## A Hidden Herbrand Theorem: Combining the Object and Logic Paradigms (1998)

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Venue: | Principles of Declarative Programming |

Citations: | 11 - 3 self |

### BibTeX

@INPROCEEDINGS{Goguen98ahidden,

author = {Joseph Goguen and Grant Malcolm and Tom Kemp},

title = {A Hidden Herbrand Theorem: Combining the Object and Logic Paradigms},

booktitle = {Principles of Declarative Programming},

year = {1998},

pages = {445--462},

publisher = {Springer}

}

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### Abstract

: The benefits of the object, logic (or relational), functional, and constraint paradigms

### Citations

1854 |
Foundations of Logic Programming
- Lloyd
- 1987
(Show Context)
Citation Context ...ied. The classical Herbrand Theorem [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, th... |

1438 | Object-oriented software construction
- Meyer
- 1997
(Show Context)
Citation Context ...is paper also contains new results on the existence of initial and final models, and on the consistency of hidden theories. 1 Introduction The object paradigm has many practical advantages (e.g., see =-=[51]-=-), including its support for code reuse through inheritance, its intuitive appeal, and its affinity for data abstraction and concurrency. However, it has not been integrated with the complementary adv... |

920 |
Categories for the Working Mathematician
- Lane
- 1971
(Show Context)
Citation Context ...dface to denote categories, e.g., C. Given morphisms f : A ! B and g : B ! C, we let g ffi f denote their composition, a morphism A ! C, and we let 1A denote the identity morphism at an object A. See =-=[2, 43]-=- for introductions to category theory. An S-indexed (or sorted) set A is a family fA s j s 2 Sg of sets indexed by the elements of S. An S-indexed (or sorted) function f : A ! B is a family ff s : A s... |

475 |
Institutions: Abstract Model Theory for Specification and Programming
- Goguen, Burstall
- 1992
(Show Context)
Citation Context ...ehavioral equivalence for states, which is also used in hidden algebra. Goguen [19] showed that hidden algebra with some intuitive restrictions on signature maps forms an institution (in the sense of =-=[22]-=-, i.e., a logical system) and used this to model objects; combined with results in this paper, this implies that parameterized programming [17, 18, 47], with its powerful generic module facilities, is... |

394 |
Category Theory for Computing Science
- Barr, Wells
- 1999
(Show Context)
Citation Context ...dface to denote categories, e.g., C. Given morphisms f : A ! B and g : B ! C, we let g ffi f denote their composition, a morphism A ! C, and we let 1A denote the identity morphism at an object A. See =-=[2, 43]-=- for introductions to category theory. An S-indexed (or sorted) set A is a family fA s j s 2 Sg of sets indexed by the elements of S. An S-indexed (or sorted) function f : A ! B is a family ff s : A s... |

273 |
An Initial Algebra Approach to the Specification, Correctness and Implementation of Abstract Data Types
- Goguen, Thatcher, et al.
- 1978
(Show Context)
Citation Context ..., we conclude that every ground term t is defined by a data value f(t) with E jj (8;) t = f(t). 2 Initial models are less important for the hidden paradigm than they are for initial algebra semantics =-=[39]-=- or the more general initial model semantics [34]. Final algebras come closer to that role, in that they capture many abstract properties of the state space. We will show that final models exist exact... |

208 | Order-sorted algebra I: Equational deduction for multiple inheritance, overloading, exceptions and partial operations
- Goguen
- 1992
(Show Context)
Citation Context ... [25] language and system implements some even more advanced techniques. A useful direction for future research is to extend our results to include the kind of subtyping given by order sorted algebra =-=[37]-=-. Burstall and Diaconescu [4] have extended the hiding process to many other institutions, and in particular, to order sorted algebra (in the sense of [37, 23]). Malcolm and Goguen [48] show that hidd... |

188 |
From Logic Programming to Prolog
- Apt
- 1997
(Show Context)
Citation Context ...ied. The classical Herbrand Theorem [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, th... |

