Abstract
The standard translation of existential fixed-point formulas into second-order logic produces strict universal formulas, that is, formulas consisting of universal quantifiers on relations (not functions) followed by an existential first-order formula. This form implies many of the pleasant properties of existential fixed-point logic, but not all. In particular, strict universal sentences can express some co-NP-complete properties of structures, whereas properties expressible by existential fixed-point formulas are always in P. We therefore investigate what additional syntactic properties, beyond strict universality, are enjoyed by the second-order translations of existential fixed-point formulas. In particular, do such syntactic properties account for polynomial-time model-checking?
To Yuri Gurevich, on the occasion of his \( 75\!{th}\) birthday.
Partially supported by NSF grant DMS-0653696
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Notes
- 1.
My talk at Yuri Gurevich’s 70th birthday conference in Brno contained much of the present paper’s material, but I had overlooked what I now call the conjunction problem in Sect. 3. The solution of that problem given here in Sect. 4 is new. This paper is, except for preliminary material, disjoint from my written contribution [2] to Yuri’s 70th birthday celebration.
- 2.
The terminology “strict \(\forall ^1_1\)” was chosen in analogy with “strict \(\varPi ^1_1\)” in [1, Sect. 8.2]. The difference is that “strict \(\varPi ^1_1\)” is used in a set-theoretic context and allows not only existential quantifiers but also bounded universal quantifiers \((\forall x\in y)\) in the first-order part of the formula.
- 3.
The expected duality between validity and satisfiability is not available for logics, like \(\exists \)FPL, that are not closed under negation.
- 4.
We are dealing here with what is often called data complexity of the model-checking problem. That is, we regard the “data” \(\mathfrak {A}\) as the input, and we measure resource usage relative to the size of \(\mathfrak {A}\), while the “query” \(\varphi \) is held fixed.
- 5.
To avoid excessive notation, we use the same symbols for these relations as for the corresponding symbols in our strict \(\exists ^1_1\) sentence.
References
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Acknowledgement
Because of the last-minute discovery of the conjunction problem, this paper was submitted after the official deadline, leaving less than the normal time for refereeing. Nevertheless, the referee provided a very useful report. I thank him or her for the report, in particular for informing me about the existence and relevance of [4].
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Blass, A. (2015). Existential Fixed-Point Logic as a Fragment of Second-Order Logic. In: Beklemishev, L., Blass, A., Dershowitz, N., Finkbeiner, B., Schulte, W. (eds) Fields of Logic and Computation II. Lecture Notes in Computer Science(), vol 9300. Springer, Cham. https://doi.org/10.1007/978-3-319-23534-9_3
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