J. Goguen and J. Meseguer. Models and Equality for Logical Programming. In G. Levi H. Ehrig, R. Kowalski and U. Montanari, editors, Proc. TAPSOFT'87, volume 250 of Lecture Notes in Computer Science, pages 1--22. Springer-Verlag, Berlin, 1987. Volume 2.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....follows the same general principle: P1) there is an underlying logic in which all basic constructs features of the language can be rigorously explained. This intimate relationship between the language and its underlying logic was called logical programming by Goguen and Meseguer in [20]. In providing the semantics for CafeOBJ we distinguish between four levels of the language: programming in the small , programming in the large , programming in the huge , and environment. Programming in the small refers to collections of program statements (as obtained by ....

Joseph Goguen and Jose Meseguer. Models and equality for logical programming. In Hartmut Ehrig, Giorgio Levi, Robert Kowalski, and Ugo Montanari, editors, Proceedings, 1987.

....semantics of pure logic programs is described by the notions of the least Herbrand model and logical consequences. In equational logic programming the declarative semantics of a program is described as the set of all equalities which are derivable by equational reasoning (see, for instance, GM87] If the rewrite relation of a term rewriting system is confluent (as usually required for equational logic programs) then the application of rewrite steps is equivalent to equational reasoning. Therefore, we use the following specialized definition of completeness. Narrowing is complete w.r.t. ....

J.A. Goguen and J. Meseguer. Models and Equality for Logical Programming. In Proc. of the TAPSOFT '87, pp. 1--22. Springer LNCS 250, 1987.

....terms are required. In the goal member(E,append( 1] 2] the second argument is equal to the list [1,2] and therefore the two answers to this goal are E=1 and E=2. This kind of amalgamated language is known as logic programming with equality and has a clearly defined declarative semantics [50, 71, 106]. It is similar to the well known Horn clause logic [83] but with the difference that the equality predicate = is always interpreted as the identity on the carrier sets in all interpretations. Therefore we omit the details of the declarative semantics in this survey. The definition of the ....

J.A. Goguen and J. Meseguer. Models and Equality for Logical Programming. In Proc. of the TAPSOFT '87, pp. 1--22. Springer LNCS 250, 1987.

....syntax and axioms, and give only a brief impression of declarative semantics. Operational semantics is not discussed at all. Details are given elsewhere [75, 60] 5. 1 Syntax of order sorted logic We start with an exposition of order sorted logic, following the expositions by Goguen and Meseguer [19, 21, 23]. An order sorted signature Sigma = S; F; P) consists of the following sets: ffl A partially ordered set (S; of sort names. We use w as a metavariable over S (strings of sort names) and extend pointwise to strings in S of equal length. Strings over S are written as s 1 Theta Delta ....

....x a variable of sort s, then 8x : s : OE is a formula. The operators , and 9 can be introduced as usual by definition. A formula is called closed if each variable in it is bound by some quantifier. For a treatment of the semantics of L Sigma (X) the reader is referred to Goguen and Meseguer [21, 23]. In particular, it is shown there that specifications over the Horn clause fragment of L Sigma (X) that contains only positive conditional formulas of the form OE with OE and conjunctions of positive atomic formulas) have an initial semantics. 5.2 Syntax of order sorted dynamic database ....

J.A. Goguen and J. Meseguer. Models and equality for logical programming. In H. Ehrig, R. Kowalski, G. Levi, and U. Montanari, editors, Proceedings of the International Joint Conference on Theory and Practice of Software Development (TAPSOFT'87) Volume 2, pages 1--22. Springer, 1987. Lecture Notes in Computer Science 250.

.... extention of programming language (Lisp [HS86] Forth [TF] ffl syntax and semantic design and development of languages (Kaleidoscope [LFB93] AKL [HJ93] Life [AKP93] FP LV [DGK93] Falcon [GY93] NUT [MT92, PT93] ffl research of different formalizations (Horn clause logic with equality [GM87], Linear logic [DGK93] OSF logic algebra [AKP93] Here a constructive base of this approarch is extended and systematized. It is formed on the basis of stratification, taking into account the heterogenity of knowledge forms in CSM. What is a basic set of constructive elements What ....

