On the expressive power of monadic least fixed point logic

Schweikardt, Nicole

doi:10.1016/j.tcs.2005.10.025

Cited by 12 publications

(7 citation statements)

References 36 publications

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“…It is known that on arbitrary structures FO(TC 1 ) < FO(LFP 1 ) < MSO (see [13] or [24]) and on trees FO(TC 1 ) < trees FO(LFP 1 ) = trees MSO (see [11] and [28]). It is also known that the (not FO definable) class of finite trees is already definable in FO(TC 1 ) (see for 6 A. GHEERBRANT AND B.…”

Section: Expressive Powermentioning

confidence: 99%

Complete Axiomatizations of Fragments of Monadic Second-Order Logic on Finite Trees

Gheerbrant¹,

Cate²

2012

Logical Methods in Computer Science

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Abstract. We consider a specific class of tree structures that can represent basic structures in linguistics and computer science such as XML documents, parse trees, and treebanks, namely, finite node-labeled sibling-ordered trees. We present axiomatizations of the monadic second-order logic (MSO), monadic transitive closure logic (FO(TC 1 )) and monadic least fixed-point logic (FO(LFP 1 )) theories of this class of structures. These logics can express important properties such as reachability. Using model-theoretic techniques, we show by a uniform argument that these axiomatizations are complete, i.e., each formula that is valid on all finite trees is provable using our axioms. As a backdrop to our positive results, on arbitrary structures, the logics that we study are known to be non-recursively axiomatizable.

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Section: Expressive Powermentioning

confidence: 99%

Complete Axiomatizations of Fragments of Monadic Second-Order Logic on Finite Trees

Gheerbrant¹,

Cate²

2012

Logical Methods in Computer Science

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“…In MSO[+], one can define a multiplication relation (see [35,Lemma 5.4]) and thus quantify over pairs of elements in [0,…”

Section: Bpfo Is Contained In Mso On Additive Structuresmentioning

confidence: 99%

Randomisation and Derandomisation in Descriptive Complexity Theory

Eickmeyer¹,

Grohe²

2011

Logical Methods in Computer Science

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Abstract. We study probabilistic complexity classes and questions of derandomisation from a logical point of view. For each logic L we introduce a new logic BPL, bounded error probabilistic L, which is defined from L in a similar way as the complexity class BPP, bounded error probabilistic polynomial time, is defined from P.Our main focus lies on questions of derandomisation, and we prove that there is a query which is definable in BPFO, the probabilistic version of first-order logic, but not in C ω ∞ω , finite variable infinitary logic with counting. This implies that many of the standard logics of finite model theory, like transitive closure logic and fixed-point logic, both with and without counting, cannot be derandomised. Similarly, we present a query on ordered structures which is definable in BPFO but not in monadic second-order logic, and a query on additive structures which is definable in BPFO but not in FO. The latter of these queries shows that certain uniform variants of AC 0 (bounded-depth polynomial sized circuits) cannot be derandomised. These results are in contrast to the general belief that most standard complexity classes can be derandomised.Finally, we note that BPIFP+C, the probabilistic version of fixed-point logic with counting, captures the complexity class BPP, even on unordered structures.

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