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A logic of nonmonotone inductive definitions
 ACM transactions on computational logic
, 2007
"... Wellknown principles of induction include monotone induction and different sorts of nonmonotone induction such as inflationary induction, induction over wellfounded sets and iterated induction. In this work, we define a logic formalizing induction over wellfounded sets and monotone and iterated i ..."
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Cited by 36 (22 self)
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Wellknown principles of induction include monotone induction and different sorts of nonmonotone induction such as inflationary induction, induction over wellfounded sets and iterated induction. In this work, we define a logic formalizing induction over wellfounded sets and monotone and iterated induction. Just as the principle of positive induction has been formalized in FO(LFP), and the principle of inflationary induction has been formalized in FO(IFP), this paper formalizes the principle of iterated induction in a new logic for NonMonotone Inductive Definitions (IDlogic). The semantics of the logic is strongly influenced by the wellfounded semantics of logic programming. This paper discusses the formalisation of different forms of (non)monotone induction by the wellfounded semantics and illustrates the use of the logic for formalizing mathematical and commonsense knowledge. To model different types of induction found in mathematics, we define several subclasses of definitions, and show that they are correctly formalized by the wellfounded semantics. We also present translations into classical first or second order logic. We develop modularity and totality results and demonstrate their use to analyze and simplify complex definitions. We illustrate the use of the logic for temporal reasoning. The logic formally extends Logic Programming, Abductive Logic Programming and Datalog, and thus formalizes the view on these formalisms as logics of (generalized) inductive definitions. Categories and Subject Descriptors:... [...]:... 1.
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 19 (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
Expressive Equivalence of Least and Inflationary FixedPoint Logic
 IN 17TH SYMP. ON LOGIC IN COMPUTER SCIENCE (LICS
, 2002
"... We study the relationship between least and inflationary fixedpoint logic. By results of Gurevich and Shelah from 1986, it has been known that on finite structures both logics have the same expressive power. On infinite structures however, the question whether there is a formula in IFP not equivale ..."
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Cited by 13 (2 self)
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We study the relationship between least and inflationary fixedpoint logic. By results of Gurevich and Shelah from 1986, it has been known that on finite structures both logics have the same expressive power. On infinite structures however, the question whether there is a formula in IFP not equivalent to any LFPformula was still open. In this
Interpolation, Preservation, and Pebble Games
 Journal of Symbolic Logic
, 1996
"... Preservation and interpolation results are obtained for L1! and sublogics L ` L1! such that equivalence in L can be characterized by suitable backandforth conditions on sets of partial isomorphisms. 1 Introduction In the heyday of infinitary logic in the 1960's and 70's, most attention ..."
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Cited by 13 (6 self)
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Preservation and interpolation results are obtained for L1! and sublogics L ` L1! such that equivalence in L can be characterized by suitable backandforth conditions on sets of partial isomorphisms. 1 Introduction In the heyday of infinitary logic in the 1960's and 70's, most attention was focused on L!1! and its fragments (see e.g. Keisler [19]), since countable formulas seemed best behaved. The past decade has seen a renewed interest in L1! and its finite variable fragments L (k) 1! (for 2 k ! !) and the modal fragment L \Pi 1! (see e.g. Ebbinghaus and Flum [17] on the former and Barwise and Moss [9] on the latter), due to various connections with topics in computer science. These logics form a hierarchy of increasingly powerful logics L \Pi 1! ae L (2) 1! ae L (3) 1! ae : : : ae L (k) 1! ae : : : ae L1! ; with each of these inclusions being proper. Moreover, there is a useful and elegant algebraic characterization of equivalence in L in each of these logics L, from b...
On Bounded Specifications
 In Proc. of the Int. Conference on Logic for Programming and Automated Reasoning (LPAR’01), LNAI
, 2002
"... Bounded model checking methodologies check the correctness of a system with respect to a given specification by examining computations of a bounded length. Results from settheoretic topology imply that sets in are both open and closed (clopen sets) are precisely bounded sets: membership of a ..."
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Cited by 11 (5 self)
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Bounded model checking methodologies check the correctness of a system with respect to a given specification by examining computations of a bounded length. Results from settheoretic topology imply that sets in are both open and closed (clopen sets) are precisely bounded sets: membership of a word in a clopen set can be determined by examining a bounded prefix of it.
Logics of imperfect information: why sets of assignments
 Proceedings of 7th De Morgan Workshop ’Interactive Logic: Games and Social Software
, 2005
"... In 1961 Leon Henkin [3] extended firstorder logic by adding partially ordered arrays of quantifiers. He proposed a semantics for sentences φ that begin with quantifier arrays of this kind: φ is true in a structure A if and only if there are a sentence φ + and a structure A + such that: • φ + comes ..."
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Cited by 8 (0 self)
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In 1961 Leon Henkin [3] extended firstorder logic by adding partially ordered arrays of quantifiers. He proposed a semantics for sentences φ that begin with quantifier arrays of this kind: φ is true in a structure A if and only if there are a sentence φ + and a structure A + such that: • φ + comes from φ by removing each existential quantifier ∃y in the partially ordered prefix, and replacing each occurrence of the variable y by a term F (¯x) where ¯x are the variables universally quantified ‘before’ ∃y in the quantifier prefix (so that the new function symbols F are Skolem function symbols), • A + comes from A by adding functions to interpret the Skolem function symbols in φ +, and • φ + is true in A +. For example the sentence
On the Boundedness Problem for TwoVariable FirstOrder Logic
 Bulletin of Symbolic Logic
, 1997
"... A positive firstorder formula is bounded if the sequence of its stages converges to the least fixed point of the formula within a fixed finite number of steps independent of the input structure. The boundedness problem for a fragment L of firstorder logic is the following decision problem: given a ..."
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Cited by 3 (3 self)
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A positive firstorder formula is bounded if the sequence of its stages converges to the least fixed point of the formula within a fixed finite number of steps independent of the input structure. The boundedness problem for a fragment L of firstorder logic is the following decision problem: given a positive formula in L, is it bounded? In this paper, we investigate the boundedness problem for twovariable firstorder logic FO 2. As a general
The Undecidability of Iterated Modal
 Relativization”, Studia Logica
"... Abstract. In dynamic epistemic logic and other fields, it is natural to consider relativization as an operator taking sentences to sentences. When using the ideas and methods of dynamic logic, one would like to iterate operators. This leads to iterated relativization. We are also concerned with the ..."
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Abstract. In dynamic epistemic logic and other fields, it is natural to consider relativization as an operator taking sentences to sentences. When using the ideas and methods of dynamic logic, one would like to iterate operators. This leads to iterated relativization. We are also concerned with the transitive closure operation, due to its connection to common knowledge. We show that for three fragments of the logic of iterated relativization and transitive closure, the satisfiability problems are Σ 1 1complete. Two of these fragments do not include transitive closure. We also show that the question of whether a sentence in these fragments has a finite (tree) model is Σ 0 1complete. These results go via reduction to problems concerning domino systems.
Boundedness of Monadic FO over Acyclic Structures
, 2007
"... We study the boundedness problem for monadic least fixed points as a decision problem. While this problem is known to be undecidable in general and even for syntactically very restricted classes of underlying firstorder formulae, we here obtain a decidability result for the boundedness issue for m ..."
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We study the boundedness problem for monadic least fixed points as a decision problem. While this problem is known to be undecidable in general and even for syntactically very restricted classes of underlying firstorder formulae, we here obtain a decidability result for the boundedness issue for monadic fixed points over arbitrary firstorder formulae in restriction to acyclic structures.