Goguen J.A., Meseguer J. Models and equality for logical programming. Proc. of TAPSOFT, 250 of Lect. Notes in Comp. Sci., 87, Springer Verlag, 1987, pp.1--22

....out applications. 4. 1 Induction Principles for Membership Equational Theories Given that membership equational logic is a subset of equational Horn logic (indeed, they can be shown to be equivalent [29] it follows immediately that any theory( E) has a unique (up to isomorphism) initial model [19]. The following is an induction principle for reasoning about properties of sorts, with respect to this model. De nition 2 (Induction over sort de nitions) Let T = E) be a theory in membership equational logic and let s be a sort in some S k . Let C [T;s] fC 1 ; Cn g be those ....

J. Goguen and J. Meseguer. Models and equality for logical programming. In H. Ehrig, G. Levi, R. Kowalski, and U. Montanari, editors, Proceedings TAPSOFT'87, volume 250 of Lecture Notes in Computer Science, pages 1-22. Springer-Verlag, 1987.

....techniques are discussed. 8.3 Further Research 156 Tools The adequacy of OBJ for implementing prototypes of simulation tools has been discussed and confirmed in [85] and [75] Here we follow a similar approach. In [12] the Eqlog language (an extension of OBJ with logic programming features) [37, 38] has been used in an interesting way to illustrate how programs written in a subset of FOOPS can be simulated, following the sheaf semantics described in [12] OBJ is a quite effective tool for formal reasoning, as reported in several places [26, 30, 21] Ideally, we should use OBJ for formally ....

Joseph Goguen and Jos'e Meseguer. Models and equality for logical programming. In Hartmut Ehrig, Giorgio Levi, Robert Kowalski, and Ugo Montanari, editors, Proceedings, 1987 TAPSOFT, pages 1--22. Springer, 1987. Lecture Notes in Computer Science, Volume 250.

....120, 24, 96, 97, 126] Formal Methods, Specification and Verification, and Parameterized Programming Methodology [96, 119, 65, 50, 66, 121] 2. Declarative Specification and Programming Languages, including: Equational Languages (OBJ) 65, 49, 85, 66] Relational and Equational Languages (Eqlog) [61, 62]; Multiparadigm Concurrent Object Oriented Languages (FOOPS and Maude) 63, 132, 121, 120, 122, 131, 89] and Reflective Languages [32, 30] In 1990 Dr. Meseguer proposed rewriting logic as a new specification and programming formalism for concurrent and distributed systems. Since that time, more ....

Joseph Goguen and Jos'e Meseguer. Models and equality for logical programming. In Hartmut Ehrig, Giorgio Levi, Robert Kowalski, and Ugo Montanari, editors, Proceedings, 1987 TAPSOFT, pages 1--22. Springer-Verlag, 1987. LNCS, Volume 250; extended version to appear in J. Logic Programming.

.... theorems for particular logics appear by specialising the abstract rules of deduction to exploit more specific assumptions and reveal the usual syntactic formulations (see [13, 14] The completeness of category based equational deduction leads to a generic Herbrand theorem in the style of [23], characterising Herbrand models as initial models of the program regarded as an equational theory (see [13, 14] When applied to constraint logics [13] this gives a Herbrand theorem for extensible constraint logic programming. This rather sophisticated new result requires minimal effort using ....

.... This example is based on the more elaborated CB semantics for constraint logic programming given in [13] This treatment of constraint logic (abbreviated CL) as CB equational logic extends Goguen Meseguer s approach to constraint logic programming developed in the context of the language Eqlog [23]. We sketch here a brief description of the main ideas within the framework of Horn clause logic with equality rather than internalising them into CBEL. 5 Let Sigma be a [Horn clause logic with equality] signature of built ins , and Sigma , Sigma 0 be an extension containing new ....

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Joseph Goguen and Jos'e Meseguer. Models and equality for logical programming. In Hartmut Ehrig, Giorgio Levi, Robert Kowalski, and Ugo Montanari, editors, Proceedings, 1987 TAPSOFT, pages 1--22. Springer, 1987. Lecture Notes in Computer Science, Volume 250.

