M. Walicki and S. Meldal. A complete calculus for the multialgebraic and functional semantics of nondeterminism. ACM Trans. Program. Lang. Syst., 17:366-393, 1997.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....e.g. Hus 92, Nip 86, Hes 88] Others also observe that nondeterminism is an extremely useful abstraction concept, and should be studied from this perspective. Thus nondeterminism may be helpful even in situations when the nal goal the program to be speci ed and developed is deterministic [WM 95] One may say that in the latter case nondeterminism is used in place of underspeci cation. The most common models of nondeterministic speci cations are multialgebras. Multialgebras di er from ordinary algebras in that they have set valued functions. In [Hus 92] and [WM 95] two logics ....

....is deterministic [WM 95] One may say that in the latter case nondeterminism is used in place of underspeci cation. The most common models of nondeterministic speci cations are multialgebras. Multialgebras di er from ordinary algebras in that they have set valued functions. In [Hus 92] and [WM 95] two logics with complete reasoning systems are presented; however, completeness is achieved only at the price of introducing certain additional limitations on the model level. Both are calculi with (at least) an inclusion operator, but the completeness results are obtained for restricted model ....

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Walicki, M., Meldal, S. (1995): A Complete Calculus for Multialgebraic and Functional Semantics of Nondeterminism. ACM TOPLAS, 17, 2, 366-393.

....ers the left bound q of the annotation is strictly smaller than the right bound q we can attain arbitrarily small, user de ned, error. 16 9 Related Work Languages for modeling nondeterminism in various other forms have been introduced within the frame of abstract data types see for example [44, 43, 22, 28]. There one uses nondeterministic speci cations to either model nondeterminism occurring in reality, or to abstract away unnecessary details of the behavior of a real or desired system; these details might be speci ed later. Also in analysis nondeterminism has been modeled in order to deal with ....

M. Walicki and S. Meldal. A complete calculus for the multialgebraic and functional semantics of nondeterminism. ACM Transactions on Programming Languages and Systems, 17(2), 1995.

....and elegant complete deduction system for the language using the R S method. In fact, this application shows a decided advantage of the R S methodology in the sense of its simplicity and power, since the only two complete deduction systems of the more widespread types presented earlier in [Hu92, WM95] for such nondeterministic speci cation logics were based on rather unnatural restrictions on the model level whereas the R S system presented here is complete for the class of arbitrary models. Now let us introduce some basic notions. Let = S; F ] be an algebraic signature, where S is the ....

Walicki, M., Meldal, S., A complete calculus for multialgebraic and functional semantics of nondeterminism, ACM TOPLAS, 17(2), 1995.

....c : s ffl an operation A : s A 1 Theta Delta Delta Delta Theta s A k P(s A ) for each symbol : s1 Theta Delta Delta Delta Theta sk s 2 Omega Operations are defined on sets by pointwise extension. 3 One sometimes demands that constants and operations are total [24, 23], i.e. never return empty set and take values only in P (s A ) the nonempty subsets of s A . We will not make this assumption. Note that for a constant c 2 Omega Gamma c A denotes a (sub)set of the carrier s A . This will allow us to use constants as predicates. As homomorphisms of ....

....the multialgebras form an institution MA. Its formal definition and the proof of the fact that it is an institution are postponed to section 5. We also recall that there exist sound and complete calculii for multialgebraic specifications: for multialgebras without operations returning empty set [23, 24], and for the ones admitting empty result sets (like in 2.3 above) 3] 4 3 Partiality handling with multialgebras The straightforward way to model partiality in a multialgebra A is to let an operation undefined on some arguments, A (a1 ; an ) be represented by a multi function ....

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Micha/l Walicki and Sigurd Meldal. A complete calculus for the multialgebraic and functional semantics of nondeterminism. ACM TOPLAS, 17(2), 1995.

....specifications, many authors have appreciated the value of nondeterminacy on the one hand, and refinement at the level of expressions on the other. Unfortunately, a satisfactory combination of both features has proved elusive. For various approaches see [Par90] NH93] LH96] Rai92] Rai95] [WM95], and [War94] The language of the CIP development method, CIP L [Par90] has constructs similar to our choice, equivalence (called strong equality ) refinement ( descendancy ) and Delta. However, its nondeterminacy is erratic ( is not a zero of choice) rather than demonic, which is ....