175 |
On the structure of abstract algebras
- Birkhoff
- 1935
(Show Context)
Citation Context ...). 2 The following completeness result for conditional many-sorted algebra was first proved by Goguen and Meseguer [33], though the unconditional one-sorted case is well known, going back to Birkhoff =-=[3]-=- in 1935: Theorem 4: Given a set E of (possibly conditional) \Sigma-equations, an unconditional \Sigma-equation is satisfied by every (\Sigma; E)-algebra iff it is derivable from E using the rules (0)... |

118 |
Parameterized Programming
- Goguen
- 1984
(Show Context)
Citation Context ... on signature maps forms an institution (in the sense of [22], i.e., a logical system) and used this to model objects; combined with results in this paper, this implies that parameterized programming =-=[17, 18, 47]-=-, with its powerful generic module facilities, is available for our new paradigm. Two restrictions are that operations should have at most one argument of hidden sort, and that a fixed visible sorted ... |

111 |
Recherches sur la Theorie de la Demonstration, Travaux de la Societe des Sciences et de Lettres de Varsovie
- Herbrand
- 1930
(Show Context)
Citation Context ...ution to such a query in an algebra A consists of values in A for each variable (some of which may range over states) such that each equation is behaviorally satisfied. The classical Herbrand Theorem =-=[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 programmi... |

110 | A hidden agenda
- Goguen, Malcolm
- 2000
(Show Context)
Citation Context ...al 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 supports behavioral correctness and refin... |

108 | A calculus of mobile processes, parts I and II
- Milner, Parrow, et al.
- 1989
(Show Context)
Citation Context ...g 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 tha... |

104 |
Algebraic Semantics of Imperative Programs
- Goguen, Malcolm
- 1996
(Show Context)
Citation Context ...on and Preliminaries We assume familiarity with many sorted algebra, but to establish notation, we will briefly review some main concepts and results. For compatible expositions with more detail, see =-=[28]-=- or [50]; this approach, based on indexed sets, originated in lectures by Joseph Goguen at the University of Chicago in 1968. Some of the examples in Sections 4.2 and 5 assume basic knowledge of term ... |

100 |
EQLOG: equality, types and generic modules for logic programming
- Goguen, Meseguer
(Show Context)
Citation Context ...ative semantics, and (for the relational case) natural integration with database query languages. Logic programming and functional programming over user-definable abstract data types were combined in =-=[34]-=- by combining their underlying logics (Horn clause logic and equational logic, respectively), to obtain Horn clause logic with equality, which in addition provides an elegant semantics for contraint l... |

99 |
Unifying functional, object-oriented and relational programming with logical semantics
- Goguen, Meseguer
- 1987
(Show Context)
Citation Context ...quational case. All this provides a semantic foundation for a novel programming paradigm, in which posing a query can activate methods that change the world so that a solution actually comes to exist =-=[36, 20, 27]-=-. For example, consider a query about a holiday package, where a customer has provided constraints on destination, cost, flight times, seat assignments, hotels, expected weather, etc.; a solution to t... |

98 |
Universal Algebra
- Meinke, Tucker
- 1992
(Show Context)
Citation Context ...eral existence of free algebras, but this follows easily from Theorem 5. Many interesting results about conditional equations and their algebras appear in the literature for the one sorted case, e.g. =-=[7, 45, 59, 49]-=-, but as far as we know, few of these have been considered carefully for the many sorted case. 3 Hidden Algebra Well designed software often fails to strictly satisfy its specifications, but instead s... |

92 |
An approach to object semantics based on terminal coalgebras
- Reichel
- 1995
(Show Context)
Citation Context ...idden model theory in Sections 3 and 4 further develops the research programme of [31], and the coinductive proof technique of Section 4 is useful in hidden algebra and related coalgebraic approaches =-=[56, 42, 46, 8]-=-. The consistency results in Section 3.3.1 are novel and useful. The hidden approach differs from classical algebraic approaches in using behavioral satisfaction except for a fixed interpretation of v... |

85 | Towards an algebraic semantics for the object paradigm
- GOGUEN, DIACONESCU
- 1994
(Show Context)
Citation Context ...al 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 supports behavioral correctness and refin... |

