R. Milner and M. Tofte. Co-induction in relational semantics. Theoretical Computer Science, 87, 209--220, 1991.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....on # terms. Recently, I have applied inductive definitions to the analysis of cryptographic protocols [29] 23 To demonstrate coinductive definitions, Frost [10] has proved the consistency of the dynamic and static semantics for a small functional language. The example is due to Milner and Tofte [18]. It concerns an extended correspondence relation, which is defined coinductively. A codatatype definition specifies values and value environments in mutual recursion. Non well founded values represent recursive functions. Value environments are variant functions from variables into values. This ....

Milner, R., Tofte, M., Co-induction in relational semantics, Theoretical Comput. Sci. 87 (1991), 209--220

....(e.g. products are the relation of behavioural equivalence) We investigate a similar topos structure in Section 3 and de ne a modality that captures behavioural equivalence. We also give a categorical account of coinduction, a proof technique commonly used in hidden algebra and coalgebra [9, 22, 14]. 2 Hidden Algebra In order to motivate the constructions of Section 3, we give a brief introduction to behavioural equality in the context of hidden algebra, which was introduced by Goguen [6] as a foundation for a semantics of the object paradigm. Hidden algebra is based on many sorted ....

Robin Milner and Mads Tofte. Co-induction in relational semantics. Theoretical Computer Science, 87(1):209-220, 1991.

....applied to a value of type , we use structural induction over k and appeal to the typing rules. The resulting proof is elementary: it proceeds only by structural induction over the terms representing values and evaluation derivations. In particular, there is no need for proofs by co induction [19, 14]. The proofs also easily extend to other polymorphic type systems for references and continuations [12] 4 Assessment Polymorphism by name supports references and continuations in a type safe way, while retaining the ML type algebra and typing rules, that are familiar and easy to understand. ....

R. Milner and M. Tofte. Co-induction in relational semantics. Theoretical Comput. Sci., 87:209-220, 1991.

....So let us consider the nature of possible worlds. Peregrin s [24] analysis concludes that a possible world in the intuitive sense can be explicated as a maximal consistent class of statements . This implies that to give the semantics of possible worlds we require techniques like coinduction [23, 17, 6] or non well founded sets [2] each of which is in some sense syntax dependent. In light of that, and Peregrin s conclusion that possible worlds are language dependent , our embedding of syntactic types (that express intension) in the semantics seems unavoidable. But the latter should not be ....

R. Milner and M. Tofte. Co-induction in relational semantics. Theoretical Computer Science, 87(1):209--220, 1991.

....context of the last recent k procedure calls (called k CFA) Other splittings, also depending on procedure calls, were proposed by Jagannathan and Weeks [14] as poly k CFA and by Wright and Jagannathan [32] as polymorphic splitting. The concept of coinduction arose from Milner and Tofte s works [18] on semantics and type systems of an extended # calculus with references. Nielson and Nielson were the first to use coinduction as a means for specifying a static analysis [22] Their work provides the theoretical framework for the specification of our analysis. 10.3 Subject reduction The ....

R. Milner and M. Tofte. Co-induction in relational semantics. Theoretical Computer Science, 87:209--220, 1991.

....is the operator on the powerset of C such that for any set A C and tuple c 2 C, we have c 2 F(A) if and only if c is a conclusion of one of the above inference rules using only tuples in A as premises. Clearly, F is monotonic: A B implies F(A) F(B) Thus, by Tarski s xed point theorem (see [MT91]) there exists a greatest xed point for F and this greatest xed point is also the greatest set A satisfying A F(A) We therefore make the following denition: Figure 3.1 Rules dening a consistency relation Pi; oe; R; Delta v : Con Int) Pi; oe; R; Delta i : int (Con Null) Pi; oe; ....

