M. Hofmann and D. Sannella. On behavioural abstraction and behavioural satisfaction in higher-order logic. Theoretical Computer Science, 167:3--45, 1996.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....# be a # formula and let A be a # Obs algebra. Then #Obs # if and only if # #. This theorem is a generalization of Theorem 3.11 in [10] The proof is done by induction on the form of the formula # (along the lines of the proof of Theorem 3. 11 in [10] Similar results are provided in [23] and in [33] Definition 2.13 (Basic observational specification) A basic observational specification Obs , Ax# consists of an observational signature # Obs = #,OP Obs ) and a set Ax of # sentences, called the axioms of SP Obs . The semantics of SP Obs is given by its signature Sig Obs (SP Obs ....

.... Observational Logic) introduced in [8] The (more general) observational equality relation used in this integrated approach takes into account also the constructor generated elements and hence is strongly related to the notion of partial observational equality considered e.g. in [10] and [23]. ....

M. Hofmann and D. Sannella. On behavioural abstraction and behavioural satisfaction in higher-order logic. In P.D. Mosses, M. Nielsen, and M.I. Schwartzbach, editors, Proc. 6th Int. Joint Conf. Theory and Practice of Software Development (TAPSOFT'95), Aarhus, Denmark, volume 915 of LNCS, pages 247--261. Springer, 1995.

....and let A be a Obs algebra. Then: A j= Obs if and only if BB Obs (A) j= This theorem is a generalization of Theorem 3.11 in [10] The proof is done by induction on the form of the formula (along the lines of the proof of Theorem 3. 11 in [10] Similar results are provided in [23] and in [33] De nition 13 (Basic observational speci cation) A basic observational speci cation SP Obs = h Obs ; Axi consists of an observational signature Obs = OP Obs ) and a set Ax of sentences, called the axioms of SP Obs . The semantics of SP Obs is given by its signature Sig ....

.... Observational Logic) introduced in [8] The (more general) observational equality relation used in this integrated approach takes into account also the constructor generated elements and hence is strongly related to the notion of partial observational equality considered e.g. in [10] and [23]. ....

M. Hofmann and D. Sannella. On behavioural abstraction and behavioural satisfaction in higher-order logic. In P.D. Mosses, M. Nielsen, and M.I. Schwartzbach, editors, Proc. 6th Int. Joint Conf. Theory and Practice of Software Development (TAPSOFT'95), Aarhus, Denmark, volume 915 of LNCS, pages 247-261. Springer, 1995.

....are admitted. Thus standard implementations which simply contain junk (like the realization of the natural numbers by the integers) are ruled out from the models of a speci cation. The ultra loose approach of [20] the notion of behavioral speci cation w.r.t. a partial observational equality in [5, 12] and the hidden algebra approach are closely related to our framework. The main di erence to [20] is that no explicit notion of observer or constructor operation is used there while in our approach they are the basic ingredients of a signature which lead to a speci cation methodology and to an ....

M. Hofmann and D. Sannella. On behavioural abstraction and behavioural satisfaction in higher-order logic. Theoretical Computer Science, 167:3-45, 1996.

....focused on a comparison of the two concepts and not on their integration. This is, of course, an important topic of further development. We believe that such an integration will be strongly related to (a generalization of) the notion of partial observational equivalence considered e.g. in [3] and [10]. 6 The (essence of the) diagram in definition 17 appeared as the definition of ( Omega ; Xi ) structures in [8] 16 ....

M. Hofmann and D.T. Sannella. On behavioural abstraction and behavioural satisfaction in higher-order logic. In TAPSOFT '95, volume 915 of LNCS, pages 247--261. Springer, 1995.

....equality) the behavioural satisfaction of equations. However the explicit de nition we have chosen (cf. De nition 3.2) is necessary to de ne the behavioural satisfaction of arbitrary formulas w.r.t. an arbitrary behavioural equality (this idea is even extended to higher order logic in [15]) On the other hand, this lemma suggests that, in the total observational framework, to prove the behavioural satisfaction of an equation l = r (between non observable terms of sort s) it is equivalent to prove the standard satisfaction of the in nite set of equations C[l] C[r] for all C 2 C ....

....for any class C of algebras the behavioural theory of C consists of all formulas whose lifted version L( belongs to the 4 We assume the reader to be familiar with the usual notions of predicate symbols and their interpretations. 5 Similar constructions, called relativizations, are used in [29,15]. 15 standard theory of L(C) Theorem 4.2 Let be a signature and be a behavioural equality. For any algebra A, class C of algebras, ASL like structured speci cation SP with signature (cf. Section 2) and formula , we have: i) A j= if and only if L(A) j= L( ii) C j= ....

