Bruno Courcelle. Fundamental properties of infinite trees. Theoretical Computer Science, 25(1):95--169, 1983.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....a preprocessing phase, in order to decrease the number of nodes in the problem that is submitted to the AC subtyping algorithm. 3 Recursive types 3. 1 Presentation Recursive types are usually given in concrete syntax as finite systems of contractive type equations, which, according to Courcelle [13], uniquely define regular trees; or as finite terms involving binders, see e.g. 18] The process of unfolding these finite representations gives rise to regular infinite trees. DEFINITION 1 (SIGNATURE) A signature is a mapping from symbols, written s, to integer arities. In this paper, we ....

B. Courcelle. Fundamental properties of infinite trees. Theoretical Comput. Sci., 25:95--169, 1983.

....terms to be a coalgebraic monad by equipping them with a notion of substitution which is the restriction of that of T . This way, one can show that the following are all coalgebraic monads over Set: i) infinite terms which contain only a finite number of variables; ii) locally finite terms [7], i.e. finite and infinite terms which have the property that from every node, there is a finite path to a leaf; iii) rational terms are terms with a finite number of subterms, or, more formally, the free iterative theory over a signature [9] However, the notion of a coalgebraic monad also ....

Bruno Courcelle. Fundamental properties of infinite trees. Theoretical Computer Science, 25(2):95--169, 1983.

.... of labeled trees form a complete partial order under the partial ordering where t 1 t 2 , i# t 1 can be obtained from t 2 by replacing any number of subtrees with the singleton tree# In this setting, normal forms can be defined as unique limits of chains of approximations, as discussed in [3,4]. The singleton tree# is smaller than all other trees and corresponds to the type defined by Type T = T which we shall refer to as the vacuous type. Note that if two type expressions are equivalent, then their corresponding structural invariants must be equal. The equivalence is unique in ....

....this confronts us with the problem of performing a countably infinite number of rewriting steps. This is, in fact, possible in the present context, since we can work with finite approximants. Definition 4. 2: t # v i# ## t : # # # # A similar method for defining functions is described in [3]. Definition 4.3: The maximal interpretation is defined by Valmax (T ) which mimics proposition 4.1. Lemma 4.4: If t 1 t 2 then t 2 # t 1 . Proof: If # ## t 1 , then # ## t 2 . Since # # # #, we are done. # Lemma 4.5: # is reflexive and transitive. 13 Proof: Reflexivity ....

Courcelle B. "Fundamental Properties of Infinite Trees" in Theoretical Computer Science Vol 25 No 1, 95-169, North-Holland 1983.

....to the well understood problem of finding a size stable partition of a graph [16] The organization of the paper: A small example is described in Section 2. This example will be used throughout the paper for illustrative purposes. In Section 3 we recall the notions of terms and term automata [1, 8, 11, 13], and we state the definitions of types and type equivalence from the paper by Palsberg and Zhao [17] In Section 4 we prove our main result. An implementation of our algorithm is discussed Section 5. Subtyping of recursive types is discussed in Section 6. Concluding remarks appear in Section 7. ....

Bruno Courcelle. Fundamental properties of infinite trees. Theoretical Computer Science, 25(1):95--169, 1983.

....is the theory of rational trees over the given signature of function symbols. We will use the following three facts about trees, the first one for the consistency test and the other two for the entailment test. The first one is the characteristic property of rational trees (cf. for example, [6]) this is what God made them for. Fact 1. A linear system is satisfiable; i.e. Delta j= 9) We use = for both the logical equality symbol and the meta level identity; no ambiguity will arise. The first fact is a logical consequence of the next one. On the other hand, given a proof of ....

Bruno Courcelle. Fundamental properties of infinite trees. Theoretical Computer Science, 25(2):95--169, 1983.

....is the theory of rational trees over the given signature of function symbols. We will use the following three facts about trees, the first one for the consistency test and the other two for the entailment test. The first one is the characteristic property of rational trees (cf. for example, [6]) Fact 1 A linear system is satisfiable; i.e. Delta j= 9) The first fact is a logical consequence of the next one. On the other hand, given a proof of the first fact, the second fact could have been proven from the first. Namely, the value of a non determined variable never contains an ....

Bruno Courcelle. Fundamental properties of infinite trees. Theoretical Computer Science, 25(2):95--169, 1983.

.... The decidability of the first order theory of term algebras follows from Mal cev s work on locally free algebras [31, Chapter 23] 39] also gives an argument for decidability of term algebra and presents a unification algorithm based on congruence closure [38] Infinite trees are studied in [12]. 30] presents a complete axiomatization for algebra of finite, infinite and rational trees. A proof in the style of [22] for an extension of free algebra with queues is presented in [43] Decidability of an extension of term algebras with membership tests is presented in [10] in the form of a ....

