P. Aczel. An introduction to inductive definitions. In J. Barwise, editor, Handbook of Mathematical Logic,volume90ofStudies in Logic andthe Foundations of Mathematics,pages 739--782. Elsevier Science Publishers B.V., 1977.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... points of such set operators in type theory and showing that the corresponding rules of the schema can be derived; iii) interpreting typetheoretic strictly positive operators as n continuous functors on the category of sets (assuming a theory without universes) In this paper I use Aczel s [1] notion of rule set rather than continuous functors. It is really only a variation, since a rule set generates a continuous operator. But it allows a direct concrete translation of the type theoretic introduction rules to set theoretic rule sets and generalizes the concrete construction of the ....

....families. A : P: J abbreviates Gamma J. P set; P = P: intro i (as; b k ) k ) P ; k : Hs ik [as] P ) k k ) k ) P ffl Hs ik [xs] is a telescope relative to T in the context xs : Gs i for each k. 3. 3 Inductive sets in set theory We shall use Aczel s [1] set theoretic notion of rule set to interpret the introduction rules for a new set former. The set defined inductively by a rule set is the least set closed under all rules in the rule set. A rule on a base set U in Aczel s sense is a pair of sets hu; vi, often written v such that u U and v ....

P. Aczel. An introduction to inductive definitions. In J. Barwise, editor, Handbook of Mathematical Logic, chapter C.7, pages 739--782. North Holland, 1977.

....Sharing Theory of Dialogue and Four Classes of Circularity Problems Norihiro Ogata University of 2kuba, Department of Literature and Linguistics, ogata stc.ipa.go. jp Abstract In Information Sharing Theory of Dialogue ( STD) 11, 12, 10] a dialogue is considered as a circular object. However, circular objects have five classes of circularity problems. In this paper we shall show and olve five cixcularity problems by means of ISTD. We shall formulate problematic objects, and show a method of const, ruction of models ....

....Sharing Theory of Dialogue and Four Classes of Circularity Problems Norihiro Ogata University of 2kuba, Department of Literature and Linguistics, ogata stc.ipa.go. jp Abstract In Information Sharing Theory of Dialogue ( STD) 11, 12, 10] a dialogue is considered as a circular object. However, circular objects have five classes of circularity problems. In this paper we shall show and olve five cixcularity problems by means of ISTD. We shall formulate problematic objects, and show a method of const, ruction of models ....

[Article contains additional citation context not shown here]

P. Aczel. An Introduction to Inductive Definition, in J. Barwise ed., Handbook of Mathematical Logic, Dor- drecht: North-Holland Publishing Company, 1977, 739-782.

....Such data structures are called datatypes below, by analogy with datatype declarations in Standard ml. Some logics take datatypes as primitive; consider Boyer and Moore s shell principle [4] and the Coq type theory [22] A datatype is but one example of an inductive definition. Such a definition [2] specifies the least set R closed under given rules: applying a rule to elements of R yields a result within R. Inductive definitions have many applications. The collection of theorems in a logic is inductively defined. A structural operational semantics [13] is an inductive definition of a ....

....lleq(A) coincides with the equality relation takes some work. 5.5 The accessible part of a relation Let be a binary relation on D ; in short, #) DD . The accessible or wellfounded part of written acc(#) is essentially that subset of D for which admits no infinite decreasing chains [2]. Formally, acc(#) is inductively defined to be the least set that contains a if it contains all # predecessors of a, for a D . Thus we need an introduction rule of the form Paulin Mohring treats this example in Coq [22] but it causes di#culties for other systems. Its premise is ....

Aczel, P., An introduction to inductive definitions, In Handbook of Mathematical Logic, J. Barwise, Ed. North-Holland, 1977, pp. 739--782

....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 ....

P. Aczel. An introduction to inductive definitions. In J. Barwise, editor, Handbook of Mathematical Logic, volume 90 of Studies in Logic, pages 739 --782. North-Holland, 1977.

....are the least set containing propositional letters and closed under #, #, Most importantly: the set of theorems in a logical system is the least set containing all axioms and closed under all applications of inference rules. As Aczel explains, this is the general form of an inductive definition [2]. What has this to do with logic programming We can regard a logic program as an inductive definition by taking its clauses as axioms and inference rules. We regard our family relationships database as a new logic with rules like parent(x, y) parent(y,z) grandparent(z,x) grandparent(z,y) ....

....grandparent(z,x) grandparent(z,y) cousin(x, y) This inductively defines various sets. The grandparent of relation is the set of all pairs #x, y# such that grandparent(x, y) follows from the database. Similarly the derivable instances of cousin(x, y) define the cousin of relation. Aczel [2] gives the semantics of an inductive definition as follows. A rule has the form p P , where P is the set of premises and p is the conclusion. Let # be a set of rules. A set A is # closed provided that for each rule p P in #,ifP A then p A. Thus if the premises are in A then so ....

[Article contains additional citation context not shown here]

Aczel, P., An introduction to inductive definitions, In Handbook of Mathematical Logic, J. Barwise, Ed. North-Holland, 1977, pp. 739--782

....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 ....

P. Aczel. An Introduction to Inductive Definitions. In J. Barwise, editor, Handbook of Mathematical Logic, pages 739--782. North-Holland Publishing Company, 1977.

....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 definitions in ....

P. Aczel. An introduction to inductive definitions. In J. Barwise, editor, Handbook of Mathematical Logic, volume 90 of Studies in Logic, pages 739 --782. NorthHolland, 1977.

