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Existential Least FixedPoint Logic and its Relatives
 Journal of Logic and Computation
, 1997
"... The main objects of our interest are the existential fragment 9LFP of least xed{point logic, stratied xed point logic SFP, which is the smallest regular logic containing 9LFP, and transitive closure logic TC. The main result of the rst part of this paper is a normal form for 9LFP, which transfers to ..."
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The main objects of our interest are the existential fragment 9LFP of least xed{point logic, stratied xed point logic SFP, which is the smallest regular logic containing 9LFP, and transitive closure logic TC. The main result of the rst part of this paper is a normal form for 9LFP, which transfers to SFP to a certain extent. We study some of the consequences of this normal form and show that TC can be seen as a natural fragment of SFP. The second part of the paper is concerned with separating the logics under consideration. Furthermore, it shows that the existential preservation theorem fails for TC and SFP (both on nite and arbitrary structures) . The method used to show this also yields a negative answer to a question posed by Rosen and Weinstein [RW95] concerning rst{order sentences preserved under extensions of nite structures. 1 Introduction Inductive denitions by positive existential formulae have rst been studied in generalized recursion theory (see e.g. [Acz77]). Chand...
On FixedPoint Logic With Counting
, 1998
"... additional, naturally ordered, domain. Analyzing the proofs of the capturing results for IFP+C mentioned above, we observe that the formulas describing polynomial time problems there work as follows: (1) They first do a fixedpoint computation on the points that is used to retrieve all relevant inf ..."
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additional, naturally ordered, domain. Analyzing the proofs of the capturing results for IFP+C mentioned above, we observe that the formulas describing polynomial time problems there work as follows: (1) They first do a fixedpoint computation on the points that is used to retrieve all relevant information the structure contains. (2) Then this information is carried over to the number domain by a firstorder process (with counting). The result of this process can be seen as a structure on the ordered domain of numbers. (3) On this ordered structure, the original polynomial time problem is now simulated by a fixedpoint formula (without counting). Our main result, Theorem 10, shows that this behavior is not only characteristic for situations where IFP+C captures polynomial time. As a matter of fact, we prove that every IFP+Cformula is equivalent to a formula working in these three steps. As a consequence, we note that mixed relat
Guarded Quantification in Least Fixed Point Logic
, 2002
"... We develop a variant of Least Fixed Point logic based on First Order logic with a relaxed version of guarded quantification. We develop a Game Theoretic Semantics of this logic, and find that under reasonable conditions, guarding quantification does not reduce the expressibility of Least Fixed Point ..."
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We develop a variant of Least Fixed Point logic based on First Order logic with a relaxed version of guarded quantification. We develop a Game Theoretic Semantics of this logic, and find that under reasonable conditions, guarding quantification does not reduce the expressibility of Least Fixed Point logic. But guarding quantification increases worstcase time complexity.