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Computing With FirstOrder Logic
, 1995
"... We study two important extensions of firstorder logic (FO) with iteration, the fixpoint and while queries. The main result of the paper concerns the open problem of the relationship between fixpoint and while: they are the same iff ptime = pspace. These and other expressibility results are obtaine ..."
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Cited by 54 (13 self)
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We study two important extensions of firstorder logic (FO) with iteration, the fixpoint and while queries. The main result of the paper concerns the open problem of the relationship between fixpoint and while: they are the same iff ptime = pspace. These and other expressibility results are obtained using a powerful normal form for while which shows that each while computation over an unordered domain can be reduced to a while computation over an ordered domain via a fixpoint query. The fixpoint query computes an equivalence relation on tuples which is a congruence with respect to the rest of the computation. The same technique is used to show that equivalence of tuples and structures with respect to FO formulas with bounded number of variables is definable in fixpoint. Generalizing fixpoint and while, we consider more powerful languages which model arbitrary computation interacting with a database using a finite set of FO queries. Such computation is modeled by a relational machine...
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 39 (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...
Computing with Infinitary Logic
 Theoretical Computer Science
, 1995
"... Most recursive extensions of relational calculus converge around two central classes of queries: fixpoint and while. Infinitary logic (with finitely many variables) is a very powerful extension of these languages which provides an elegant unifying formalism for a wide variety of query languages. ..."
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Cited by 9 (6 self)
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Most recursive extensions of relational calculus converge around two central classes of queries: fixpoint and while. Infinitary logic (with finitely many variables) is a very powerful extension of these languages which provides an elegant unifying formalism for a wide variety of query languages. However, neither the syntax nor the semantics of infinitary logic are effective, and its connection to practical query languages has been largely unexplored. We relate infinitary logic to another powerful extension of fixpoint and while, called relational machine, which highlights the computational style of these languages. Relational machines capture the kind of computation occurring when a query language is embedded in a host programming language, as in C+SQL. The main result of this paper is that relational machines correspond to the natural effective fragment of infinitary logic. Other wellknown query languages are related to infinitary logic using syntactic restrictions formula...
The Power of Reflective Relational Machines
 IN PROC. IEEE SYMP. ON LOGIC IN COMPUTER SCIENCE
, 1994
"... A model of database programming with reflection, called reflective relational machine, is introduced and studied. The reflection consists here of dynamic generation of queries in a host programming language. The main results characterize the power of the machine in terms of known complexity classes. ..."
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Cited by 6 (0 self)
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A model of database programming with reflection, called reflective relational machine, is introduced and studied. The reflection consists here of dynamic generation of queries in a host programming language. The main results characterize the power of the machine in terms of known complexity classes. In particular, the polynomialtime restriction of the machine is shown to express PSPACE, and to correspond precisely to uniform circuits of polynomial depth and exponential size. This provides an alternative, logicbased formulation of the uniform circuit model, more convenient for problems naturally formulated in logic terms. Since time in the polynomiallybounded machine coincides with time in the uniform circuit model, this also shows that reflection allows for more "intense" parallelism, which is not attainable otherwise (unless P = PSPACE). Other results concern the power of the reflective relational machine subject to restrictions on the number of variables used.
Databases and FiniteModel Theory
 IN DESCRIPTIVE COMPLEXITY AND FINITE MODELS
, 1997
"... Databases provide one of the main concrete scenarios for finitemodel theory within computer science. This paper presents an informal overview of database theory aimed at finitemodel theorists, emphasizing the specificity of the database area. It is argued that the area of databases is a rich sourc ..."
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Cited by 5 (1 self)
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Databases provide one of the main concrete scenarios for finitemodel theory within computer science. This paper presents an informal overview of database theory aimed at finitemodel theorists, emphasizing the specificity of the database area. It is argued that the area of databases is a rich source of questions and vitality for finitemodel theory.
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
How hard is positive quantification
 In preparation
, 2014
"... Abstract. We show that the constructive predicate logic with positive (covariant) quantification is hard for doubly exponential universal time, i.e., for the class co2Nexptime. Our approach is to represent proofsearch as computation of an alternating automaton. The memory of the automaton is stru ..."
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Cited by 1 (1 self)
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Abstract. We show that the constructive predicate logic with positive (covariant) quantification is hard for doubly exponential universal time, i.e., for the class co2Nexptime. Our approach is to represent proofsearch as computation of an alternating automaton. The memory of the automaton is structured in a way that strictly corresponds to scopes of the binders used in the constructed proof. This provides an application of automatatheoretic techniques in proof theory. 1
Restricted Positive Quantification Is Not Elementary∗
"... We show that a restricted variant of constructive predicate logic with positive (covariant) quantification is of superelementary complexity. The restriction is to limit the number of eigenvariables used in quantifier introductions rules to a reasonably usable level. This construction suggests that ..."
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We show that a restricted variant of constructive predicate logic with positive (covariant) quantification is of superelementary complexity. The restriction is to limit the number of eigenvariables used in quantifier introductions rules to a reasonably usable level. This construction suggests that the known nonelementary decision algorithms for positive logic may actually be best possible.