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M. Hofmann and D. Sannella. On behavioural abstraction and behavioural satisfaction in higher-order logic. In Proc. of TAPSOFT'95, pages 247-261. Springer-Verlag L.N.C.S. 915, 1995. 55

....; if s 2 S n In. 6. For each sort s 2 S, Ax[ s ] is the in nitary sentence de ned by: Ax[ s ] def = 8x; y : s: h x s y , V c2T (X In [fz s g) 8Var(c) c[x] c[y] i if s 2 In, 23 Another approach which axiomatizes the observational equality in higher order logic is presented in [HS96]. Michel Bidoit, Mar a Victoria Cengarle, and Rolf Hennicker Ax[ s ] def = 8x; y : s: h x s y , T s (x) T s (y) V c2T (X In [fz s g) 8Var(c) c[x] c[y] i if s 2 S n In, where T (X In [ fz s g) is the set of the observable contexts, with context variable z s of sort s (see ....

....adding the predicate symbols T s and s and the axioms AX[L] to SP. Hence this extension of SP is de ned in the institution IFOLEq. This leads to the following proof rule: quot) SP hL( AX[L]i s L( SP= Obs;In s 25 Similar constructions, called relativizations, are used in [Wir91,HS96]. Michel Bidoit, Mar a Victoria Cengarle, and Rolf Hennicker where = Sig(SP) and L( AX[L] L( are de ned in Proposition 11.63. For the observational behavior operator we use the fact that Beh Obs;In and : Obs;In form a Galois connection. Hence, for any speci cation SP, we know ....

M. Hofmann and D. Sannella. On behavioural abstraction and behavioural satisfaction in higher-order logic. Theoretical Computer Science, 167, 1996.

..... 1. A j= Sigma Obs if and only if FA Sigma Obs (A) j= 2. SP Obs j= Sigma Obs if and only if [ SP Obs ] j= This theorem shows the adequacy of the black box semantics in the observational case. The theorem is a variant of Theorem 3. 11 in [4] and it is related to similar results in [10]. 5.2 Black Box Semantics of Constructor Based Specifications Let Sigma Cons be a constructor based signature. Since for any Sigma Cons algebra A, the reachable part hAi Cons of A is a Sigma algebra, which by definition contains only 11 those elements that are generated by the given ....

....of the two concepts and not on their integration. The combination of the two concepts offers a promising perspective of future reserach. We believe that such an integration will be strongly related to (a generalization of) the notion of partial observational equivalence considered e.g. in [4] and [10]. 15 ....

M. Hofmann and D.T. Sannella. On behavioural abstraction and behavioural satisfaction in higher-order logic. In TAPSOFT '95, volume 915 of LNCS, pages 247--261. Springer, 1995.

....of proceeding which we believe is applicable to many other similar frameworks. For instance, these ideas should work as well for order sorted and for error algebras. Another framework with behavioural semantics likely to t into the mould presented here is that of higher order logic, as studied in [HS95]. Of course, there also exist frameworks where some of our basic assumptions are not satis ed, and some expected facts and constructions break down. As shown in [BT95b] this is the case for instance for continuous algebras with some relief we have realised that for regular algebras, which may ....

Hofmann, M., Sannella, D. On behavioural abstraction and behavioural satisfaction in higher-order logic. Proc. 6th Intl. Conf. CAAP/FASE Theory and Practice of Software Development TAPSOFT'95, Springer LNCS 915, 247-261, 1995.

.... (C) Th L (L(C) Moreover, Th L (L(C) Th(C) because for any fully abstract algebra A, A j= if and only if L(A) j= L( Hence we obtain in this particular case again our previous result Th (C) Th(C) 6 Similar constructions, called relativizations, were used in [12] and recently in [6]. Theorem 15 provides a means for reducing the set of behavioural theorems over an arbitrary class C of algebras to a set of standard theorems over a corresponding class L(C) of L( algebras. 7 However, the usefulness of this reduction still depends on the possibility to perform proofs of ....

.... conditions) Then we have: Th (C) Th (Abs (C) Th (Beh (C) Th(FA (C) 5 Axiomatization of the Behavioural Equality In the previous section we have shown how to replace the behavioural theory of some given class C of algebras by (a subset of) the standard theory of another 7 In [6] it is shown how a similar result for expressible behavioural equalities can be obtained in the framework of higher order logic. related class of algebras. The next step is to provide a characterization of this class of algebras in terms of an axiomatization of the behavioural equality. De ....

