## Many facets of complexity in logic

### BibTeX

@MISC{Kolokolova_manyfacets,

author = {Antonina Kolokolova},

title = {Many facets of complexity in logic},

year = {}

}

### OpenURL

### Abstract

Abstract. There are many ways to define complexity in logic. In finite model theory, it is the complexity of describing properties, whereas in proof complexity it is the complexity of proving properties in a proof system. Here we consider several notions of complexity in logic, the connections among them, and their relationship with computational complexity. In particular, we show how the complexity of logics in the setting of finite model theory is used to obtain results in bounded arithmetic, stating which functions are provably total in certain weak systems of arithmetic. For example, the transitive closure function (testing reachability between two given points in a directed graph) is definable using only NL-concepts (where NL is non-deterministic log-space complexity class), and its totality is provable within NL-reasoning. 1

### Citations

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(Show Context)
Citation Context ...em for the logic is solvable in the complexity class and every problem in the class is representable in the logic.A classical reference on the subject is the book “Descriptive complexity” by Immerman =-=[Imm99]-=-. Definition 1 (Capture by a logic). Let C be a complexity class, L a logic and K a class of finite structures. Then L captures C on K if 1. For every L-sentence φ and every A ∈ K, testing if A |= φ w... |

239 | Nondeterministic space is closed under complementation - Immerman - 1988 |

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(Show Context)
Citation Context ...s line of research is related to other meta-questions in complexity theory such as what proof techniques are needed to prove complexity separations. For example, natural proofs of Razborov and Rudich =-=[RR97]-=- address the question of NP vs. P/poly, motivated by the P vs. NP question. 7 Conclusions In this paper we touch upon one example of a connection between two different notions of hardness: the hardnes... |

157 |
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(Show Context)
Citation Context ...ystem of arithmetic captures a function class if it proves totality of all and only functions in this class. Traditionally, functions are introduced by their recursion-theoretic characterization (see =-=[Cob65]-=- for the original such result or [Zam96]). For example, Cobham’s characterization of P uses limited recursion on notation: F (0, ¯x, ¯ Y ) = G(¯x, ¯ Y ) (1) F (z + 1, ¯x, ¯ Y ) = cut(p(z, ¯x, ¯ Y ), H... |

145 | Undirected st-connectivity in log-space
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(Show Context)
Citation Context ...class? We were able to prove the basic properties of P and NL with reasoning no more complex then the class itself. However, for classes such as symmetric logspace, now proven to be equal to logspace =-=[Rei04]-=-, it is not clear whether the proof of complementation (or the proof of equivalence with logspace) are formalizable using only reasoning within this class. This line of research is related to other me... |

123 |
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(Show Context)
Citation Context ...l-time reasoning [Coo75]. P V is an equational system with a function for every polynomial-time computable function.The major development in bounded arithmetic came in the 1985 PhD thesis of S. Buss =-=[Bus86]-=-. There, several (classes of) systems of bounded arithmetic were described, capturing major complexity classes such as P and EXP (viewed as classes of functions). The best known system is S1 2, which ... |

119 |
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(Show Context)
Citation Context ...ower bounds for the Parity Principle, which implied lower bounds for the complexity class AC 0 . A different approach was used by Cook: in 1975 he presented a system P V for polynomial-time reasoning =-=[Coo75]-=-. P V is an equational system with a function for every polynomial-time computable function.The major development in bounded arithmetic came in the 1985 PhD thesis of S. Buss [Bus86]. There, several ... |

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(Show Context)
Citation Context ... of weak systems of arithmetic concentrated on restricted fragments of Peano Arithmetic, e.g., I∆0 in which induction is over bounded first-order formulae [Par71]. This system, I∆0, was used by Ajtai =-=[Ajt83]-=- to obtain lower bounds for the Parity Principle, which implied lower bounds for the complexity class AC 0 . A different approach was used by Cook: in 1975 he presented a system P V for polynomial-tim... |

61 | Notes on polynomially bounded arithmetic
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(Show Context)
Citation Context ...class if it proves totality of all and only functions in this class. Traditionally, functions are introduced by their recursion-theoretic characterization (see [Cob65] for the original such result or =-=[Zam96]-=-). For example, Cobham’s characterization of P uses limited recursion on notation: F (0, ¯x, ¯ Y ) = G(¯x, ¯ Y ) (1) F (z + 1, ¯x, ¯ Y ) = cut(p(z, ¯x, ¯ Y ), H(z, ¯x, ¯ Y , F (z, ¯x, ¯ Y ))). (2) Her... |

44 |
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(Show Context)
Citation Context ... – SO∃-Krom: ψ is a CNF with no more than two Pi literals per clause. In particular, the formula for Parity in the example above is both a secondorder Horn and a second-order Krom formula. Theorem 1 (=-=[Grä92]-=-). Over structures with successor, SO∃-Horn and SO∃-Krom capture complexity classes P and NL, respectively. 4 Bounded arithmetic Just like in complexity classes P and NP the computation length is boun... |

33 | Theories for complexity classes and their propositional translations
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(Show Context)
Citation Context ...re operator to first-order logic, and P is captured by first-order logic together with a least fixed point operator [Imm83,Imm82,Var82]. Here we will concentrate on the second approach, due to Grädel =-=[Grä91]-=-. Definition 2. Restricted second-order formulae are of the form ∃P1 . . . Pk∀x1 . . . xlψ( ¯ P , ¯x, ā, ¯ Y ), where ψ is quantifier-free. Two important types of restricted second-order formulae are ... |

31 |
RSUV isomorphisms
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(Show Context)
Citation Context ...finite structures. In this new setting many of the standard techniques of model theory, most notably compactness, do not apply. Since testing if a firstorder formula has a finite model is undecidable =-=[Tra50]-=-, our focus will be on the complexity of model checking: given a finite structure and formula of some logic, decide if this structure is a model of this formula. Considering both the formula and the s... |

19 | Constraint propagation as a proof system
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(Show Context)
Citation Context ...del theory to obtain resolution proof system lower bounds [Ats02], and, in his later work with Kolaitis and Vardi, uses constraint satisfaction problems as a generic basis for a class of proof system =-=[AKV04]-=-. Complexity in logic is a broad area of research, with many problems still unsolved. Little is known about connections among different settings and notions of hardness. Here we have given one such ex... |

17 | A second-order system for polytime reasoning based on Grädel’s theorem - Cook, Kolokolova |

10 | Relational queries computable in polytime - Immerman - 1982 |

7 | Systems of bounded arithmetic from descriptive complexity - Kolokolova - 2004 |

5 | A second-order system for polynomial-time reasoning based on Grädel’s theorem - Cook, Kolokolova - 2001 |

3 | Fixed-point logics, descriptive complexity and random satisfiability
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(Show Context)
Citation Context ...ne of research exploring connections between finite model theory and proof complexity is pursued by Atserias. He uses his results in finite model theory to obtain resolution proof system lower bounds =-=[Ats02]-=-, and, in his later work with Kolaitis and Vardi, uses constraint satisfaction problems as a generic basis for a class of proof system [AKV04]. Complexity in logic is a broad area of research, with ma... |