....also contains nondeterministic expressions, but most of the proof rules given in [Rai95] do not apply to them. In contrast, our work provides the logical basis for nondeterminacy at the level of expressions, and potentially could be used as the logic underlying RAISE. Walicki and Meldal s work [WM95] treats nondeterministic operators in the context of algebraic specifications. It gives completeness results with respect to a set based semantics and a computational semantics, in which specifications of nondeterministic functions are transformed to under specifications of deterministic ....

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M. Walicki and S. Meldal. A complete calculus for the multialgebraic and functional semantics of nondeterminism. ACM Transactions on Programming Languages and Systems, 17:366--393, 1995. Joseph M. Morris and Alexander Bunkenburg A. Some theorems that follow from the axioms of

....as set inclusion. In spite of such limited means (no logical connectives at all ) there is a concise R S deduction system for the logic. This shows a decided advantage of the R S formalism in this case, since the only two other complete deduction systems for similar kind of logics presented in [6, 13] assumed rather unnatural restrictions on the model level. Let = S; F ] be an algebraic signature, where S is the set of sorts, and F the set of function symbols, and let X be an S sorted set of variables. A multialgebra over is a tuple A = S A ; F A ] where S A = fA s g s2S is ....

....identity valuation with Dom(v) S fX s : H s 6= g; and v(x) 0 x 0 for x 2 Dom(v) It is easy to check that H is indeed a counter model for . To end the proof, it remains to show that MH is a counter model for too. This is done in a standard way (see again the proof of Thm. 1 and [13]) by induction on the rank of a formula. Q.E.D. 8 Conclusions From the examples presented above it clearly follows that Rasiowa Sikorski deduction mechanism is indeed well suited to many applications in software sepci cation and AI logics, allowing us to develop complete systems for various ....

Walicki, M., Meldal, S., A complete calculus for multialgebraic and functional semantics of nondeterminism, ACM TOPLAS, 17(2), 1995.

....operations is quite well understood; on the other hand many other approaches stick to standard terms and address the problems sketched above directly in the design of the logical system. As a typical example, substitutivity is only admitted for deterministic terms in the calculi proposed, e.g. in [28,42] and in the variation of term rewriting discussed in [23] In the rest of this section we will sketch how the framework set up in Section 6 can be applied to multi algebras, showing that term graphs play for multi algebras the same r ole that standard terms play for total algebras. We think that ....

....which satisfy the inequality. We leave as a topic for future work the formal de nition of this calculus of inequalities for multi algebras, as well as the comparison between the expressive power of term graph inequalities and other speci cation techniques developed for multi algebras [41,42]. Let us conclude this section with the sketch of the proof of Theorem 7.1, that we are now able to present having introduced term graphs. Proof outline of the last statement of Theorem 7.1. We have to show that category C Term( of conditioned terms is isomorphic to the g monoidal theory ....

M. Walicki and S. Meldal. A complete calculus for the multialgebraic and functional semantics of nondeterminism. ACM Trans. Program. Lang. Syst., 17:366-393, 1997. 32

....algebras here. Many extensions of speci cation techniques to nondeterminism are non intrusive or conservative, namely, they do not change the existing framework for deterministic speci cations, and in particular they reduce to the standard theory when only deterministic operations are considered [WM97b] Possibly, the success of multi algebras in the algebraic speci cation community (with respect to power algebras, but also with respect to (heterogeneous) relations, which are essentially the same as multi algebras) can be explained in this perspective by the fact that they are closer to ....

....is quite well understood; on the other hand many other approaches stick to standard terms and address the problems sketched above directly in the design of the logical system. As a typical example, substitutivity is only admitted for deterministic terms in the calculi proposed, e.g. in [WM97b,KB99] and in the variation of term rewriting discussed in [Hus92] We understand that there are many good reasons for refraining from using terms with sharing in the formal speci cation of nondeterministic systems: they are intrinsically more complex than standard terms; they force one to depart ....

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M. Walicki and S. Meldal. A complete calculus for the multialgebraic and functional semantics of nondeterminism. ACM Trans. Program. Lang. Syst., 17:366-393, 1997. 16

....be allowed to apply a deterministic operation to the result of a non deterministic term because of type conflicts. An interesting approach taking care of these problematics is the multialgebraic semantics extending conservatively the traditional algebraic semantics that has been developed in [927, 928, 929] and related to alternative semantics in [927, 924, 925, 664] Variants of the multialgebraic semantics have been studied with the restriction to generated models [928] and for the plural (run time choice) semantics of parameters [931] 929] The above works focus also on the logic of ....

....of a non deterministic term because of type conflicts. An interesting approach taking care of these problematics is the multialgebraic semantics extending conservatively the traditional algebraic semantics that has been developed in [927, 928, 929] and related to alternative semantics in [927, 924, 925, 664]. Variants of the multialgebraic semantics have been studied with the restriction to generated models [928] and for the plural (run time choice) semantics of parameters [931] 929] The above works focus also on the logic of nondeterminism addressing in particular the problem of restricted ....

M. A. Walicki and S. Meldal. A complete calculus for the multialgebraic and functional semantics of nondeterminism. ACM Transactions on Programming Languages and Systems, 17(2):366--393, 1995. \Phi.

....as set inclusion. In spite of such limited means (no logical connectives at all ) there is a concise R S deduction system for the logic. This shows a decided advantage of the R S formalism in this case, since the only two other complete deduction systems for similar kind of logics presented in [6, 13] assumed rather unnatural restrictions on the model level. Let Sigma = S; F ] be an algebraic signature, where S is the set of sorts, and F the set of function symbols, and let X be an S sorted set of variables. A multialgebra over Sigma is a tuple A = S A ; F A ] where S A = ....

....valuation with Dom(v) S fX s : H s 6= g; and v(x) 0 x 0 for x 2 Dom(v) It is easy to check that H is indeed a counter model for Delta. To end the proof, it remains to show that MH is a counter model for Omega too. This is done in a standard way (see again the proof of Thm. 1 and [13]) by induction on the rank of a formula. Q.E.D. 8 Conclusions From the examples presented above it clearly follows that Rasiowa Sikorski deduction mechanism is indeed well suited to many applications in software sepcification and AI logics, allowing us to develop complete systems for various ....

Walicki, M., Meldal, S., A complete calculus for multialgebraic and functional semantics of nondeterminism, ACM TOPLAS, 17(2), 1995.

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Micha#l Walicki and Sigurd Meldal. A complete calculus for the multialgebraic and functional semantics of nondeterminism. ACM TOPLAS, 17(2), March 1995.

....= fhy; xi : hx; yi 2 OEg. 2 Multialgebras Our interest in relational structures originates from earlier studies of multialgebras [4, 5, 10, 13, 17, 21, 27, 30, 32] which provide means for modeling nondeterminism in the context of algebraic specification of abstract data types [13, 17 19, 31 33] Multilagebras can be viewed as relational structures with a specific composition of relations of arbitrary arities. According to definition 1.2, relations are viewed as set valued functions where the last, n th argument corresponds to an element of the result set obtained by applying the ....

Walicki, M. and Meldal, S. A complete calculus for the multialgebraic and functional semantics of nondeterminism. ACM ToPLaS 17, 2. (1995).

....one, we are striving for a semantic system which allows us to state its properties (and consequences of use) in a similarly straightforward and uncomplicated manner. The presentation focuses on the concepts and general phenomena rather than technicalities. More technical details can be found in [7, 5]. In particular, the issue of formal reasoning with the specifications of nondeterminsm is not addressed at all the reader is referred to [6, 1] 2 UNDERSPECIFICATION The choice operation may serve as a (paradigmatic) example: assume that we want an operation t : E Theta E E which, for ....

....even intuitively clear that the only way of selecting elements is by picking them arbitrarily. The importance of our approach consists in that it offers the means for both modelling such operations and reasoning about them. Although we do not discuss reasoning systems here, the reader may consult [6, 5, 1]. This reasoning preserves then the appropriate level of abstraction addressing only the properties which emerge under arbitrary choices or else under arbitrary deterministic implementations. 4 CONCLUSIONS We have argued that nondeterminism provides adequate means for specifying operations on ....

Micha/l Walicki and Sigurd Meldal. A complete calculus for the multialgebraic and functional semantics of nondeterminism. ACM TOPLAS, 17(2), 1995.