M. Hofmann and D. Sannella. On behavioural abstraction and behavioural satisfaction in higher-order logic. In Proc. of TAPSOFT'95. Springer-Verlag L.N.C.S., 1995. To appear.

....way by which quotienting is dealt with is by introducing a predicate symbol and explicitly axiomatising the congruence in terms of that symbol [20] This also goes for the behavioural equalities viz. congruences dealt with in [7] where the axiomatisations are in general infinitary, although in [9] this problem is taken to higher order logic and finitary axiomatisation is then possible. Our approach is beneficial to mechanised reasoning because it remains finitary, first order and purely equational. In some cases it also allows one to do behavioural verification more directly because now we ....

M. Hofmann and D. Sannella. On behavioural abstraction and behavioural satisfaction in higher-order logic. Theoretical Computer Science, 167:3--45, 1996.

....( Prof SP ) We must derive 9B:9b:T [B] packBb) u Phi[B; b] assuming 9A:9a:T [A] packAa) u Phi 0 [A; a] Let a and A denote the witnesses projected out from that assumption. Now, Beh is in general infinitary. However, with higher order logic one gets a finitary Beh equivalent to Beh [12]. Thus we form type theoretically by def = a: A; a 0 : A) Beh A (a; a 0 ) Since is an axiomatisation of a partial congruence, we have a T [ a. We use Sub to get SA , s a and 0 ae SA Theta SA and mono:SA A s.t. we can derive (s1) a (T [ a: A; s: SA ) a =A (mono s) s a (s2) ....

M. Hofmann and D. Sannella. On behavioural abstraction and behavioural satisfaction in higher-order logic. Theoretical Computer Science, 167:3--45, 1996.

....of the same abstract type signature observationally indistinguishable Even for abstract types with first order signatures, these questions present a significant challenge. There is already a substantial body of work in this area from the algebraic specification community (see e.g. [TZ88, BT95, GM82, BHW95, HS96]) However, as far as we are aware, there has so far been very little work on truly denotational models for abstract types that capture notions of computability and behavioural equivalence (as achieved for simply typed languages by the models in [Plo77, AJM96] for example) We will seek natural ....

M. Hofmann and D. Sannella. On behavioural abstraction and behavioural satisfaction in higher-order logic. Theoretical Comp. Sci., 167, 1996.

....derived functions (or into methods and attributes for the object paradigm) and reduces the need for complicated inductive proofs. In [644] it is shown that this proof technique is an example of coinduction (cf. the comments on final and behavioural semantics in Section 2. 1) Hofmann and Sannella [502, 500] have studied behavioural abstraction and behavioural satisfaction in the context of specifications with higher order logical formulae as axioms. The central result is a generalization of [113] to this context, where the role of equality in the context of behavioural semantics is revealed as ....

M. Hofmann and D. T. Sannella. On behavioural abstraction and behavioural satisfaction in higher-order logic. In P. D. Mosses, M. Nielsen, and M. I. Schwartzbach, editors, TAPSOFT'95: Theory and Practice of Software Development. 6th International Joint Conference CAAP/FASE, pages 247--261. Springer LNCS 915, 1995. \Phi.

....specification. 113] provides necessary and sufficient conditions for the semantical equivalence of both kinds of specifications and shows that behavioral specifications can be characterized by an abstractor construction and vice versa. These results are generalized to higher order logic in [501] and to an even more general setting based on institutions in [115] Constraints In some approaches, restrictions to the class of models of a given specification are given explicitly using socalled constraints. A constraint is a semantic construction fixing or restricting the interpretation ....

M. Hofmann and D. T. Sannella. On behavioural abstraction and behavioural satisfaction in higher-order logic. In P. D. Mosses, M. Nielsen, and M. I. Schwartzbach, editors, TAPSOFT'95: Theory and Practice of Software Development. 6th International Joint Conference CAAP/FASE, pages 247--261. Springer LNCS 915, 1995. \Phi.

....derived functions (or into methods and attributes for the object paradigm) and reduces the need for complicated inductive proofs. In [644] it is shown that this proof technique is an example of coinduction (cf. the comments on final and behavioural semantics in Section 2. 1) Hofmann and Sannella [502, 500] have studied behavioural abstraction and behavioural satisfaction in the context of specifications with higher order logical formulae as axioms. The central result is a generalization of [113] to this context, where the role of equality in the context of behavioural semantics is revealed as ....

M. Hofmann and D. Sannella. On behavioural abstraction and behavioural satisfaction in higherorder logic. Theoretical Computer Science, 1996. To appear, \Phi.

