4 research outputs found
MAXIMALITY OF LOGIC WITHOUT IDENTITY
Lindström’s theorem obviously fails as a characterization of first-order logic without identity ( L
−
ωω
). In this note, we provide a fix: we show that L
−
ωω
is a maximal abstract logic satisfying a weak form of the isomorphism property (suitable for identity-free languages and studied in [11]), the Löwenheim–Skolem property, and compactness. Furthermore, we show that compactness can be replaced by being recursively enumerable for validity under certain conditions. In the proofs, we use a form of strong upwards Löwenheim–Skolem theorem not available in the framework with identity
Regular Representations of Uniform TC^0
The circuit complexity class DLOGTIME-uniform AC^0 is known to be a modest
subclass of DLOGTIME-uniform TC^0. The weakness of AC^0 is caused by the fact
that AC^0 is not closed under restricting AC^0-computable queries into simple
subsequences of the input. Analogously, in descriptive complexity, the logics
corresponding to DLOGTIME-uniform AC^0 do not have the relativization property
and hence they are not regular. This weakness of DLOGTIME-uniform AC^0 has been
elaborated in the line of research on the Crane Beach Conjecture. The
conjecture (which was refuted by Barrington, Immerman, Lautemann, Schweikardt
and Th{\'e}rien) was that if a language L has a neutral letter, then L can be
defined in first-order logic with the collection of all numerical built-in
relations, if and only if L can be already defined in FO with order.
In the first part of this article we consider logics in the range of AC^0 and
TC^0. First we formulate a combinatorial criterion for a cardinality quantifier
C_S implying that all languages in DLOGTIME-uniform TC^0 can be defined in
FO(C_S). For instance, this criterion is satisfied by C_S if S is the range of
some polynomial with positive integer coefficients of degree at least two. In
the second part of the paper we first adapt the key properties of abstract
logics to accommodate built-in relations. Then we define the regular interior
R-int(L) and regular closure R-cl(L), of a logic L, and show that the Crane
Beach Conjecture can be interpreted as a statement concerning the regular
interior of first-order logic with built-in relations B. We show that if B={+},
or B contains only unary relations besides the order, then R-int(FO_B)
collapses to FO with order. In contrast, our results imply that if B contains
the order and the range of a polynomial of degree at least two, then R-cl(FO_B)
includes all languages in DLOGTIME-uniform TC^0