....A lg S;E (A; preserves filtered colimits. 4.4. A Herbrand Theorem An important consequence of the most abstract version of the completeness result (Theorem 2) is a Herbrand theorem for CBEL. Our approach uses the categorical characterisation of Herbrand universes as initial models suggested in (Goguen and Meseguer, 1987). Herbrand theorem for CBEL admits various formulations corresponding to different abstraction levels. The one that fits best the level of presentation of this paper is the following: 9 See (Diaconescu, 1995; Diaconescu, 1994) for a more general version of this result using a finiteness condition ....

....iff BM j= U (lA)hs; ti : 5. Category based Equational) Constraint Logic Constraint logic is central to our approach to constraint logic programming in that it is the logic underlying this programming paradigm. This matches the principle of logical programming introduced by Goguen and Meseguer in (Goguen and Meseguer, 1987). This section does actually more than setting up the logic underlying ECLP; it also shows how constraint logic is a CBEL, which means ECLP is semantically integrated to the equational logic programming paradigm. On the other hand, we develop constraint logic internally to CBEL (we might thus call ....

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Goguen, J. and Meseguer, J. (1987). Models and equality for logical programming. In Ehrig, H., Levi, G., Kowalski, R., and Montanari, U., editors, Proceedings, 1987 TAPSOFT, pages 1--22. Springer. Lecture Notes in Computer Science, Volume 250.

....follows the same general principle: P1) there is an underlying logic 1 in which all basic constructs features of the language can be rigorously explained. This intimate relationship between the language and its underlying logic was called logical programming by Goguen and Meseguer in [25]. In providing the semantics for CafeOBJ we distinguish between four levels of the language: 2 programming in the small , programming in the large , programming in the huge , and environment. Programming in the small refers to collections of program statements (as obtained by ....

Joseph Goguen and Jos'e Meseguer. Models and equality for logical programming. In Hartmut Ehrig, Giorgio Levi, Robert Kowalski, and Ugo Montanari, editors, Proceedings, 1987 TAPSOFT, pages 1--22. Springer, 1987. Lecture Notes in Computer Science, Volume 250.

....2 Through the ORS scheme. 1 1 Introduction This thesis is mainly about equational logic programming. It belongs to the tradition of equational and constraint logic programming started by Goguen and Meseguer in their pioneering work on the programming language Eqlog during the mid eighties [38, 39]. Eqlog has been implemented in Oxford by the author of this thesis as an extension of the SRI implementation of OBJ3. 3 1.1 The Equational Logic Programming Paradigm 1.1.1 A historical perspective Equational logic programming can be regarded as joining two major cultures in Computing: ....

....paradigm unifies logic programming based on Horn clause logic and equational (i.e. functional) programming based on equational logic, i.e. the logic of substituting equals for equals. One of the earliest contributions to this field was [76] As Goguen and Meseguer repeatedly pointed out [38, 39], the best way to achieve this goal should be to unify the two logics involved. However, because equational logic is more fundamental than Horn clause logic 6 , it is enough to base the new paradigm only on equational logic. The main difference between equational logic programming and equational ....

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Joseph Goguen and Jos'e Meseguer. Models and equality for logical programming. In Hartmut Ehrig, Giorgio Levi, Robert Kowalski, and Ugo Montanari, editors, Proceedings, 1987 TAPSOFT, pages 1--22. Springer, 1987. Lecture Notes in Computer Science, Volume 250.

....can be regarded as a model (not necessarily the standard one) for a certain data type specification. This paper presents a model theoretic semantics for constraint logic programming, without directly addressing the computational aspect. 3 Our approach departs from the usual ones by following [19] in proving a Herbrand theorem for constraint logic, which is the logic underlying ECLP. As with the CLP approach of Jaffar and Lassez [20] both constraint relations and programs are (sets of) sentences in the same logical system. But our constraint logics are much more general than Horn clause ....

....U is finitary if it preserves filtered colimits. 8 3.5 A Herbrand Theorem An important consequence of the most abstract version of the completeness result (3) is a Herbrand theorem for CBEL. Our approach uses the categorical characterisation of Herbrand universes as initial models suggested in [19]. Herbrand theorem for CBEL admits various formulations corresponding to different abstraction levels. The one that fits best the level of presentation of this paper is the following: Corollary 4. Herbrand Theorem Assume the Adjointness Framework and the Simplifying Assumption. If A has an ....