....on the contrary, defines the stepwise evolution of a negative agent. Our rule is a real transition rule , in the SOS sense, and does not contain negative premises. The resulting operational semantics actually is an inductive operational semantics ( 4] i.e. defined by an inductive system ([3]) with the advantages that this approach implies (see for instance [4] and [5] Moreover, we show that our treatment of negation allows a natural declarative interpretation of the operational semantics. 2 Constraint System The concept of constraint is central for the paradigm of clp and ....

P. Aczel. An introduction to inductive definitions. In J. Barwise, editor, Handbook of Mathematical Logic, pages 739--782. Elsevier, 1977.

....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 definitions in ....

P. Aczel. An introduction to inductive definitions. In J. Barwise, editor, Handbook of Mathematical Logic, volume 90 of Studies in Logic, pages 739 --782. NorthHolland, 1977.

.... and existential quantification) a Theorem 4 runs against the suggestion from Theorem B 0 that mechanical transitions generally depend on more than the first order theory of the data model, although perhaps the underlying logic is best associated with positive existential induction (Aczel [3], section 3.2) in view of the Record Property, Lemma 1. At any rate, by equating bisimilarity with isomorphism under L, Corollary 5 reduces L(S) to the sequences l 1 ; l n of labels for which the initial state of S fails to make a transition. The most technically involved part of the ....

Peter Aczel. An introduction to inductive definitions. In J. Barwise, editor, Handbook of Mathematical Logic. North-Holland, Amsterdam, 1977.

....will each give rise to two recursive calls involving Sn Gamma2 and Tn Gamma2 , and so on. The total number of recursive calls will clearly be proportional to 2 n . 11 Related Work and Further Reading For background on induction and coinduction, readers are referred to Aczel s handbook article [Acz77] and Gordon s tutorial on coinduction and functional programming [Gor94] Basic information on fixed points can be found in Davey and Priestley s text [DP90] Properties of infinite and regular trees are surveyed by Courcelle [Cou83] Recursive types in computer science go back to (at least) ....

Peter Aczel. An introduction to inductive definitions. In Jon Barwise, editor, Handbook of Mathematical Logic, number 90 in Studies in Logic and the FOundations of Mathematics, pages 739--782. North Holland, 1977.

....no atom can be true without a cause and a cause for a true atom can never depend on the atom itself. 3. 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) [3, 23, 1]) We formalise this semantics in a different way and extend it to non stratified definitions. We need the following concepts. Definition 3.1 (VP ,F ) Given a set of ground atoms P, the set VP of (3valued) valuations on P is the set of all functions P ft; u; fg. On VP , a partial order F is ....

P. Aczel. An Introduction to Inductive Definitions. In J. Barwise, editor, Handbook of Mathematical Logic, pages 739--782. North-Holland Publishing Company, 1977.

No context found.

P. Aczel. An introduction to inductive definitions. In J. Barwise, editor, Handbook of Mathematical Logic,volume90ofStudies in Logic andthe Foundations of Mathematics,pages 739--782. Elsevier Science Publishers B.V., 1977.

No context found.

P. Aczel, 1977, 'An Introduction to Inductive Definitions', in J. Barwise, ed., Handbook of Mathematical Logic, North-Holland, 739--782, Amsterdam.

No context found.

Peter Aczel. An introduction to inductive definitions. In J. Barwise, editor, Handbook of Mathematical Logic, pages 739--782. North-Holland, 1977.

No context found.

Peter Aczel. An introduction to inductive definitions. In J. Barwise, editor, Handbook of Mathematical Logic, pages 739--782. North-Holland, 1977.

No context found.

P. Aczel. An introduction to inductive definitions. In J. Barwise, editor, Handbook of Mathematical Logic, pages 739--782. North-Holland, Amsterdam, 1977.

No context found.

P. Aczel. An introduction to inductive definitions. In J. Barwise, editor, Handbook of Mathematical Logic, pages 739-- 782. North-Holland, Amsterdam, 1977.

No context found.

P. Aczel. An introduction to inductive definitions. In J. Barwise, editor, Handbook of Mathematical Logic, vol 90 of Studies in Logic and the Foundations of Mathematics, pp 739--782. North-Holland, Amsterdam, 1977.

No context found.

P. Aczel. An introduction to inductive definitions. In J. Barwise, editor, The Handbook of Mathematical Logic, volume 90 of Studies in Logic and Foundations of Mathematics, pages 739--782. North Holland, 1977.

No context found.

Aczel, P. An Introduction to Inductive Definitions. In J. Barwise, editor, Handbook of Mathematical Logic, pages 739--782, North-Holland Publishing Company, 1977.

No context found.

Peter Aczel. An introduction to inductive definitions. In J. Barwise, editor, Handbook of Mathematical Logic, volume 90 of Studies in Logic and the Foundations of Mathematics, chapter C.7, pages 739--782. North-Holland, Amsterdam, 1977.

No context found.

P. Aczel, An introduction to inductive definitions, Handbook of mathematical logic (J. Barwise, editor), North Holland, 1977, pp. 739--782.

No context found.

Peter Aczel. An introduction to inductive definitions. In Jon Barwise, editor, Handbook of Mathematical Logic, volume 90 of Studies in Logic and the Foundations of Mathematics, chapter C.7, pages 739--782. North-Holland Publishing Company, 1977.

No context found.

P. Aczel. An Introduction to Inductive Definition, in J. Barwise ed., Handbook of Mathematical Logic, Dordrecht: North-Holland Publishing Company, 1977, 739-782.

First 50 documents  Next 50

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC