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18
The complexity of theoremproving procedures
 In STOC
, 1971
"... It is shown that any recognition problem solved by a polynomial timebounded nondeterministic Turing machine can be “reduced ” to the problem of determining whether a given propositional formula is a tautology. Here “reduced ” means, roughly speaking, that the first problem can be solved determinist ..."
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Cited by 775 (4 self)
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It is shown that any recognition problem solved by a polynomial timebounded nondeterministic Turing machine can be “reduced ” to the problem of determining whether a given propositional formula is a tautology. Here “reduced ” means, roughly speaking, that the first problem can be solved deterministically in polynomial time provided an oracle is available for solving the second. From this notion of reducible, polynomial degrees of difficulty are defined, and it is shown that the problem of determining tautologyhood has the same polynomial degree as the problem of determining whether the first of two given graphs is isomorphic to a subgraph of the second. Other examples are discussed. A method of measuring the complexity of proof procedures for the predicate calculus is introduced and discussed. Throughout this paper, a set of strings 1 means a set of strings on some fixed, large, finite alphabet Σ. This alphabet is large enough to include symbols for all sets described here. All Turing machines are deterministic recognition devices, unless the contrary is explicitly stated. 1 Tautologies and Polynomial ReReducibility. Let us fix a formalism for the propositional calculus in which formulas are written as strings on Σ. Since we will require infinitely many proposition symbols (atoms), each such symbol will consist of a member of Σ followed by a number in binary notation to distinguish that symbol. Thus a formula of length n can
On the expressive power of database queries with intermediate types
 Journal of Computer and System Sciences
, 1991
"... The setheight of a complex object type is defined to be its level of nesting of the set construct. In a query of the complex object calculus which maps a database D to an output type T,anintermediate type is a type which is used by some variable of the query, but which is not present in D or T.Fore ..."
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Cited by 44 (2 self)
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The setheight of a complex object type is defined to be its level of nesting of the set construct. In a query of the complex object calculus which maps a database D to an output type T,anintermediate type is a type which is used by some variable of the query, but which is not present in D or T.Foreachk, i ≥ 0 we define CALCk,i to be the family of calculus queries mapping from and to types with setheight ≤ k and using intermediate types with setheight ≤ i. In particular, CALC0,0 is the classical relational calculus, and CALC0,1 is equivalent to the family of secondorder (relational) queries. Several results concerning these families of languages are obtained. A primary focus is on the families CALC0,i, which map relations to relations. Upper and lower bounds in terms of hyperexponential time and space on the complexity of these families are provided. The CALC0,i hierarchy does not collapse with respect to expressive power. The union ∪0≤iCALC0,i is exactly the family of elementary queries, i.e., queries with hyperexponential complexity. The expressive power of queries from the complex object calculus interpreted using semantics based on the use of arbitrarily large finite or infinite set of invented values is studied. Under these semantics, the expressive power of the relational calculus is not increased, and the CALC0,i hierarchy collapses at CALC0,1. In general, queries with these semantics may not be computable. We also consider an alternative semantics which yields a family of queries equivalent to the computable queries. 1
TimeSpace Tradeoffs for Satisfiability
 Journal of Computer and System Sciences
, 1997
"... We give the first nontrivial modelindependent timespace tradeoffs for satisfiability. Namely, we show that SAT cannot be solved simultaneously in n 1+o(1) time and n 1\Gammaffl space for any ffl ? 0 on general randomaccess nondeterministic Turing machines. In particular, SAT cannot be solved ..."
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Cited by 29 (1 self)
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We give the first nontrivial modelindependent timespace tradeoffs for satisfiability. Namely, we show that SAT cannot be solved simultaneously in n 1+o(1) time and n 1\Gammaffl space for any ffl ? 0 on general randomaccess nondeterministic Turing machines. In particular, SAT cannot be solved deterministically by a Turing machine using quasilinear time and p n space. We also give lower bounds for logspace uniform NC 1 circuits and branching programs. Our proof uses two basic ideas. First we show that if SAT can be solved nondeterministically with a small amount of time then we can collapse a nonconstant number of levels of the polynomialtime hierarchy. We combine this work with a result of Nepomnjascii that shows that a nondeterministic computation of super linear time and sublinear space can be simulated in alternating linear time. A simple diagonalization yields our main result. We discuss how these bounds lead to a new approach to separating the complexity classes NL a...
Tractable Query Languages for Complex Object Databases
, 1995
"... The expressiveness and complexity of several calculusbased query languages for complex objects is considered. Unlike previous investigations, we are concerned with the complexity of queries on databases of complex objects, rather than flat databases. This raises new issues specific to complex objec ..."
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Cited by 26 (4 self)
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The expressiveness and complexity of several calculusbased query languages for complex objects is considered. Unlike previous investigations, we are concerned with the complexity of queries on databases of complex objects, rather than flat databases. This raises new issues specific to complex objects. For instance, it is shown that the way the database makes use of its higherorder types has direct impact on query complexity. The use of fixpoint operators is shown to yield languages wellbehaved with respect to complexity and expressiveness. In particular, an extension of the fixpoint queries to complex objects is shown to express precisely the PTIME queries, under the assumption that the database makes "full" use of all its types. Similar results involve rangerestricted queries. 1 Introduction Complex objects are increasingly part of advanced database systems. They provide the structural core of objectoriented databases. Several query languages for complex objects have been propo...