[Article contains additional citation context not shown here]

Joseph Goguen and Jos'e Meseguer. Models and equality for logical programming. In Hartmut Ehrig, Giorgio Levi, Robert Kowalski, and Ugo Montanari, editors, Proceedings, 1987 TAPSOFT, pages 1--22. Springer, 1987. Lecture Notes in Computer Science, Volume 250.

....Its results apply to multi paradigm logical computing languages. A logical language is a specification and or programming language having an underlying logic 1 in which all its basic constructs features can be rigorously explained. This concept was first formulated by Goguen and Meseguer in [21] under the name of logical programming . Examples of logical languages include most of the OBJ family of languages, such as OBJ3 [24] Eqlog [20] Maude [28] CafeOBJ [11] etc. but they might also include (pure) Prolog and (pure) Lisp. Multi paradigm logical languages admit institution ....

....paradigm, the basic specifications are assimilated to the theories generated in the corresponding institution. This intimate relationship between the language and its underlying logic (in this case given by the lattice of institution embeddings) was first conceptualized by Goguen and Meseguer [21] under the name of logical programming. As mentioned in the Introduction, such logical languages include most of the OBJ family of languages. In the Appendix we illustrate our logical semantics with the example of CafeOBJ [12] a modern successor of OBJ. 4.1 Basic Specifications At the level ....

Joseph Goguen and Jose Meseguer. Models and equality for logical programming. In Hartmut Ehrig, Giorgio Levi, Robert Kowalski, and Ugo Montanari, editors, Proceedings, 1987 TAPSOFT, pages 1--22. Springer, 1987. Lecture Notes in Computer Science, Volume 250.

.... Logic This example is based on the more elaborated CB semantics for constraint logic programming given in [8, 10] This treatment of constraint logic (abbreviated CL) as CBEL extends Goguen Meseguer s approach to constraint logic programming developed in the context of the language Eqlog [20]. We sketch here a brief simplified description of the main ideas within the framework of Horn clause logic with equality rather than internalising them into CBEL. Let 6 be a [Horn clause logic with equality] signature of built ins , and 6 , 6 0 be an extension containing new sorts, new ....

.... presentation of these inference rules within CBEL the reader may consult [8] or [9] An important consequence of the completeness of CBEL proof theory is a CB Herbrand theorem (see [8, 9] This approach uses the categorical characterisation of Herbrand universes as initial models suggested in [20]. 8] and [9] also prove a non empty sorts version of Herbrand theorem, for which the domain of the initial model has all carriers non empty. This provides foundations for solving queries using techniques like resolution and paramodulation. 11 3.3 Consequences of Freeness So far, we have not ....

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Joseph Goguen and Jos'e Meseguer. Models and equality for logical programming. In Hartmut Ehrig, Giorgio Levi, Robert Kowalski, and Ugo Montanari, editors, Proceedings, 1987 TAPSOFT, pages 1--22. Springer, 1987. Lecture Notes in Computer Science, Volume 250.

.... [27] rigourously based on order sorted conditional equational logic [23] Even computation in Horn clause logics (as in pure Prolog) can be regarded as equational deduction in a precise way [11] Thus, equational logic provides an adequate framework for unifying functional and logic programming [21, 22]. Equational logic can also be used as a meta programming language, by encoding the inference rules of any logical system as rewrite rules [24] Powerful computational techniques have been developed for implementing different programming systems based on different versions of equational logic. ....

.... with respect to modularisation in the style of OBJ (based on the perspective on the forgetful functors from models to domains as signatures for an institution of category based equational logic) and a treatment of constraint logic programming along the lines sketched by Goguen and Meseguer in [22] (with a semantics based on comma categories of models over a built in model, so as to reduce them to our theory of category based equational logic) 3 Institutions for which there exists a forgetful functor from signatures to sets ( extracting the sort component of the signatures ) that has a ....

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Joseph Goguen and Jos'e Meseguer. Models and equality for logical programming. In Hartmut Ehrig, Giorgio Levi, Robert Kowalski, and Ugo Montanari, editors, Proceedings, 1987 TAPSOFT, pages 1--22. Springer, 1987. Lecture Notes in Computer Science, Volume 250.

