211 research outputs found
The prospects for mathematical logic in the twenty-first century
The four authors present their speculations about the future developments of
mathematical logic in the twenty-first century. The areas of recursion theory,
proof theory and logic for computer science, model theory, and set theory are
discussed independently.Comment: Association for Symbolic Logi
On Symmetric Circuits and Fixed-Point Logics
We study properties of relational structures such as graphs that are decided
by families of Boolean circuits. Circuits that decide such properties are
necessarily invariant to permutations of the elements of the input structures.
We focus on families of circuits that are symmetric, i.e., circuits whose
invariance is witnessed by automorphisms of the circuit induced by the
permutation of the input structure. We show that the expressive power of such
families is closely tied to definability in logic. In particular, we show that
the queries defined on structures by uniform families of symmetric Boolean
circuits with majority gates are exactly those definable in fixed-point logic
with counting. This shows that inexpressibility results in the latter logic
lead to lower bounds against polynomial-size families of symmetric circuits.Comment: 22 pages. Full version of a paper to appear in STACS 201
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