The complexity of propositional proofs
 Bulletin of Symbolic Logic
"... Abstract. Propositional proof complexity is the study of the sizes of propositional proofs, and more generally, the resources necessary to certify propositional tautologies. Questions about proof sizes have connections with computational complexity, theories of arithmetic, and satisfiability algorit ..."
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Abstract. Propositional proof complexity is the study of the sizes of propositional proofs, and more generally, the resources necessary to certify propositional tautologies. Questions about proof sizes have connections with computational complexity, theories of arithmetic, and satisfiability algorithms. This is article includes a broad survey of the field, and a technical exposition of some recently developed techniques for proving lower bounds on proof sizes. Contents
FiniteModel Theory  A Personal Perspective
 Theoretical Computer Science
, 1993
"... Finitemodel theory is a study of the logical properties of finite mathematical structures. This paper is a very personalized view of finitemodel theory, where the author focuses on his own personal history, and results and problems of interest to him, especially those springing from work in his Ph ..."
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Cited by 20 (0 self)
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Finitemodel theory is a study of the logical properties of finite mathematical structures. This paper is a very personalized view of finitemodel theory, where the author focuses on his own personal history, and results and problems of interest to him, especially those springing from work in his Ph.D. thesis. Among the topics discussed are:
Bounded Arithmetic and Propositional Proof Complexity
 in Logic of Computation
, 1995
"... This is a survey of basic facts about bounded arithmetic and about the relationships between bounded arithmetic and propositional proof complexity. We introduce the theories S 2 of bounded arithmetic and characterize their proof theoretic strength and their provably total functions in terms of t ..."
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This is a survey of basic facts about bounded arithmetic and about the relationships between bounded arithmetic and propositional proof complexity. We introduce the theories S 2 of bounded arithmetic and characterize their proof theoretic strength and their provably total functions in terms of the polynomial time hierarchy. We discuss other axiomatizations of bounded arithmetic, such as minimization axioms. It is shown that the bounded arithmetic hierarchy collapses if and only if bounded arithmetic proves that the polynomial hierarchy collapses. We discuss Frege and extended Frege proof length, and the two translations from bounded arithmetic proofs into propositional proofs. We present some theorems on bounding the lengths of propositional interpolants in terms of cutfree proof length and in terms of the lengths of resolution refutations. We then define the RazborovRudich notion of natural proofs of P NP and discuss Razborov's theorem that certain fragments of bounded arithmetic cannot prove superpolynomial lower bounds on circuit size, assuming a strong cryptographic conjecture. Finally, a complete presentation of a proof of the theorem of Razborov is given. 1 Review of Computational Complexity 1.1 Feasibility This article will be concerned with various "feasible" forms of computability and of provability. For something to be feasibly computable, it must be computable in practice in the real world, not merely e#ectively computable in the sense of being recursively computable.
Sharply Bounded Alternation within P
, 1996
"... We define the sharply bounded hierarchy, SBH (QL), a hierarchy of classes within P , using quasilineartime computation and quantification over values of length log n. It generalizes the limited nondeterminism hierarchy introduced by Buss and Goldsmith, while retaining the invariance properties. T ..."
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Cited by 5 (3 self)
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We define the sharply bounded hierarchy, SBH (QL), a hierarchy of classes within P , using quasilineartime computation and quantification over values of length log n. It generalizes the limited nondeterminism hierarchy introduced by Buss and Goldsmith, while retaining the invariance properties. The new hierarchy has several alternative characterizations. We define both SBH (QL) and its corresponding hierarchy of function classes, FSBH(QL),and present a variety of problems in these classes, including ql m complete problems for each class in SBH (QL). We discuss the structure of the hierarchy, and show that certain simple structural conditions on it would imply P 6= PSPACE. We present characterizations of SBH (QL) relations based on alternating Turing machines and on firstorder definability, as well as recursiontheoretic characterizations of function classes corresponding to SBH (QL).
From bounded arithmetic to second order arithmetic via automorphisms
 Logic in Tehran, Lect. Notes Log
"... Abstract. In this paper we examine the relationship between automorphisms of models of I∆0 (bounded arithmetic) and strong systems of arithmetic, such as P A, ACA0 (arithmetical comprehension schema with restricted induction), and Z2 (second order arithmetic). For example, we establish the following ..."
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Cited by 5 (1 self)
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Abstract. In this paper we examine the relationship between automorphisms of models of I∆0 (bounded arithmetic) and strong systems of arithmetic, such as P A, ACA0 (arithmetical comprehension schema with restricted induction), and Z2 (second order arithmetic). For example, we establish the following characterization of P A by proving a “reversal ” of a theorem of Gaifman: Theorem. The following are equivalent for completions T of I∆0: (a) T ⊢ P A; (b) Some model M = (M, · · ·) of T has a proper end extension N which satisfies I∆0 and for some automorphism j of N, M is precisely the fixed point set of j. Our results also shed light on the metamathematics of the QuineJensen system NF U of set theory with a universal set. 1.