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On the Parameterised Intractability of Monadic Second-Order Logic
"... Abstract. One of Courcelle’s celebrated results states that if C is a class of graphs of bounded tree-width, then model-checking for monadic second order logic (MSO2) is fixed-parameter tractable (fpt) on C by linear time parameterised algorithms. An immediate question is whether this is best possib ..."
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Abstract. One of Courcelle’s celebrated results states that if C is a class of graphs of bounded tree-width, then model-checking for monadic second order logic (MSO2) is fixed-parameter tractable (fpt) on C by linear time parameterised algorithms. An immediate question is whether this is best possible or whether the result can be extended to classes of unbounded tree-width. In this paper we show that in terms of tree-width, the theorem can not be extended much further. More specifically, we show that if C is a class of graphs which is closed under colourings and satisfies certain constructibility conditions such that the tree-width of C is not bounded by log 16 n then MSO2-model checking is not fpt unless SAT can be solved in sub-exponential time. If the tree-width of C is not poly-log. bounded, then MSO2-model checking is not fpt unless all problems in the polynomial-time hierarchy can be solved in sub-exponential time. 1
Fixed-Point Definability and Polynomial Time
"... Abstract. My talk will be a survey of recent results about the quest for a logic capturing polynomial time. In a fundamental study of database query languages, Chandra and Harel [4] first raised the question of whether there exists a logic that captures polynomial time. Actually, Chandra and Harel p ..."
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Abstract. My talk will be a survey of recent results about the quest for a logic capturing polynomial time. In a fundamental study of database query languages, Chandra and Harel [4] first raised the question of whether there exists a logic that captures polynomial time. Actually, Chandra and Harel phrased the question in a somewhat disguised form; the version that we use today goes back to Gurevich [15]. Briefly, but slightly imprecisely, 1 a logic L captures a complexity class K if exactly those properties of finite structures that are decidable in K are definable in L. The existence of a logic capturing PTIME is still wide open, and it is viewed as one of the main open problems in finite model theory and database theory. One reason the question is interesting is that we know from Fagin’s Theorem [9] that existential second-order logic captures NP, and we also know that there are logics capturing most natural complexity classes above NP. Gurevich conjectured that there is no logic capturing PTIME. If this conjecture was true, this would not only imply that PTIME ̸ = NP, but it would also show that NP and the complexity
On the Complexity of Gödel’s Proof Predicate
, 2009
"... The undecidability of first-order logic implies that there is no computable bound on the length of shortest proofs of valid sentences of first-order logic. Some valid sentences can only have quite long proofs. How hard is it to prove such “hard ” valid sentences? The polynomial time tractability of ..."
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The undecidability of first-order logic implies that there is no computable bound on the length of shortest proofs of valid sentences of first-order logic. Some valid sentences can only have quite long proofs. How hard is it to prove such “hard ” valid sentences? The polynomial time tractability of this problem would imply the fixed-parameter tractability of the parameterized problem that, given a natu-ral number n in unary as input and a first-order sentence ϕ as parameter, asks whether ϕ has a proof of length ≤ n. As the underlying classical problem has been considered by Gödel we denote this problem by p-GÖDEL. We show that p-GÖDEL is not fixed-parameter tractable if DTIME(hO(1)) 6 = NTIME(hO(1)) for all time constructible and increasing functions h. Moreover we analyze the complexity of the con-struction problem associated with p-GÖDEL. 1.
On optimal proof systems and logics for PTIME
"... We prove that TAUT has a p-optimal proof system if and only if a logic related to least fixed-point logic captures polynomial time on all finite structures. Furthermore, we show that TAUT has no effec-tive p-optimal proof system if NTIME(hO(1)) 6 ⊆ DTIME(hO(log h)) for every time constructible and i ..."
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We prove that TAUT has a p-optimal proof system if and only if a logic related to least fixed-point logic captures polynomial time on all finite structures. Furthermore, we show that TAUT has no effec-tive p-optimal proof system if NTIME(hO(1)) 6 ⊆ DTIME(hO(log h)) for every time constructible and increasing function h. 1.