81 |
Initiality, Induction and Computability
- Meseguer, Goguen
- 1985
(Show Context)
Citation Context ...onsist of applying a sequence of methods and then examining an attribute. Hidden algebra originated in [19], extending earlier work of Goguen and Meseguer on (what they then called) abstract machines =-=[33, 50]-=-, mainly through using behavioral satisfaction for equations, an idea introduced by Reichel [54] in the context of partial algebras. Reichel [55] later introduced behavioral equivalence for states, wh... |

77 |
Co-induction in relational semantics
- Milner, Tofte
- 1991
(Show Context)
Citation Context ...g 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 tha... |

74 |
Universal Algebra for Computer Scientists
- Wechler
- 1992
(Show Context)
Citation Context ...eral existence of free algebras, but this follows easily from Theorem 5. Many interesting results about conditional equations and their algebras appear in the literature for the one sorted case, e.g. =-=[7, 45, 59, 49]-=-, but as far as we know, few of these have been considered carefully for the many sorted case. 3 Hidden Algebra Well designed software often fails to strictly satisfy its specifications, but instead s... |

72 |
Types as theories
- Goguen
- 1991
(Show Context)
Citation Context ...al 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 supports behavioral correctness and refin... |

65 |
Universal Realization, Persistent Interconnection and Implementation of Abstract Modules
- Goguen, Meseguer
- 1982
(Show Context)
Citation Context ...onsist of applying a sequence of methods and then examining an attribute. Hidden algebra originated in [19], extending earlier work of Goguen and Meseguer on (what they then called) abstract machines =-=[33, 50]-=-, mainly through using behavioral satisfaction for equations, an idea introduced by Reichel [54] in the context of partial algebras. Reichel [55] later introduced behavioral equivalence for states, wh... |

46 | Circular Coinductive Rewriting
- Goguen, Lin, et al.
- 2000
(Show Context)
Citation Context ...ates in a more natural way, and also supports behavioral correctness and refinement proofs for systems with objects, inheritance, nondeterminism and concurrency, in addition to functional programming =-=[15, 38, 57, 58, 25]-=-. The main result of this paper is a Herbrand theorem for hidden Horn clause logic, allowing solutions of queries to be constructed in a term algebra; it is obtained by applying the extended lifting r... |

46 |
José Meseguer, Kokichi Futatsugi and JeanPierre Jouannaud. Introducing OBJ
- Goguen, Winkler
- 2000
(Show Context)
Citation Context ...mple 12: We specify a simple flag object, where intuitively a flag is either up or down, and there are methods to put it up, to put it down, and to reverse its state. We use the notation of OBJ3 (see =-=[40]-=-, although here equality for hidden sorts has a very different meaning than in standard OBJ): 7 th FLAG is sort Flag . pr DATA . ops (up) (dn) (rev) : Flag -? Flag . op up? : Flag -? Bool . var F : Fl... |

42 |
Models and equality for logical programming
- Goguen, Meseguer
- 1987
(Show Context)
Citation Context ...all P -algebras behaviorally satisfy it; and (2) a final P -algebra behaviorally satisfies a query iff some P -algebra behaviorally satisfies it. Proof: Immediate from Lemma 67. 2 Goguen and Meseguer =-=[35, 34]-=- gave a Herbrand theorem for Horn clause logic with equality, which states that an existential query is satisfied by the initial model of a specification iff it is satisfied by all models of that spec... |

42 | Hiding more of hidden algebra
- Goguen, Ro¸su
- 1999
(Show Context)
Citation Context ...ates in a more natural way, and also supports behavioral correctness and refinement proofs for systems with objects, inheritance, nondeterminism and concurrency, in addition to functional programming =-=[15, 38, 57, 58, 25]-=-. The main result of this paper is a Herbrand theorem for hidden Horn clause logic, allowing solutions of queries to be constructed in a term algebra; it is obtained by applying the extended lifting r... |

