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117
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
A New RecursionTheoretic Characterization Of The Polytime Functions
 COMPUTATIONAL COMPLEXITY
, 1992
"... We give a recursiontheoretic characterization of FP which describes polynomial time computation independently of any externally imposed resource bounds. In particular, this syntactic characterization avoids the explicit size bounds on recursion (and the initial function 2 xy ) of Cobham. ..."
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Cited by 179 (7 self)
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We give a recursiontheoretic characterization of FP which describes polynomial time computation independently of any externally imposed resource bounds. In particular, this syntactic characterization avoids the explicit size bounds on recursion (and the initial function 2 xy ) of Cobham.
Functional interpretations of feasibly constructive arithmetic
 Annals of Pure and Applied Logic
, 1993
"... i ..."
Notes on Polynomially Bounded Arithmetic
"... We characterize the collapse of Buss' bounded arithmetic in terms of the provable collapse of the polynomial time hierarchy. We include also some general modeltheoretical investigations on fragments of bounded arithmetic. Contents 0 Introduction and motivation. 1 1 Preliminaries. 3 1.1 The polyno ..."
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Cited by 58 (1 self)
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We characterize the collapse of Buss' bounded arithmetic in terms of the provable collapse of the polynomial time hierarchy. We include also some general modeltheoretical investigations on fragments of bounded arithmetic. Contents 0 Introduction and motivation. 1 1 Preliminaries. 3 1.1 The polynomially bounded hierarchy. : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 4 1.2 The axioms of secondorder bounded arithmetic. : : : : : : : : : : : : : : : : : : : : : : : : : : : 5 1.3 Rudimentary functions. : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 5 1.4 Other fragments. : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 6 1.5 Polynomial time computable functions. : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 7 1.6 Relations among fragments. : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 8 1.7 Relations with Buss' bounded arithmetic. : : : :...
Algebras of feasible functions
 in "Proc. 24th Annual IEEE Sympos. Found. Comput. Sci
, 1983
"... What happens if we interpret the syntax of primitive recursive functions in finite domains rather than in the (Platonic) realm of all natural numbers? The answer is somewhat surprising: primitive recursiveness coincides with LOGSPACE computability. Analogously, recursiveness coincides with PTIME com ..."
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Cited by 50 (5 self)
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What happens if we interpret the syntax of primitive recursive functions in finite domains rather than in the (Platonic) realm of all natural numbers? The answer is somewhat surprising: primitive recursiveness coincides with LOGSPACE computability. Analogously, recursiveness coincides with PTIME computability on finite domains (cf. [Sa]). Inductive definitions for some other complexity classes are discussed too.
The History and Status of the P versus NP Question
, 1992
"... this article, I have attempted to organize and describe this literature, including an occasional opinion about the most fruitful directions, but no technical details. In the first half of this century, work on the power of formal systems led to the formalization of the notion of algorithm and the re ..."
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Cited by 50 (0 self)
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this article, I have attempted to organize and describe this literature, including an occasional opinion about the most fruitful directions, but no technical details. In the first half of this century, work on the power of formal systems led to the formalization of the notion of algorithm and the realization that certain problems are algorithmically unsolvable. At around this time, forerunners of the programmable computing machine were beginning to appear. As mathematicians contemplated the practical capabilities and limitations of such devices, computational complexity theory emerged from the theory of algorithmic unsolvability. Early on, a particular type of computational task became evident, where one is seeking an object which lies
Predicative Recursion and Computational Complexity
, 1992
"... The purpose of this thesis is to give a "foundational" characterization of some common complexity classes. Such a characterization is distinguished by the fact that no explicit resource bounds are used. For example, we characterize the polynomial time computable functions without making any direct r ..."
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Cited by 45 (3 self)
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The purpose of this thesis is to give a "foundational" characterization of some common complexity classes. Such a characterization is distinguished by the fact that no explicit resource bounds are used. For example, we characterize the polynomial time computable functions without making any direct reference to polynomials, time, or even computation. Complexity classes characterized in this way include polynomial time, the functional polytime hierarchy, the logspace decidable problems, and NC. After developing these "resource free" definitions, we apply them to redeveloping the feasible logical system of Cook and Urquhart, and show how this firstorder system relates to the secondorder system of Leivant. The connection is an interesting one since the systems were defined independently and have what appear to be very different rules for the principle of induction. Furthermore it is interesting to see, albeit in a very specific context, how to retract a second order statement, ("inducti...
A New Characterization Of Type 2 Feasibility
, 1996
"... . K. Mehlhorn introduced a class of polynomial time computable operators in order to study poly time reducibilities between functions. This class is defined using a generalization of A. Cobham's definition of feasibility for type 1 functions to type 2 functionals. Cobham's feasible functions are equ ..."
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Cited by 36 (6 self)
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. K. Mehlhorn introduced a class of polynomial time computable operators in order to study poly time reducibilities between functions. This class is defined using a generalization of A. Cobham's definition of feasibility for type 1 functions to type 2 functionals. Cobham's feasible functions are equivalent to the familiar poly time functions. We generalize this equivalence to type 2 functionals. This requires a definition of the notion `poly time in the length of type 1 inputs'. The proof of this equivalence is not a simple generalization of the proof for type 1 functions; it depends on the fact that Mehlhorn's class is closed under a strong form of simultaneous limited recursion on notation, and requires an analysis of the structure of oracle queries in time bounded computations. Key words. type 2 computability, polynomial time, notational recursion, oracle Turing machine AMS subject classifications. 68Q05,68Q15,03D65,03D20 1. Introduction. A type 1 function is a mapping from N to ...
Resource Control for Synchronous Cooperative Threads
 In CONCUR, volume 3170 of LNCS
, 2004
"... We develop new methods to statically bound the resources needed for the execution of systems of concurrent, interactive threads. ..."
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Cited by 34 (4 self)
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We develop new methods to statically bound the resources needed for the execution of systems of concurrent, interactive threads.