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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...
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 Underlying Logic of Hoare Logic
 IN CURRENT TRENDS IN THEORETICAL COMPUTER SCIENCE
, 1997
"... Formulas of Hoare logic are asserted programs # # # where # is a program and #, # are assertions. The language of programs varies; in the survey [Apt 1980], one finds the language of while programs and various extensions of it. But the assertions are traditionally expressed in firstorder logic ..."
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Cited by 7 (3 self)
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Formulas of Hoare logic are asserted programs # # # where # is a program and #, # are assertions. The language of programs varies; in the survey [Apt 1980], one finds the language of while programs and various extensions of it. But the assertions are traditionally expressed in firstorder logic (or extensions of it). In that sense, firstorder logic is the underlying logic of Hoare logic. We question the tradition and demonstrate, on the simple example of while programs, that alternative assertion logics have some advantages. For some natural assertion logics, the expressivity hypothesis in Cook's completeness theorem is automatically satisfied.
Program Schemes, Queues, the Recursive Spectrum and ZeroOne Laws
"... We prove that a very basic class of program schemes augmented with access to a queue and an additional numeric universe within which counting is permitted, so that the resulting class is denoted NPSQ+ (1), is such that the class of problems accepted by these program schemes is exactly the class of r ..."
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Cited by 1 (1 self)
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We prove that a very basic class of program schemes augmented with access to a queue and an additional numeric universe within which counting is permitted, so that the resulting class is denoted NPSQ+ (1), is such that the class of problems accepted by these program schemes is exactly the class of recursively solvable problems. The class of problems accepted by the program schemes of the class NPSQ(1) where only access to a queue, and not the additional numeric universe, is allowed is exactly the class of recursively solvable problems that are closed under extensions. We dene an innite hierarchy of classes of program schemes for which NPSQ(1) is the rst class and the union of the classes of which is the class NPSQ. We show that the class of problems accepted by the program schemes of NPSQ has a zeroone law and is the union of the classes of problems dened by the sentences of all vectorized Lindstrom logics formed using operators whose corresponding problems are recursively solvab...
unknown title
, 2001
"... Program schemes with binary writeonce arrays and the complexity classes they capture ..."
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Program schemes with binary writeonce arrays and the complexity classes they capture