....of nondeterministic behavior [5, 21, 20] A nondeterministic operation returns the set of all possible outcomes, so it is interpreted as a function from the carrier to the powerset of the carrier. We follow, and in few places generalize, the definitions of multialgebras from earlier works like [20, 19]. We summarize earlier results on multialgebras in section 2 leading to the fact that they form an exact institution [15] This result underlies the study of parameterized specifications and specification of parameterized data types, reported in [8, 9] it will be used as the current paper ....

...., where P(s ) denotes the power set of s . Composition of operations is defined by pointwise extension, i.e. f (g (x) y#g A (x) f (y) The disjoint union of the carrier set(s) of a multialgebra A is denoted by A . One sometimes demands that constants and operations are total [21, 19], i.e. never return the empty set we will not make this assumption. An operation is total if it returns nonempty result set for all arguments (a partial operation returns empty result set for some arguments) An operation returning not more than one value for any argument is deterministic (a ....

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Micha#l Walicki and Sigurd Meldal. A complete calculus for the multialgebraic and functional semantics of nondeterminism. ACM TOPLAS, 17(2), March 1995.

....The specific feature of the calculus which we will study is the accommodation of such a restricted substitutivity. In section 2 we introduce a sound and complete calculus NEQ for the multialgebraic semantics of nondeterminism (originally from [Wal93] and whose variants were studied in [WM95a] [WM95b]) Using techniques of [Kan63] Pli71] Pli73] and their Department of Mathematical Logic, Vilnius Institute of Mathematics and Informatics, Lithuania, fregis, jurates, aidag ktl.mii.lt Department of Informatics, University of Bergen, 5020 Bergen, Norway, michal ii.uib.no The problem ....

....since variables are assigned only individual elements, we do have NEQ 7 x = x for any variable x. Consequently, since terms may denote sets of individuals, unrestricted substitutivity would not be sound wrt. the multialgebraic semantics. For motivations and examples the reader is referred to [WM95a, WM95b, WB95]. Theorem 2.5 [WM95a] NEQ is sound and complete with respect to the multialgebraic semantics, i.e. for any sequent S: MMod( Sigma; A) j= S ( A; NEQ S A; NEQ S, indicating that S is derivable from A using the rules of NEQ, might be written in a more standard fashion as A S. However, ....

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M. Walicki, S. Meldal, A Complete Calculus for the Multialgebraic and Functional Semantics of Nondeterminism, ACM ToPLaS, vol. 17, no. 2, (1995).

....of the introduced rewriting technique. 1 Introduction Reasoning with sets becomes an important issue in different areas of computer science. Its relevance can be noticed in constraint and logic programming e.g. SD86, DO92, Jay92, Sto93] in algebraic approach to nondeterminism e.g. [Hus93, Hes88, WM95], in term rewriting e.g. LA93, Kap88, Hus93] Our interest in the set concepts originates from an earlier study of specifications of nondeterministic operations. Such operations are naturally modelled as set valued functions. The semantic structures serving this purpose multialgebras ....

.... structures serving this purpose multialgebras generalize the traditional algebras allowing operations which, for a given argument, return not necessarily a single value but a set of values (namely, the set of all possible values returned by an arbitrary application of the operation) In [WM95, Wal93] we defined a specification language using set relations and its multialgebraic semantics. The set relations we considered were: inclusion, intersection and identity of 1 element sets. The first two are the usual set relations. Inclusion allows one to define set equality which, for that reason, is ....

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M. Walicki, S. Meldal. A Complete Calculus for Multialgebraic and Functional Semantics of Nondeterminism. [to appear in ACM ToPLaS, (1995).]

....nondeterminism, completeness, ordered superposition 1 Introduction Reasoning with sets becomes an important issue in different areas of computer science. Its relevance can be noticed in constraint and logic programming e.g. SD86, DO92, Jay92] in algebraic approach to nondeterminism e.g. [Hus93, Hes88, WM95b], in term rewriting e.g. LA93, Kap88, Hus93] The set relations we are considering are not congruences not even equivalences: singleton identity is not reflexive, inclusion is not symmetric, and nonempty intersection is not transitive. We study the rewriting proofs in the presence of these ....

