We establish a general hierarchy theorem for quantifier classes in the infinitary logic L∞ωωon finite structures. In particular, it is shown that no infinitary formula with bounded number of universal quantifiers can express the negation of a transitive closure.This implies the solution of several open problems in finite model theory: On finite structures, positive transitive closure logic is not closed under negation. More generally the hierarchy defined by interleaving negation and transitive closure operators is strict. This proves a conjecture of (...) Immerman.We also separate the expressive power of several extensions of Datalog, giving new insight in the fine structure of stratified Datalog. (shrink)
When a structure or class of structures admits an unbounded induction, we can do arithmetic on the stages of that induction: if only bounded inductions are admitted, then clearly each inductively definable relation can be defined using a finite explicit expression. Is the converse true? We examine evidence that the converse is true, in positive elementary induction . We present a stronger conjecture involving the language L consisting of all L∞ω formulas with a finite number of variables, and examine a (...) combinatorial property equivalent to “all L-definable relations are elementary”. (shrink)
We develop a variant of Least Fixed Point logic based on First Orderlogic with a relaxed version of guarded quantification. We develop aGame Theoretic Semantics of this logic, and find that under reasonableconditions, guarding quantification does not reduce the expressibilityof Least Fixed Point logic. But we also find that the guarded version ofa least fixed point algorithm may have a greater time complexity thanthe unguarded version, by a linear factor.
“Logic” entails both a toolkit for dealing with situations requiring precision, and a prescription for a type of public reasoning. A sufficiently extended society facing a stream of genuinely novel opportunities and challenges will benefit from an ability to generate and encourage the use of such reasoning systems to deal with these opportunities and challenges. The study of “logic” is the result of using the toolkit on itself, which would appear to be a necessary and not unnatural step for a (...) society developing sufficient familiarity with the toolkit. Many societies have developed something like logic, and past and present use of logic-like toolkits in learning situations and transmission of skills suggests that many societies will develop something like logic. (shrink)
It is known that for two given countable sets of unary relations A and B on ω there exists an infinite set H ⫅ ω on which A and B are the same. This result can be used to generate counterexamples in expressibility theory. We examine the sharpness of this result.
We present game-theoretic characterizations of the complexity/expressibility measures “dimension” and “the number of variables” as Least Fixed Point queries. As an example, we use these characterizations to compute the dimension and number of variables of Connectivity and Connectivity.