Moschovakis 1974 Yiannis N. Moschovakis, "Elementary Induction on Abstract Structures" North-Holland, 1974.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....for some a i 2 R i n R i . A node labelled h R 1 : Rn i has a child labelled h S 1 : Sn i for each possible S 1 ; Sn . is R dense if in nitely many of its nodes are labelled with tuples whose rst element is R. Then, nio (F 1 ; F n ) It is well known (see [Mos1974]) that any query de nable by a simultaneous least xed point of monotone formulae is also de nable by a formula using only simple xed points (i.e. xed points of single formulae) The analogous result also holds for simultaneous nio de nitions. Theorem 12: Let nioR1 ; R n ; x ( 1 ; ....

Y.N. Moschovakis, Elementary Induction on Abstract Structures, North Holland, 1974.

....xed point logic In this section we prove the main result of this paper, the separation of PFP gen and IFP. As we are not considering the nite model semantics anymore, we simply write PFP and pfp instead of PFP gen and pfp . We rst present a class of structures called acceptable (See [Mos74, Chapter 5]. These structures are particularly well suited to be used with diagonalisation arguments. 4.1 Acceptable structures De nition 4.1. Let A be an in nite set. A coding scheme on A is a triple (N ; for some N A, where the structure (N ; is isomorphic to ( and is an injective ....

....acceptable. We call A quasi acceptable if there exists an acceptable expansion A of A by a nite set of PFP de nable relations. 10 Observe that quasi acceptable structures are those which admit an PFPde nable coding scheme, i.e. one where the relations , seq, lh, and q are PFPde nable. See [Mos74, Chapter 5] for more on elementary and inductive coding schemes. 4.2 Coding and Diagonalisation We show now how formulae can be encoded by elements of acceptable structures. For the rest of this section let A be an acceptable structure, where : rel [ const is the disjoint union of a nite set ....

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Y.N. Moschovakis. Elementary Induction on Abstract Structures. North Holland, 1974. ISBN 0 7204 2280 9.

....and for pointing out to us the connection between public announcement and relativisation. We will answer the question in Section 5. 4. Fixed Point Extensions of First Order Logic The first systematic studies of least and inflationary fixed points on abstract structures appeared in the 1970s, see [1, 16, 17]. At that time the focus was on monotone and non monotone inductions over first order formulae. No explicit fixed point operators were added to the language of first order logic, fixed points were not being nested, and not interleaved with other logical operations. Despite these differences with ....

....: A j= R; a)g. Example 4.6. For the formula (T; x; y) Exy 9z(Exz T zy) the relation on a graph (V; E) is distance comparison: a; b) c; d) iff dist(a; b) dist(c; d) Stage Comparison Theorems deal with the definability of stage comparison relations. For instance, Moschovakis [16] proved that the stage comparison relations and of any positive first order formula are definable by a simultaneous induction over positive first order formulae. For the equivalence results on IFP and LFP one needs a Stage Comparison Theorem for IFP inductions. We first observe that the ....

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Y. Moschovakis. Elementary induction on abstract structures. North Holland, 1974.

....model theory; for a description of Finite Model Theory, see [EF95] But we are interested in representing algorithms as games. ffl In [Mo72] a game quantifier was introduced, and expanded on in [Ac75] From these arises the game for (positive elementary) least fixed point logic described in [Mo74]. ffl In [HaK84] and [AbV89] game semantics based on Datalog but resembling the Game Theoretic Semantics of Hintikka (see [HiS97] are introduced; HaK84] presents a variant of the game for (positive elementary) least fixed point logic. ffl In [Mc95a] a game in the tradition of [HaK84] is ....

....We will presume that the readers are familiar with First Order (FO) logic on these structures, and proceed directly to more expressive logics. 1. 1 Least Fixed Point Logic Perhaps the most popular fixed point logic is the First Order (positive) Least Fixed Point (FO pos LFP) logic of [Mo74] and [AhU79] First, given a relation variable symbol S, a second order formula ( S; is S positive if it has no second order quantifications and if all occurrences of S in are positive, viz. none are within any negated subformulas. And given a tuple S = S 0 ; S 1 ; S , ....