....subtyping. We define the first order structure of structural subtyping of recursive types similarly to the corresponding structure for non recursive types in Section 4; the only di#erence is that the domain contains both finite and infinite terms. Infinite terms correspond to infinite trees [12, 30]. We define infinite trees as follows. We use alphabet to denote paths in the tree. A tree domain D is a finite or infinite subset of the set r # such that: 1. D is prefix closed: if w r # , x r then D implies w D; D then exactly one of the following two properties ....

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Bruno Courcelle. Fundamental properties of infinite trees. Theoretical Computer Science, 25(2):95--169, March 1983. 3.4, 7.1, 7.4

....products, and the constants and . We argue that this simple setting is general enough in section 6. Section 7 concludes. 2 Recursive types 2. 1 Presentation Recursive types are usually given in concrete syntax as finite systems of contractive type equations, which, according to Courcelle [13], uniquely define regular trees; or as finite terms involving binders, see e.g. 18] The process of unfolding these finite representations gives rise to regular infinite trees. Definition 1 (Signature) A signature is a mapping from symbols, written s, to integer arities. In this paper, we ....

B. Courcelle. Fundamental properties of infinite trees. Theoretical Comput. Sci., 25:95--169, 1983.

....framework. Regular trees, on the contrary, naturally express deep invariants of the concrete values. A sound mathematical basis comes along with them. Analyses using regular trees should be considered. They have been used by Aiken (and collaborators) in [5, 4, 6] and presented by Courcelle in [18]. The results by Aiken showed an impressive representation power but did not seem to be efficient enough. 7.3.3 Extensions Other Languages Although we explicitly aim at analyzing dynamically typed languages, we believe that the type analysis could be useful in some statically typed languages, ....

Bruno Courcelle. Fundamental properties of infinite trees. Theoretical Computer Science, 25(2):95-169, mar 1983.

....First, agents can cope with infinite belief hierarchies by reducing them to finite graphs. Second, agents can apply to infinite beliefs the same techniques they use to handle finite beliefs. Definition 3. An infinite tree is regular if and only if the number of its distinct subtrees is finite [5]. It is evident that a regular belief tree is a repetition of a finite number of subtrees. This means that by extending a regular tree to infinity we do not add new strategically relevant information. The following proposition states that for every regular belief tree there exists some finite ....

Courcelle B. Fundamental Properties of Infinite Trees. Theoretical Computer Science, 25: 95-169, 1983.

....(lazy) values. The variety of different interpretations is investigated in [8] 4 2.1 Type Equivalence Several type expressions may be taken to denote the same type. These can be identified by an equivalence relation , which is defined as the identity of normal forms, using the techniques of [4,5]. To each type expression T we associate a unique normal form nf (T ) which is a possibly infinite labeled tree. Informally, the tree is obtained by repeatedly unfolding the type expression. Formally, we use the fact that the set of labeled trees form a complete partial order under the partial ....

....optimal. Proof: Assume all(F ; A) We shall construct an ff that always works; as we shall see, this ff will be an appropriate mixture of formal and actual types. Being regular, the type has finitely many different subtypes 1 ; 2 ; k , where = 1 . The i s can be uniquely defined [5] through a set of type equations of the form i = f i ( 1 ; 2 ; k ) Now, the type ff = ff 1 is defined by the equations ff i = A# oe if F # oe = i f i (ff 1 ; ff 2 ; ff k ) otherwise 27 This is well defined since, because all holds, F # oe = F # oe = i implies A# ....

Courcelle, B. "Fundamental Properties of Infinite Trees" in Theoretical Computer Science Vol 25 No 1, North-Holland 1983.

.... the change from by name to by structure matching entails some loss of precision in the sortings (we give up the ability to prevent mixing of accidentally isomorphic data structures) it substantially simplifies the technical aspects of the system by allowing sorts to be treated as regular trees [Cou83]. Our subsort relation can be motivated by the common situation in which two processes must cooperate in the use of a shared resource such as a printer. The printer provides a request channel p carrying values of some sort T , which represent data sent for printing by the client processes. If one ....

B. Courcelle. Fundamental properties of infinite trees. Theoretical Computer Science, 25:95--169, 1983.

....clear) we will do this in a particular way. First, we wish to keep the constraint set finite and simple, so we cannot mention the (possibly infinite) trees top s(a) directly in C . However, as regular trees, the elements top s(a) can all be defined by sets of regular and contractive equations (see [16]) of the form 7 = 71 x 72 or 7 = 71 72, which are evidently expressible by simple in equalities. Such sets have unique solutions (they define contractions in the complete metric space of trees, see [16] We can assume, therefore, that for any structural constraint set C and any shape ....

