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Languages That Capture Complexity Classes
 SIAM Journal of Computing
, 1987
"... this paper a series of languages adequate for expressing exactly those properties checkable in a series of computational complexity classes. For example, we show that a property of graphs (respectively groups, binary strings, etc.) is in polynomial time if and only if it is expressible in the first ..."
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Cited by 230 (21 self)
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this paper a series of languages adequate for expressing exactly those properties checkable in a series of computational complexity classes. For example, we show that a property of graphs (respectively groups, binary strings, etc.) is in polynomial time if and only if it is expressible in the first order language of graphs (respectively groups, binary strings, etc.) together with a least fixed point operator. As another example, a property is in logspace if and only if it is expressible in first order logic together with a deterministic transitive closure operator. The roots of our approach to complexity theory go back to 1974 when Fagin showed that the NP properties are exactly those expressible in second order existential sentences. It follows that second order logic expresses exactly those properties which are in the polynomial time hierarchy. We show that adding suitable transitive closure operators to second order logic results in languages capturing polynomial space and exponential time, respectively. The existence of such natural languages for each important complexity class sheds a new light on complexity theory. These languages reaffirm the importance of the complexity classes as much more than machine dependent issues. Furthermore a whole new approach is suggested. Upper bounds (algorithms) can be produced by expressing the property of interest in one of our languages. Lower bounds may be demonstrated by showing that such expression is impossible.
Languages which capture complexity classes
 SIAM J. on Computing
, 1987
"... We present in this paper a series of languages adequate for expressing exactly those properties checkable in a series of computational complexity classes. For example, we show that a graph property is in polynomial time if and only if it is ..."
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Cited by 52 (5 self)
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We present in this paper a series of languages adequate for expressing exactly those properties checkable in a series of computational complexity classes. For example, we show that a graph property is in polynomial time if and only if it is
Descriptive and Computational Complexity
 COMPUTATIONAL COMPLEXITY THEORY, PROC. SYMP. APPLIED MATH
, 1989
"... Computational complexity began with the natural physical notions of time and space. Given a property, S, an important issue is the computational complexity of checking whether or not an input satisfies S. For a long time, the notion of complexity referred to the time or space used in the computatio ..."
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Cited by 47 (0 self)
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Computational complexity began with the natural physical notions of time and space. Given a property, S, an important issue is the computational complexity of checking whether or not an input satisfies S. For a long time, the notion of complexity referred to the time or space used in the computation. A mathematician might ask, "What is the complexity of expressing the property S?" It should not be surprising that these two questions  that of checking and that of expressing  are related. However it is startling how closely tied they are when the second question refers to expressing the property in firstorder logic. Many complexity classes originally defined in terms of time or space resources have precise definitions as classes in firstorder logic. In 1974 Fagin gave a characterization of nondeterministic polynomial time (NP) as the set of properties expressible in secondorder existential logic
Fixpoint Logics, Relational Machines, and Computational Complexity
 In Structure and Complexity
, 1993
"... We establish a general connection between fixpoint logic and complexity. On one side, we have fixpoint logic, parameterized by the choices of 1storder operators (inflationary or noninflationary) and iteration constructs (deterministic, nondeterministic, or alternating). On the other side, we have t ..."
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Cited by 36 (5 self)
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We establish a general connection between fixpoint logic and complexity. On one side, we have fixpoint logic, parameterized by the choices of 1storder operators (inflationary or noninflationary) and iteration constructs (deterministic, nondeterministic, or alternating). On the other side, we have the complexity classes between P and EXPTIME. Our parameterized fixpoint logics capture the complexity classes P, NP, PSPACE, and EXPTIME, but equality is achieved only over ordered structures. There is, however, an inherent mismatch between complexity and logic  while computational devices work on encodings of problems, logic is applied directly to the underlying mathematical structures. To overcome this mismatch, we develop a theory of relational complexity, which bridges tha gap between standard complexity and fixpoint logic. On one hand, we show that questions about containments among standard complexity classes can be translated to questions about containments among relational complex...
Weighted NP Optimization Problems: Logical Definability and Approximation Properties
 In Proceedings of the 10th IEEE Conference on Structure in Complexity Theory
, 1995
"... . Extending a wellknown property of NP optimization problems in which the value of the optimum is guaranteed to be polynomially bounded in the length of the input, it is observed that, by attaching weights to tuples over the domain of the input, all NP optimization problems admit a logical characte ..."
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Cited by 5 (0 self)
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. Extending a wellknown property of NP optimization problems in which the value of the optimum is guaranteed to be polynomially bounded in the length of the input, it is observed that, by attaching weights to tuples over the domain of the input, all NP optimization problems admit a logical characterization. It is shown that any NP optimization problem can be stated as a problem in which the constraint conditions can be expressed by a \Pi 2 firstorder formula. The paper analyzes the weighted analogue of all syntactically defined classes of optimization problems that are known to have good approximation properties in the nonweighted case. Dramatic changes occur when negative weights are allowed. KeywordsNP optimization problems, logical definability, approximation properties, multiprover interactive systems. AMS Classification 68Q15, 68Q25. 1 Introduction Recent years have seen considerable progress in the understanding of the approximation properties of NP optimization problems. Tw...
Computing on Structures
"... this paper various devices operating directly on structures, without encoding. The motivation and benefits for doing this are manyfold. On a fundamental level, encodings of structures seem to be a technical device rather than an intrinsic feature. This point has already been made by several mathemat ..."
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Cited by 3 (1 self)
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this paper various devices operating directly on structures, without encoding. The motivation and benefits for doing this are manyfold. On a fundamental level, encodings of structures seem to be a technical device rather than an intrinsic feature. This point has already been made by several mathematicians such as Tarski [Tar86], and Harvey Friedman [Fri71] (see Section 5). It has come up more recently in the context of databases, where devices computing on structures model more acurately database computation carried out against an abstract interface hiding the internal representation of data. Thus, the primary benefit of studying devices and languages computing on structures is that they clarify issues which are obscured in classical devices such as Turing machines. For example, they yield new notions of complexity, quite different from classical computational complexity. They reflect more acurately the actual complexity of computation, which, like database computation, cannot take advantage of encodings of structures. An example is provided by the query even on a set
On Bijections vs. Unary Functions
 In Proc. 13th Symposium on Theoretical Aspects of Computer ScienceSTACS 96
, 1996
"... . A set of finite structures is in Binary NP if it can be characterized by existential second order formulas in which second order quantification is over relations of arity 2. In [DLS95] subclasses of Binary NP were considered, in which the second order quantifiers range only over certain classes of ..."
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Cited by 2 (1 self)
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. A set of finite structures is in Binary NP if it can be characterized by existential second order formulas in which second order quantification is over relations of arity 2. In [DLS95] subclasses of Binary NP were considered, in which the second order quantifiers range only over certain classes of relations. It was shown that many of these subclasses coincide and that all of them can be ordered in a threelevel linear hierarchy, the levels of which are represented by bijections, successor relations and unary functions respectively. In this paper it is shown that  Graph Connectivity is expressible by bijections, thereby showing that the two lower levels of the hierarchy coincide;  the set of graphs with exactly as many vertices as arcs is expressible by unary functions but not by bijections. This shows that level 3 is strictly stronger than the other two levels. 1 Introduction Fagin [Fag74] showed that NP coincides with the class of sets of finite structures that are characteri...