....words, any consistent address can be updated with a consistent value of the proper type, and the address in the new store is still consistent. This is not trivial, since the update may introduce a circular reference. To show this lemma we use co induction. We try to use the notational style of [MT91] and let C denote the set of tuples ( Pi; oe; R; Delta; v; let F denote a functional version of our consistency relation, and S is the maximum x point of F. For any Q U , in order to prove Q S, it is suOEcient to prove that Q is F consistent, i.e. that Q F(Q) Given Pi; oe; R; Delta; ....

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Robin Milner and Mads Tofte. Co-induction in relational semantics. Theoretical Computer Science, 87(1):209220, September 1991. 148 BIBLIOGRAPHY 149

....(f x) La m6me mthode utilise pour dcrire des objets peut donc 6tre utilise galement pour dcrire des preuves. Dans la pratique, la construction de preuves infinis devient ainsi loutil de base pour raisonner sur des objet infinis. Le principe de co induction. Le principe de co induction [57, 73, 68] est souvent invoqua cornroe un principe de base pour montrer l galit des oh jets infinies. Ce principe postule que deux listes sont extensionnellement gales si on peut trouver une relation binaire entre listes telle que 1. soit une bisimulation. Cela signifie que si deux listes satisfont ....

....with inductive types, we can thus recognize the existence of certain proof principles associated to co inductive types, which act as introduction combinatm, which correspond to certain positive construction schema. One of these proof principles is Milner and Tofte s principle of co induction [57], a variant of David Park s principle of fixpoint induction [63] The principle of co induction (PCI) is used to prove the extensional equality of two potentially infinite lists from the existence of a certain invariant relating their construction, called a bisimulation. A bisimulation is a ....

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R. Milner and M. Tofte. Co-induction in relational semantics. Theoretical Computer Scietce, 87, 1991.

....c safe V al a [13, 43, 58] The definitions for traceC (p0 ; c) and traceA(p0 ; a) are in Section 2.3. 2. 2 Inductively and Coinductively Defined Sets The flowchart traces in the previous section can be infinite, and proofs on infinite traces are best worked with coinductive techniques [2, 54, 40], which we now review. The following presentation is summarized from Cousot and Cousot [14] We begin with the classical inductive definition. Let U be a universe of terms, and let F : P(U) P(U) be continuous with respect to the powerset lattice hP(U) i. The set defined inductively by F is ....

R. Milner and M. Tofte. Co-induction in relational semantics. Theoretical Computer Science, 17:209--220, 1992.

....test data for the meson proof procedure. These are mostly taken from Pelletier [19] File set.ML proves Cantor s Theorem, which is presented in 2.12 below, and the Schroder Bernstein Theorem. Theory MT contains Jacob Frost s formalization [4] of Milner and Tofte s coinduction example [8]. This substantial proof concerns the soundness of a type system for a simple functional language. The semantics of recursion is given by a cyclic environment, which makes a coinductive argument appropriate. 2.12 Example: Cantor s Theorem Cantor s Theorem states that every set has more subsets ....

Robin Milner and Mads Tofte. Co-induction in relational semantics. Theoretical Computer Science, 87:209--220, 1991.

....definition of the bisimilarity relation on a labelled transition system. It is defined as the greatest fixed point of a monotone function on the lattice of relations on the states of this transition system (see [Mil89] An example of the above co induction proof principle can be found in [MT91], where it is used to prove the consistency of the static and the dynamic semantics of a simple functional programming language with recursive functions. By generalizing preorders to categories C and monotone functions to functors F : C C, a coinduction principle can be obtained for recursive ....

R. Milner and M. Tofte. Co-induction in relational semantics. Theoretical Computer Science, 87:209--220, 1991.

....could be called a coinduction principle for final F coalgebras: let (A; ff) be a final F 50 coalgebra and let (B; fi) be any F coalgebra. If : B; fi) A; ff) is a mapping between F algebras and is epic (the generalization of surjective) then is an isomorphism. See also [Smy92] In [MT91], this principle is used in the basic case where the category under consideration is a lattice and the functor F a monotonic operation. At the same time, the fact that an F coalgebra (A; ff) is final implies the principle of strong extensionality (stating that on (A; ff) equality and ....