....top s(a) can all be defined by sets of regular and contractive equations (see [16] of the form 7 = 71 x 72 or 7 = 71 72, which are evidently expressible by simple in equalities. Such sets have unique solutions (they define contractions in the complete metric space of trees, see [16]. We can assume, therefore, that for any structural constraint set C and any shape s, a 6 Vat(C) there is a simple constraint set C s of regular equations defining top s . We assume that the defining equations in the C s use variables, ranged over by 7 which occur nowhere else, and we let F ....

B. Courcelle. Fundamental properties of infinite trees. Theoretical Com- puter Science, 25:95-169, 1983.

....datatype Maybe and (K1 Snd) encodes the datatype of Peano numerals. One can reasonably argue that the definition of unary functors via minimalization of bifunctors is not the most direct way to go. An alternative that we explore in this article is to define functor expressions as rational trees [6] over the following grammar of unary functors. F = KT j Id j F F j F Theta F j F Delta F This approach is, of course, inspired by the way types are introduced in most functional programming languages. Type definitions usually take the form of recursion equations. If we interpret these ....

Bruno Courcelle. Fundamental properties of infinite trees. Theoretical Computer Science, 25(2):95-- 169, March 1983.

.... introduction, it is not difficult to see that each processor can anonymously build its own view truncated at any desired depth (and k steps are needed to obtain k levels) Note that views of finite (strongly connected) graphs are exactly the regular trees without leaves, in the sense of Courcelle [10]. Consider now a graph B such that every fibration with total graph B is an isomorphism. Such a graph is called fibration prime: intuitively, fibration prime graphs cannot be collapsed by a fibration. It is worth observing that they are node rigid (i.e. all automorphisms act on the nodes as ....

Bruno Courcelle. Fundamental properties of infinite trees. Theoret. Comput. Sci., 25(2):95-- 169, 1983.

....# for the empty address and set t # = t. An address, #, is valid in t if t # denotes a subterm as above. If # is a valid address in t we denote by t(#) the root label of t #. We can view the term as a function from the set of valid addresses to # as in the following definition due to Courcelle [3]. Definition 4.1 Let # denote the set of natural numbers and let # be a finite ranked alphabet. A (# )term is a partial function t : # # # # whose domain is nonempty, prefix closed, and respects rank in the sense that if t(#) is defined then i t(#i) is defined = 1. rank(t(#) A finite or ....

Bruno Courcelle. Fundamental properties of infinite trees. Theoretical Computer Science, 25(1):95--169, 1983.

....brevity, we deal in this paper with just three type constructors: Theta, and Top. We define types as (possibly infinite) trees with nodes labeled by one of the symbols , Theta, or Top. The definition is specialized to our present needs; for a general treatment of infinite labeled trees see [Cou83]. 3.1 Definition: Write N for the set of sequences of natural numbers. The empty sequence is written ffl. If and oe are sequences, then Deltaoe denotes the concatenation of and oe. 2 3.2 Definition: A tree type, or, simply, a tree, is a partial function T 2 N f ; Theta; Topg satisfying ....

.... 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) Morris [Mor68] Basic syntactic and semantic properties (without subtyping) are collected in Cardone and Coppo [CC91] Amadio and Cardelli [AC93] gave the first subtyping algorithm for recursive types. Their paper defines three ....

B. Courcelle. Fundamental properties of infinite trees. Theoretical Computer Science, 25:95--169, 1983.

....assume these are rational trees, i.e. trees with only finitely many subtrees. A rational tree is a good model for data structures with pointers since the tree can be represented (though not uniquely represented) by a rooted directed graph. Unfolding the graph to remove its cycles yields the tree [Courcelle 1983; Podelski and Smolka 1997] The constraints C model bindings; we assume they are equalities between terms that describe sets of rational trees. For example, the constraint x = f(y) means that the trees described by the variable x all have a root labeled f and a single subtree, which is a tree ....

Courcelle, B. 1983. Fundamental properties of infinite trees. Theoretical Computer Science 25, 95--169.

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Bruno Courcelle. Fundamental properties of infinite trees. Theoretical Computer Science, 25(1):95--169, 1983.

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Courcelle, B.: Fundamental properties of infinite trees. Theoretical Computer Science 25 (1983) 95--169

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B. Courcelle. Fundamental properties of infinite trees. Theoretical Computer Science, 25:95--169, 1983.

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B. Courcelle. Fundamental properties of infinite trees. Theoretical Computer Science, 25(2): 95--169, March 1983. ISSN 0304-3975.

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Courcelle B. "Fundamental Properties of Infinite Trees." Journal of TCS, n 17, pp. 95--169. 1983.

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Bruno Courcelle. Fundamental properties of infinite trees. Theoretical Computer Science, 25:96--169, 1983.

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B. Courcelle. Fundamental properties of infinite trees. Theor. Comp. Sc., 25:95--169, 1983.

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