Y. Moschovakis, Elementary induction on abstract structures, (North-Holland, 1974).

....each e#ect is caused by the next e#ect in the chain. To adequately model this propagation process, a constructive semantics with the same features as the process itself seems appropriate. We base our approach on the main mathematical constructive principle: the principle of inductive definition ([2, 14, 1]) We define a conservative extension of this principle which allows for dealing with non stratified definitions, using techniques inspired by those used in the definition of logic programming semantics. In the following section, we formally define the principle of inductive definition and show ....

....rules is to read them as an inductive definition of predicates Caus and Init. This is formalised below. 2. 1 Principle of Inductive Definition The semantics and expressiveness of inductive definitions are studied in a subarea of mathematical logic, the area of Iterated Inductive Definitions (IID) [2, 14, 1]) The semantics proposed there require the definitions to be stratified. For our purposes this is not su#cient, as indicated above. However, as it appears, the problems of defining the semantics of non stratified inductive definitions are analogous to the problems of defining the semantics of ....

Y. N. Moschovakis. Elementary Induction on Abstract Structures. NorthHolland Publishing Company, Amsterdam- New York, 1974.

.... 57 is from [Meyer and Parikh, 1981] Theorem 60 is from [Harel, 1979] see also [Harel, 1984] and [Harel and Kozen, 1984] it is similar to the expressiveness result of [Cook, 1978] Theorem 61 and Corollary 62 are from [Harel and Kozen, 1984] Arithmetical structures were first defined by [Moschovakis, 1974] under the name acceptable structures. In the context of logics of programs, they were reintroduced and studied in [Harel, 1979] 1 completeness of DL was first proved by Meyer, and Theorem 63 appears in [Harel et al. 1977] An alternative proof is given in [Harel, 1985] see [Harel et al. ....

Y. N. Moschovakis. Elementary Induction on Abstract Structures. NorthHolland, 1974.

....has a long history in logic and computer science. Beginning with the work of Kleene and others on inductive definitions on the structure of arithmetic in recursion theory, inductive definitions on abstract structures have been studied since the early seventies, most notably by Moschovakis [9, 10], Aczel [1] and others. Whereas in the seventies the study of inductive definitions focused on monotone or non monotone inductions of first order formulae on infinite structures, the rise of database and finite model theory in the eighties gave birth to a renewed interest in such kinds of ....

....must also be the inflationary fixed point of F . Therefore it is clear, that regarding expressive power MFP IFP. 3. Comparing the stages of inductive definitions In this section we introduce the stage comparison method which has been the key to many results about fixedpoint logics. See [9, 5] for instance. The method will be essential for the explorations in the later sections. Let #(R, x) be a formula, say in first order logic. As mentioned above, if # is positive in R, then its least fixed point is obtained as the fixed point of the sequence of sets as defined by (1) Thus we ....

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Y.N. Moschovakis. Elementary Induction on Abstract Structures. North Holland, 1974.

....quanti ed) variables. Relationally, a CQ is a project join query. By adding union and recursion to conjunctive queries, one gets Datalog, the language of logic programs (known also as Horn clause programs) without function symbols [15] which is essentially a fragment of xpoint logic [16, 17]. Datalog consists, in a pure way, only of the most fundamental elements of relational queries: join, projection, union, and recursion. With respect to query containment, CQs and Datalog span the spectrum in terms of computational complexity. In [14] it is shown that CQ containment is equivalent ....

Moschovakis, Y.N.: Elementary Induction on Abstract Structures. North-Holland Publ. Co., Amsterdam (1974)

....more readable form. Positive LFP. While LFP and the modal calculus allow for arbitrary nesting of least and greatest fixed points, and for arbitrary interleaving of fixed points with Boolean operations and quantifiers, classical studies of inductive definability over first order logic (such as [78]) focus on a more restricted logic. Let LFP1 (sometimes also called positive LFP) be the extension of first order logic that is obtained by taking least fixed points of positive first order formulae (without parameters) and closing them under disjunction, conjunction, and under existential and ....

....(R, 6) Example 3.46. For the formula 9(T, x, y) Exy V Sz(Exz A Tyz) the relation on a graph (V, E) is distance comparison: a, b) c, d) iff dist(a, b) dist(c, d) Stage Comparison Theorems are results about the definability of stage comparison relations. For instance, Moschovakis [78] proved that the stage comparison relations and of any positive first order formula 9 are definable by a simultaneous induction over positive first order formulae. For the equivalence results on IFP and LFP one needs a stage comparison theorem for IFP inductions. We first observe that the ....