R. Milner and M. Tofte. Co-induction in relational semantics. Theoretical Computer Science, 87:209--220, 1991. 52

....of the function F : Judgments Judgments derived from Figure 4 [3] F (Q) gives us a set of left hand side judgments asserted by the rules of Figure 4 assuming that judgments in Q hold. If we find a set Q of judgments such that (S j= 2 Q and Q F (Q) then by the co induction principle [9], Q is included in the greatest fixed point of F and S j= holds. Therefore, the module variant 0CFA s solution, which is defined as the least X such that X j= is included in the modularized solution Sol 0CFA m (M 1 ; Delta Delta Delta ; M n ) The detailed proof is in [10] 2 ....

R. Milner, M. Tofte, Co-induction in relational semantics, Theoretical Comput. Sci. 87 (1991) 209--220.

....be the equality relation on visible sorts. This means that the least Gamma congruence extending R is in fact a behavioral Gamma congruence. Coinduction proofs can be considered to generalize the bisimilarity proofs used in process algebra [46] since bisimilarity is the greatest bisimulation [52, 53], one can prove that two states are bisimilar by exhibiting any bisimulation that relates them. Similarly, behavioral equivalence of two states can be shown by exhibiting any behavioral congruence that relates them. A common technique for bisimilarity proofs is to extend some relation that relates ....

Robin Milner and Mads Tofte. Co-induction in relational semantics. Theoretical Computer Science, 87(1):209--220, 1991.

....E(x) Gamma(x) for any x 2 Dom(E) The Definitions 5 and 6 above do not define the consistency relation in an inductive fashion as the cases for channels and functions refer to the consistency of semantic objects which we cannot recover from the structure of the channel or the function value. In [22, 25] Tofte, Talpin and Jouvelot discuss in detail why Definitions such as 5 and 6 should be regarded as the property of a consistency relation rather than the relation itself. We define the consistency relation as the maximal fixed point of the property stated in Definition 5 as is typically the ....

....of Proposition 1 are given in Appendix B. The method used in the proof of consistency depends on the approach used in the specification of the dynamic and static semantics. Relative merits of different methods in proving the two semantics consistent have been discussed in the related literature [12, 31, 22, 25]. Our formulation can be likened to that of [25] in the sense that we use the formalism of relational semantics in specifying the evaluation rules and the static semantics rules involve effect inference. It can also be likened to that of [12] in the sense that the formalism of relational semantics ....

R.Milner and M.Tofte. Co-induction in relational semantics. Theoretical Computer Science, 87(1):209--220, Sep 1991.

....Consistency of Dynamic and Static Semantics The method used in the proof of consistency depends on the approach used in the specification of the dynamic and static semantics. Relative merits of different methods in proving the two semantics consistent have been discussed in the related literature [23, 24, 25, 20]. Our formulation can be likened to that of [20] in the sense that we use the formalism of relational semantics in specifying the evaluation rules and the static semantics rules involve effect inference. It can also be likened to that of [23] in the sense that the formalism of relational semantics ....

....system is one of the major attractions of a typical functional programming language. Although polymorphism becomes a more delicate issue in the presence of non applicative features such as references and communication 11 channels, the literature includes satisfactory treatments of these problems [25, 23, 22, 21]. Extending our type system with polymorphism seems to be a matter of adapting to our setting the previous work by other authors. As it stands, the monomorphic effect based type system we have presented not only suggests the main idea for a sound method of detecting transmissible functions but ....

R.Milner and M.Tofte. Co-induction in relational semantics. Theoretical Computer Science, 87(1):209--220, Sep 1991.