.... Both authors gratefully acknowledge the financial support received from the Norwegian Research Council. In an earlier paper [KW94] we proved ground completeness of reasoning system for these three set relations. Their use originated from the study of specification of nondeterminism in [Wal93, WM95a, WM95b]. The reader is referred to these works for more detailed motivation and background. In the present paper we extend rather than apply these results to the non ground case. In a standard completeness proof one can utilize completeness of the ground case by establishing a lifting lemma. It ....

M. Walicki, S. Meldal. A Complete Calculus for Multialgebraic and Functional Semantics of Nondeterminism. ACM Trans. on Programming Languages and Systems, vol. 17, No. 2, pp. 366--393 (1995).

....also its technical price. While underspecification fits into the concepts of classical logic and model theory smoothly, the treatment of nondeterminism leads to complications. Classical concepts of logical calculi such as substitutivity might no longer be valid. This was treated in full detail in [WM95a], WM95b] and will be taken into account in the following. Of course, we may (and in this paper we do) work with a specification method that supports both nondeterminism and underspecification. Since, on the one hand, underspecification is a simple, commonly used abstraction mechanism in ....

.... Gamma E3) However, implies SEQ(E) j= 9s j fflxy 9h : P fst (h) h(s) x, from which we can conclude that SEQ(E) j j= 9h : P fst (h) h( fflxy] x. 4. 4 Oracles In the context of algebraic specifications, the oracle semantics of nondeterminism is studied, for instance, in [Wal93] [WM95a]. It models a nondeterministic operation by a function with additional parameter called oracle or index. A possible result of N f (x) corresponds to picking a particular value i of the oracle argument and evaluating f(i; x) N f (s) t = 8i : f(i; s) t N f (s) t = 9i : ....

M. Walicki, S. Meldal. A Complete Calculus for Multialgebraic and Functional Semantics of Nondeterminism. To appear in ACM ToPLaS, (1995).

....of the introduced rewriting technique. 1 Introduction Reasoning with sets becomes an important issue in different areas of computer science. Its relevance can be noticed in constraint and logic programming e.g. SD86, DO92, Jay92, Sto93] in algebraic approach to nondeterminism e.g. [Hus93, Hes88, MW93], in term rewriting e.g. LA93, Kap88, Hus93] Our interest in the set concepts originates from an earlier study of specifications of nondeterministic operations. Such operations are naturally modelled as set valued functions. The semantic structures serving this purpose multialgebras ....

.... structures serving this purpose multialgebras generalize the traditional algebras allowing operations which, for a given argument, return not necessarily a single value but a set of values (namely, the set of all possible values returned by an arbitrary application of the operation) In [MW93, Wal93] we defined a specification language using set relations and its multialgebraic semantics. The set relations we considered were: inclusion, intersection and identity of 1 element sets. The first two are the usual set relations. Inclusion allows one to define set equality which, for that reason, is ....

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S. Meldal, M. Walicki. A Complete Calculus for Multialgebraic and Functional Semantics of Nondeterminism. Submitted to ACM TOPLAS, (1993).

....allowing nondeterminism is quite distinct from underspecification. The latter, though admitting 1 distinct models, still insists on a function always returning a particular value, given a particular list of arguments. In the case of nondeterministic operators this is no longer true. In [9, 10] we presented a function oriented view of nondeterministic operators. The result of a (non )deterministic operator is considered a single (rather than set) value, namely the result of a particular application. Distinct occurrences of a particular term each denote a single value, but not ....

....occurrences of a particular term each denote a single value, but not necessarily the same one for each. For instance, specifying an operator t:s which chooses nondeterministically an element from the set s, one needs to say that for any application i, t i :f0; 1g = 0or t i :f0; 1g = 1. In [9,10] it is shown how viewing nondeterministic operations as determinisitc functions dependent on an additional index (application) argument, allows one to translate nondeterministic specifications into deterministic ones using the language of conditional equations extended with disjunction. Then we ....

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M. Walicki, S. Meldal, A complete calculus for the multialgebraic and functional semantics of nondeterminism, (submitted for publ.) 1993. 15

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M. Walicki and S. Meldal. A complete calculus for the multialgebraic and functional semantics of nondeterminism. ACM Trans. Program. Lang. Syst., 17:366-393, 1997.

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