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Y. MOSCHOVAKIS, Elementary induction on abstract structures, North Holland, 1974.

....number of genuine alternations between least and greatest xed points in a formula. For LFP formulae of bounded width and bounded alternation depth the model checking problem can be solved in polynomial time. The fragments of bounded alternation depth in LFP induce a strict semantical hierarchy [3, 15]. On nite structures, this remains true for the modal calculus (since it has the nite model property) but not for LFP. Every LFP formula is equivalent, over nite structures, to an alternation free one, indeed to a formula with a single application of an lfp operator to a rstorder formula ....

Y. Moschovakis, Elementary induction on abstract structures, North Holland, 1974.

....has a long history in logic and computer science. Beginning with the work of Kleene and others on inductive definitions on the structure of arithmetic in recursion theory, inductive definitions on abstract structures have been studied since the early seventies, most notably by Moschovakis [9, 10], Aczel [1] and others. Whereas in the seventies the study of inductive definitions focused on monotone or non monotone inductions of first order formulae on infinite structures, the rise of database and finite model theory in the eighties gave birth to a renewed interest in such kinds of ....

....also be the inflationary fixed point of F . Therefore it is clear, that regarding expressive power LFP MFP IFP: 3. Comparing the stages of inductive definitions In this section we introduce the stage comparison method which has been the key to many results about fixedpoint logics. See [9, 5] for instance. The method will be essential for the explorations in the later sections. Let (R; x) be a formula, say in first order logic. As mentioned above, if is positive in R, then its least fixed point is obtained as the fixed point of the sequence of sets as defined by (1) Thus we ....

[Article contains additional citation context not shown here]

Y.N. Moschovakis. Elementary Induction on Abstract Structures. North Holland, 1974.

....is monotone, but, obviously discontinuous, if A is infinite. Kleene [4,5] used such mappings to develop his higher type recursion, later (and much better) formulated in terms of monotone, least fixed point recursion by Platek [15] The theory of inductive definability on first order structures [10] is also a chapter of monotone, least fixed point recursion, and it, too, has applications, to Proof Theory, Set Theory and Computer Science. Now, the basic theory of continuous recursors and algorithms, outlined in Sects. 4 6, never uses the continuity hypotheses, except to infer that certain ....

Yiannis N. Moschovakis. Elementary Induction on Abstract Structures. Studies in Logic, No. 77. North Holland, Amsterdam, 1974.

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Moschovakis 1974 Yiannis N. Moschovakis, "Elementary Induction on Abstract Structures" North-Holland, 1974.

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Moschovakis 1974 Yiannis N. Moschovakis, "Elementary Induction on Abstract Structures", North-Holland, Amsterdam, 1974.

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Y. N. Moschovakis, Elementary induction on abstract structures, North Holland, 1974.

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Y. N. Moschovakis. Elementary Induction on Abstract Structures. North Holland, 1974.

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Y. N. Moschovakis. Elementary Induction on Abstract Structures. North Holland, 1974.

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Y. N. Moschovakis, Elementary Induction on Abstract Structures (North-Holland, 1974); MR 53 #2661.

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Y. N. Moschovakis. Elementary Induction on Abstract Structures. North-Holland, 1974.

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Y.N. Moschovakis. Elementary Induction on Abstract Structures. Amsterdam, North Holland, 1974.

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Yiannis N. Moschovakis, Elementary induction on abstract structures, Studies in Logic and the Foundations of Mathematics, no. 77, North-Holland and Elsevier, Amsterdam, London and New York, 1974.

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Y.N. Moschovakis. Elementary Induction on Abstract Structures. North Holland, 1974.

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Y. MOSCHOVAKIS, Elementary induction on abstract structures, Studies in Logic series, vol. 77, North-Holland, Amsterdam, 1974.

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Yiannis Moschovakis, Elementary induction on abstract structures, North-Holland, Amsterdam, 1974.

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Yiannis Moschovakis. Elementary Induction on Abstract Structures. North-Holland, 1974.

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