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14
Infinitary Logic and Inductive Definability over Finite Structures
 Information and Computation
, 1995
"... The extensions of firstorder logic with a least fixed point operator (FO + LFP) and with a partial fixed point operator (FO + PFP) are known to capture the complexity classes P and PSPACE respectively in the presence of an ordering relation over finite structures. Recently, Abiteboul and Vianu [Abi ..."
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Cited by 56 (6 self)
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The extensions of firstorder logic with a least fixed point operator (FO + LFP) and with a partial fixed point operator (FO + PFP) are known to capture the complexity classes P and PSPACE respectively in the presence of an ordering relation over finite structures. Recently, Abiteboul and Vianu [Abiteboul and Vianu, 1991b] investigated the relationship of these two logics in the absence of an ordering, using a machine model of generic computation. In particular, they showed that the two languages have equivalent expressive power if and only if P = PSPACE. These languages can also be seen as fragments of an infinitary logic where each formula has a bounded number of variables, L ! 1! (see, for instance, [Kolaitis and Vardi, 1990]). We investigate this logic of finite structures and provide a normal form for it. We also present a treatment of the results in [Abiteboul and Vianu, 1991b] from this point of view. In particular, we show that we can write a formula of FO + LFP that defines ...
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...
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:
Higher Order Logic
 In Handbook of Logic in Artificial Intelligence and Logic Programming
, 1994
"... Contents 1 Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : : 2 2 The expressive power of second order Logic : : : : : : : : : : : 3 2.1 The language of second order logic : : : : : : : : : : : : : 3 2.2 Expressing size : : : : : : : : : : : : : : : : : : : : : : : : 4 2.3 Definin ..."
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Cited by 18 (0 self)
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Contents 1 Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : : 2 2 The expressive power of second order Logic : : : : : : : : : : : 3 2.1 The language of second order logic : : : : : : : : : : : : : 3 2.2 Expressing size : : : : : : : : : : : : : : : : : : : : : : : : 4 2.3 Defining data types : : : : : : : : : : : : : : : : : : : : : 6 2.4 Describing processes : : : : : : : : : : : : : : : : : : : : : 8 2.5 Expressing convergence using second order validity : : : : : : : : : : : : : : : : : : : : : : : : : 9 2.6 Truth definitions: the analytical hierarchy : : : : : : : : 10 2.7 Inductive definitions : : : : : : : : : : : : : : : : : : : : : 13 3 Canonical semantics of higher order logic : : : : : : : : : : : : 15 3.1 Tarskian semantics of second order logic : : : : : : : : : 15 3.2 Function and re
Queries Are Easier Than You Thought (probably)
, 1992
"... The optimization of a large class of queries is explored, using a powerful normal form recently proven. The queries include the fixpoint and while queries, and an extension of while with arithmetic. The optimization method is evaluated using a probabilistic analysis. In particular, the average compl ..."
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Cited by 11 (5 self)
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The optimization of a large class of queries is explored, using a powerful normal form recently proven. The queries include the fixpoint and while queries, and an extension of while with arithmetic. The optimization method is evaluated using a probabilistic analysis. In particular, the average complexity of fixpoint and while is considered and some surprising results are obtained. They suggest that the worstcase complexity is sometimes overly pessimistic for such queries, whose average complexity is often much more reasonable than the provably rare worst case. Some computational properties of queries are also investigated. A probabilistic notion of boundedness is defined, and it is shown that all programs in the class considered are bounded almost everywhere. An effective way of using this fact is provided. 1 Introduction The complexity of query languages has traditionally been investigated using worstcase bounds. We argue that this approach provides an overly pessimistic picture o...
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...
Asymptotic Probabilities of Languages with Generalized Quantifiers
 In Proc. IEEE Symp. of Logic in Computer Science
, 1994
"... We study the impact of adding certain families of generalized quantifiers to firstorder logic (FO) on the asymptotic behavior of sentences. All our results are stated and proved for languages disallowing free variables in the scope of generalized quantifiers. For a class K of finite structures clos ..."
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Cited by 8 (2 self)
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We study the impact of adding certain families of generalized quantifiers to firstorder logic (FO) on the asymptotic behavior of sentences. All our results are stated and proved for languages disallowing free variables in the scope of generalized quantifiers. For a class K of finite structures closed under isomorphism, the quantifier QK is said to be strongly monotonic, sm, if membership in the class is preserved under a loose form of extensions. Our first theorem (0/1 law for FO with any set of sm quantifiers) subsumes a previous criterion for proving that almost no graphs satisfy a given property [BH79]. We also establish a 0/1 law for FO with Hartig quantifiers (equicardinality quantifiers) and a limit law for a fragment of FO with Rescher quantifiers (expressing inequalities of cardinalities) . The proofs of these last two results combine standard combinatorial enumerations with more sophisticated techniques from complex analysis. We also prove that the 0/1 law fails for the exten...
Infinitary Queries and Their Asymptotic Probabilities I: Properties Definable in Transitive Closure Logic
 Proc. Computer Science Logic '91, LNCS 626
, 1991
"... We present new general method for proving that for certain classes of finite structures the limit law fails for properties expressible in transitive closure logic. In all such cases also all associated asymptotic problems are undecidable. 1 Introduction The problems considered in this paper belo ..."
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Cited by 7 (3 self)
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We present new general method for proving that for certain classes of finite structures the limit law fails for properties expressible in transitive closure logic. In all such cases also all associated asymptotic problems are undecidable. 1 Introduction The problems considered in this paper belong to the research area called random structure theory, and, more specifically, to its logical aspect. To explain (very imprecisely and incompletely) what does it mean, let us imagine that we have a class of some structures (say: finite ones over some fixed signature), equipped with a probability space structure (this probability is usually assumed to be only finitely additive). Then we draw one structure at random and ask: what does the drawn structure look like? does the drawn structure have some particular property? Those questions are typical in random structure theory. To turn to the logical part of it, look at the drawn structure through logical glasses: we can only notice properti...
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.
ZeroOne Laws For Modal Logic
 Annals of Pure and Applied Logic 69
, 1994
"... We show that a 01 law holds for propositional modal logic, both for structure validity and frame validity. In the case of structure validity, the result follows easily from the wellknown 01 law for firstorder logic. However, our proof gives considerably more information. It leads to an elegant ax ..."
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Cited by 6 (1 self)
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We show that a 01 law holds for propositional modal logic, both for structure validity and frame validity. In the case of structure validity, the result follows easily from the wellknown 01 law for firstorder logic. However, our proof gives considerably more information. It leads to an elegant axiomatization for almostsure structure validity and to sharper complexity bounds. Since frame validity can be reduced to a \Pi 1 1 formula, the 01 law for frame validity helps delineate when 01 laws exist for secondorder logics. A preliminary version of this paper appears in Proceedings of the Seventh Annual IEEE Symposium on Logic in Computer Science, 1992. This version is almost identical to one that appears in a special issue of Annals of Pure and Applied Logic (vol. 69, 1994, pp. 157193) devoted to the papers of this conference. y Part of the work of the first author was performed while he was on sabbatical at the University of Toronto. z The work of the second author was com...