....cumulative hierarchy. As an example, Barwise [Bar86] shows that to give a set theoretical model of shared information in certain types of real situations requires the use of sets in which membership loops do exist. Another interesting application of hypersets is given by Milner and Tofte ([MT91] and [BM86] Section 3.3) to the semantics of a simple functional language allowing recursive definitions. In [LS97] a promising hyperset view of weblike databases is presented. 4 Representing hypersets In this section we show how hypersets are introduced in the logic framework we consider in ....

R. Milner and M. Tofte. Co-induction in relational semantics. Theoretical Computer Science, 87:209--220, 1991.

....hierarchy. As an example, Barwise has shown ( 6] that to give a set theoretical model of shared information in certain types of real situations requires the 10 use of sets in which membership loops do exist. Another interesting application of hypersets has been given by Milner and Tofte ([33] and [8] x 3.3) to the semantics of a simple functional language allowing recursive de nitions. In [27] a promising hyperset view of web like databases has been presented. 4 Representing hypersets In this section we show how hypersets are introduced in the logic framework we are considering in ....

R. Milner and M. Tofte. Co-induction in relational semantics. Theoretical Computer Science, 87:209-220, 1991. 46

....notion: two expressions are observationally noncongruent if there is a finite experiment under which the two expressions give different, observable answers. It is thus not surprising that coinduction principles play an important role in proving program properties such as program equivalences [MT91, Gor95, HL95, Len96]. Indeed for many programming languages observational congruence can be characterized by a notion of bisimulation; see e.g. Mil77, Abr90] This means that, both in theory and practice, observational congruences can be proved by establishing bisimulations based directly on the operational ....

Robin Milner and Mads Tofte. Co-induction in relational semantics. Theoretical Computer Science, 87(1):209--220, 1991. Note.

....on # terms. Recently, I have applied inductive definitions to the analysis of cryptographic protocols [29] To demonstrate coinductive definitions, Frost [10] has proved the consistency of the dynamic and static semantics for a small functional language. The example is due to Milner and Tofte [18]. It concerns an extended correspondence relation, which is defined coinductively. A codatatype definition specifies values and value environments in mutual recursion. Non well founded values represent recursive functions. Value environments are variant functions from variables into values. This ....

Milner, R., Tofte, M., Co-induction in relational semantics, Theoretical Comput. Sci. 87 (1991), 209--220

....co inductive definitions, instead of inductive definitions. This would allow us to define the value of a recursively defined function to be its infinite unfolding. We could then use a single application, Rule 4.4. But then all our proofs would have to be co inductions instead of inductions, as in [46]. 28 The variable fi is an element of the set Loc, the set of memory locations. In SML, exceptions are generative: every time an exception declaration is evaluated, the constructor D is mapped to a new unique element from the set Delta, the set of exception constructor values. An exception ....

R. Milner and M. Tofte. Co--induction in relational semantics. Theoretical Computer Science, 87:209--220, 1991.

....this relation as the improvement simulation. The importance of this maximal fixed point definition is that it comes with a useful proof technique, sometimes referred to in the context of process algebra as Park Induction (in reference to [Par80] or in a more general setting as co induction ([MT90]) To show that R , for some binary relation R on T (L) ffi , it is sufficient to show that R is an improvement simulation. So, in order to show that two closed terms are related by it is sufficient to exhibit any improvement simulation which relates them. In what follows we will ....

R. Milner and M. Tofte. Co-induction in relational semantics. Theoretical Computer Science, 1990. (to appear).

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Milner, R., and Tofte, M. Co-induction in relational semantics. Theoretical Computer Science, 87:209-220, 1991.

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R. Milner and M. Tofte. Co-induction in relational semantics. Theoretical Computer Science, 87, 209--220, 1991.

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Robin Milner and Mads Tofte. Co-induction in relational semantics. Theoretical Computer Science, 87:209--220, 1991.

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Robin Milner and Mads Tofte. Co-induction in relational semantics. Theoretical Computer Science, 17:209--220, 1992.

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