem for linear equation systems over finite fields. Solvability logic can easily be embedded into rank logic, but it is open whether it is a strict fragment. In our second main result we give a partial answer to this question: in the absence of counting, rank operators are strictly more expressive than solvability quantifiers
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
Definability of linear equation systems over groups and rings
Motivated by the quest for a logic for PTIME and recent insights that the
descriptive complexity of problems from linear algebra is a crucial aspect of
this problem, we study the solvability of linear equation systems over finite
groups and rings from the viewpoint of logical (inter-)definability. All
problems that we consider are decidable in polynomial time, but not expressible
in fixed-point logic with counting. They also provide natural candidates for a
separation of polynomial time from rank logics, which extend fixed-point logics
by operators for determining the rank of definable matrices and which are
sufficient for solvability problems over fields. Based on the structure theory
of finite rings, we establish logical reductions among various solvability
problems. Our results indicate that all solvability problems for linear
equation systems that separate fixed-point logic with counting from PTIME can
be reduced to solvability over commutative rings. Moreover, we prove closure
properties for classes of queries that reduce to solvability over rings, which
provides normal forms for logics extended with solvability operators. We
conclude by studying the extent to which fixed-point logic with counting can
express problems in linear algebra over finite commutative rings, generalising
known results on the logical definability of linear-algebraic problems over
finite fields
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