....of quantification and logical equality. It is first necessary to show that there is a satisfactory relationship between what this would yield and the behavioural equivalence relation used for the foundations of formal development, following [BHW94] a first step in this direction is taken in [HS95]. The next major step in work on EML is to develop a sound proof theory, which would provide the user with some formal proof rules and proof tactics to verify the correctness conditions arising in the process of program development. Given the complexity of SML and hence of EML, it may be ....

M. Hofmann and D. Sannella. On behavioural abstraction and behavioural satisfaction in higher-order logic. Proc. 6th Intl. Joint Conf. on Theory and Practice of Software Development, Aarhus. Springer LNCS 915, 247--261 (1995).

....equality) the behavioural satisfaction of equations. However the explicit definition we have chosen (cf. Definition 2) is necessary to define the behavioural satisfaction of arbitrary Sigma formulas w.r.t. an arbitrary behavioural equality (this idea is even extended to higher order logic in [12]) On the other hand, this lemma suggests that, in the total observational framework, to prove the behavioural satisfaction of an equation l = r (between non observable terms) it is equivalent to prove the standard satisfaction of the infinite set of equations C[l] C[r] for all C 2 C Obs ....

....lemmas: Lemma25. Let A be a Sigma algebra and OE be a Sigma formula. For all valuations ff : X Dom( A ) the following conditions are equivalent: 1. A; ff j= OE 2. L(A) ff j= OE 3. L(A) ff j= L(OE) 8 Similar constructions, called relativizations, were used in [23] and recently in [12]. Behavioural Theories and The Proof of Behavioural Properties 16 Proof. Since A and L(A) have the same carrier sets, a valuation from X to Dom( A ) is a special case of a valuation from X to L(A) Moreover, for any valuation ff : X Dom( A ) and any variable y of sort s, we have L(A) ff j= D ....

M. Hofmann and D. Sannella. On behavioural abstraction and behavioural satisfaction in higher-order logic. In Proc. of TAPSOFT'95. Springer-Verlag L.N.C.S., 1995. To appear.

....to as indistinguishability, written #. In both of the latter two sections, the power of System F would allow the opposite approach to be taken. Then the question of the relationship between the resulting definitions arises. This question has been investigated in a number of simpler frameworks in [BHW95,HS96,Kat03], where the connection is given by a factorisability result of the form i# A # # B #. It is likely that the same applies in the context of System F, and this might in turn help to shed light on the relationship between the semantic and syntactic worlds. 5.3 Finiteness Pre logical ....

M. Hofmann and D. Sannella. On behavioural abstraction and behavioural satisfaction in higher-order logic. Theoretical Computer Science 167:3--45 (1996).

....higher order parameterised programs. Second, we can use higher order logic in axioms. Nothing above depends on the choice of the language of axioms, but the details of the treatment of behavioural equivalence is sensitive to this choice. The treatment in [BHW95] extends smoothly to this case, see [HS96]. Finally, we can allow higher typed functions in signatures and algebras. Again, the only thing that depends on this is the details of the treatment of behavioural equivalence. Behavioural equivalence of such algebras is characterized by existence of a so called pre logical relation between them ....

....Acknowledgements: Hardly any of the above is new, and all of it is the product of collaboration. Thanks to Martin Wirsing for starting me off in this direction in [SW83] to Andrzej Tarlecki (especially) for close collaboration on most of the remainder, to Martin Hofmann for collaboration on [HS96] and to Furio Honsell for collaboration on [HS99] Finally, thanks to the LOPSTR 99 organizers for the excuse to visit Venice. ....

M. Hofmann and D. Sannella. On behavioural abstraction and behavioural satisfaction in higher-order logic. Theoretical Computer Science 167:3--45 (1996).

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M. Hofmann and D. Sannella. On behavioural abstraction and behavioural satisfaction in higher-order logic. Theoretical Computer Science, 167:3--45, 1996.

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M. Hofmann and D. Sannella. On behavioural abstraction and behavioural satisfaction in higher-order logic. Theoretical Computer Science, 167:3--45, 1996.

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Martin Hofmann and Donald Sannella. On behavioural abstraction and behavioural satisfaction in higher-order logic. Theoretical Computer Science, 167:3-45, 1996.

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Martin Hofmann and Donald Sannella. On behavioural abstraction and behavioural satisfaction in higher-order logic. Theoretical Computer Science, 167:3-45, 1996.

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Martin Hofmann and Donald Sannella. On behavioural abstraction and behavioural satisfaction in higher-order logic. Theoretical Computer Science, 167:3-45, 1996.

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Martin Hofmann and Donald Sannella. On behavioural abstraction and behavioural satisfaction in higher-order logic. Theoretical Computer Science, 167:3--45, 1996.

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