....extensions. 6. 1 Induction Principles for Membership Equational Theories Given that membership equational logic is a subset of equational Horn logic (indeed, they can be shown to be equivalent [26] it follows immediately that any theory( E) has a unique (up to isomorphism) initial model [17]. The following is an induction principle for reasoning about properties of sorts, with respect to this model. 15 De nition 3 (Induction over sort de nitions) Let( E) be a theory in membership equational logic and let s be a sort in some S k . Let fC 1 ; Cn g be those sentences in E ....

J. Goguen and J. Meseguer. Models and equality for logical programming. In H. Ehrig, G. Levi, R. Kowalski, and U. Montanari, editors, Proceedings TAPSOFT'87, volume 250 of Lecture Notes in Computer Science, pages 1-22. Springer-Verlag, 1987.

....Menlo Park CA 94025, USA. Japan Advanced Institute of Science and Technology, 1 1 Asahidai, Tatsunokuchi, Ishakawa, Japan. Universit e de Paris Sud, 91405 Orsay, France. OBJ3 has been used for building FOOPS, an object oriented specification and programming system [74, 87] the Eqlog system [71, 72, 25] for equational logic (or relational) programming, OOZE [3] an object oriented specification language influenced by Z [142] the 2OBJ metalogical framework theorem prover [81] and TOOR [130] a system for tracing requirements. OBJ has been used for many applications, including debugging ....

....addition, there is a Franz Lisp OBJ2 done at Washington State University [143] UMIST OBJ has been made available as a proprietary software product from Gerrard Software, under the name ObjEx. OBJ has been extended in many directions, including logic (or relational) programming (the Eqlog system [71, 72, 25]) object oriented programming (the FOOPS system [74, 87] object oriented specification (OOZE [3] requirements tracing (TOOR [130] higher order functional programming [97, 110] and LOTOS style specification for communication protocols [127, 128] Recent developments within the OBJ ....

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Joseph Goguen and Jos'e Meseguer. Models and equality for logical programming. In Hartmut Ehrig, Giorgio Levi, Robert Kowalski, and Ugo Montanari, editors, Proceedings, 1987.

....relations as well as operations, leading to various forms of first order logic. Horn clause logic with equality is especially convenient, because it still has initial models for all theories, as shown in work on the semantics of the language Eqlog, which combines functional and logic programming [21, 20]. Yet another extension considers machines that may have internal states, identifying two such machines iff their behaviour is equivalent [19, 46] This gives a semantics for abstract objects which generalises that of abstract data types, as in the semantics of FOOPS [23] An early attempt to ....

Joseph Goguen and Jos'e Meseguer. Models and equality for logical programming. In Hartmut Ehrig, Giorgio Levi, Robert Kowalski, and Ugo Montanari, editors, Proceedings,

....Although these rules are rather compactly formulated, they correspond exactly to intuitions that we feel should be expected for equational deduction. Of course, there are many possible variations on this rule set; for example, see [72] Also, order sorted Horn clause logic is discussed in [28], and [27] gives an overview of the equational case. 3.2 Completeness and Initiality Theorems We now show that the above rules are sound and complete for deriving all the unconditional equations that hold in the class of all algebras that satisfy Gamma. We then obtain initial and free algebras ....

Joseph Goguen and Jos'e Meseguer. Models and equality for logical programming. In Hartmut Ehrig, Giorgio Levi, Robert Kowalski, and Ugo Montanari, editors, Proceedings,

.... In addition, membership equational logic is naturally a special case of, and as we shall see in Section 3 actually equivalent to, many sorted Horn logic with equality, a logic that Joseph Goguen and I studied in detail as the natural setting for unifying equational and relational programming [15]. Therefore, there is nothing particularly surprising about it, except perhaps for how naturally it can be used as an equational logical framework, how well it satisfies the goals listed above, and how long it has taken some of us to recognize it for what it is worth. The adequacy of membership ....

....as a sublogic of many sorted Horn logic with equality, all the well known results for many sorted Horn logic with equality hold true when restricted to the sublogic MEqtl. In particular, the proof of soundness and completeness for the rules of deduction of order sorted Horn logic with equality in [15], of which many sorted Horn logic with equality is a special case, yields the soundness and completeness of the following rules of deduction for a theory ( Omega ; Gamma ) in membership equational logic as an immediate corollary (where all the quantifications are assumed to involve sets of ....

[Article contains additional citation context not shown here]

J. Goguen and J. Meseguer. Models and equality for logical programming. In H. Ehrig, G. Levi, R. Kowalski, and U. Montanari, editors, Proceedings TAPSOFT '87, volume 250 of Lecture Notes in Computer Science, pages 1--22. SpringerVerlag, 1987.

....provides an elegant semantics for contraint logic programming. Following the suggestion in [34] that the best way to combine paradigms is to combine their underlying logics, we here extend the combination of relational and functional paradigms, by extending Horn clause logic with equality (as in [34, 35]) to hidden Horn clause logic with equality, building on prior work on hidden algebra as a foundation for the object paradigm [19, 24, 31] Hidden algebra is a natural extension of the initial algebra approach to abstract data types (ADTs) 39] that handles states in a more natural way, and also ....

.... [41] says that for the models of a set of Horn clauses, existential queries can be answered by examining a particular term model, called the Herbrand universe (see [44, 1] for overviews of logic programming) This result was generalized to Horn clause logic with equality by Goguen and Meseguer [34, 35], showing that it suffices to examine a term model, and moreover, that this model is initial. The advantage of a term model is the well established techniques for equational computation that are available for it. Our hidden Herbrand Theorem states that if a query is behaviorally satisfied by a ....

[Article contains additional citation context not shown here]

Joseph Goguen and Jos'e Meseguer. Models and equality for logical programming. In Hartmut Ehrig, Giorgio Levi, Robert Kowalski, and Ugo Montanari, editors, Proceedings, 1987 TAPSOFT, pages 1--22. Springer, 1987. Lecture Notes in Computer Science, Volume 250.

....applications. 4. 1 Induction Principles for Membership Equational Theories Given that membership equational logic is a subset of equational Horn logic (indeed, they can be shown to be equivalent [29] it follows immediately that any theory ( E) has a unique (up to isomorphism) initial model [20]. The following is an induction principle for reasoning about properties of sorts, with respect to this model. De nition 2 (Induction over sort de nitions) Let T = E) be a theory in membership equational logic and let s be a sort in some S k . Let C [T;s] fC 1 ; Cn g be those ....

J. Goguen and J. Meseguer. Models and equality for logical programming. In H. Ehrig, G. Levi, R. Kowalski, and U. Montanari, editors, Proceedings TAPSOFT'87, volume 250 of Lecture Notes in Computer Science, pages 1-22. Springer-Verlag, 1987.

....improve reliability and predictability. Following [18] we believe that the best way to combine paradigms is to combine their underlying logics. Consequently, we extend the combination of relational and functional paradigms to the object paradigm by extending Horn clause logic with equality (as in [18, 19]) to hidden Horn clause logic with equality, building on prior work on hidden sorted logic as a foundation for the object paradigm [10] We first study existential queries in a hidden equational setting, and then lift to hidden Horn clause logic. This provides a semantic foundation for FOOPlog ....

....overloading. From this discussion, we conclude that strong overloading is needed to formalise and mechanise certain aspects of actual practice in computing and mathematics. This helps to explain why the equational programming language OBJ [26] and its extensions to logic programming (Eqlog [18, 19]) object oriented programming (FOOPS [21] and theorem proving (the 2OBJ metalogical framework theorem prover [23] are based on overloaded OSA. Of course, overloaded OSA also provides an elegant approach to the issues that originally motivated its development, including partial operations, ....

Joseph Goguen and Jos'e Meseguer. Models and equality for logical programming. In Hartmut Ehrig, Giorgio Levi, Robert Kowalski, and Ugo Montanari, editors, Proceedings, 1987 TAPSOFT, pages 1--22. Springer, 1987. Lecture Notes in Computer Science, Volume 250.

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J. Goguen and J. Meseguer. Models and Equality for Logical Programming. In G. Levi H. Ehrig, R. Kowalski and U. Montanari, editors, Proc. TAPSOFT'87, volume 250 of Lecture Notes in Computer Science, pages 1--22. Springer-Verlag, Berlin, 1987. Volume 2.

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