38 |
Behavioural equivalence – a unifying concept for initial and final specifications
- Reichel
- 1981
(Show Context)
Citation Context ...in [19], extending earlier work of Goguen and Meseguer on (what they then called) abstract machines [33, 50], mainly through using behavioral satisfaction for equations, an idea introduced by Reichel =-=[54]-=- in the context of partial algebras. Reichel [55] later introduced behavioral equivalence for states, which is also used in hidden algebra. Goguen [19] showed that hidden algebra with some intuitive r... |

36 |
Behavioural validity of conditional equations in abstract data types
- Reichel
- 1985
(Show Context)
Citation Context ...eguer on (what they then called) abstract machines [33, 50], mainly through using behavioral satisfaction for equations, an idea introduced by Reichel [54] in the context of partial algebras. Reichel =-=[55]-=- later introduced behavioral equivalence for states, which is also used in hidden algebra. Goguen [19] showed that hidden algebra with some intuitive restrictions on signature maps forms an institutio... |

33 |
Theorem Proving and Algebra
- Goguen
- 1994
(Show Context)
Citation Context ...University of Chicago in 1968. Some of the examples in Sections 4.2 and 5 assume basic knowledge of term rewriting, such as confluence and termination. Introductions to term rewriting may be found in =-=[9, 21]-=-, among other places. Occasionally it is convenient to express a result or construction in the vocabulary of category theory, but we use only very basic notions like category, functor, and initial obj... |

32 |
and Răzvan Diaconescu. An Oxford survey of order sorted algebra
- Goguen
- 1994
(Show Context)
Citation Context ...d of subtyping given by order sorted algebra [37]. Burstall and Diaconescu [4] have extended the hiding process to many other institutions, and in particular, to order sorted algebra (in the sense of =-=[37, 23]-=-). Malcolm and Goguen [48] show that hidden order sorted logic forms an institution, using a construction that differs from Burstall and Diaconescu's in its treatment of error-handling; yet another tr... |

29 | Proving Correctness of Refinement and Implementation
- Malcolm, Goguen
- 1994
(Show Context)
Citation Context ... conditional equations. This section also characterizes behavioral equivalence in a way that serves as a basis for coinduction, a powerful behavioral proof technique, as in work of Goguen and Malcolm =-=[26, 48, 31]-=- and Goguen and Rosu [58, 38, 57, 25]. Section 3 also contains new results on the consistency of hidden theories. Section 4 presents basic results for reasoning about hidden algebraic specifications, ... |

27 | Proof of correctness of object representations
- Goguen, Malcolm
- 1994
(Show Context)
Citation Context ...) = B c (f(a 0 )) for all c 2 L \Delta [z], which is by definition f(a) j \Delta f(a 0 ). 2 Our statement of a fundamental result justifying several key techniques for proving behavioral satisfaction =-=[26, 48, 31]-=- uses the following: Definition 23: Given \Delta ` \Sigma, a behavioral \Delta-congruence on a hidden \Sigma-algebra A is a \Delta-congruence j which is equality on visible sorts, i.e., for v 2 V and ... |

27 | Hidden congruent deduction
- Rosu, Goguen
(Show Context)
Citation Context ...dels are less common in the recent more general hidden framework where operations may have more than one hidden argument, which has motivated ways to obtain similar results without using final models =-=[38, 58, 25]-=-. Definition 47: Let \Sigma \Pi denote the signature obtained from \Sigma by forgetting all generalized constants in \Sigma; i.e., \Sigma \Pi w;s = ; if w 2 V and s 2 H , and \Sigma \Pi w;s = \Sigma w... |

26 | Inheritance and cofree constructions
- Jacobs
- 1996
(Show Context)
Citation Context ...idden model theory in Sections 3 and 4 further develops the research programme of [31], and the coinductive proof technique of Section 4 is useful in hidden algebra and related coalgebraic approaches =-=[56, 42, 46, 8]-=-. The consistency results in Section 3.3.1 are novel and useful. The hidden approach differs from classical algebraic approaches in using behavioral satisfaction except for a fixed interpretation of v... |

25 | Incompleteness of behavioral logics
- Buss, Ro¸su
(Show Context)
Citation Context ...and attributes act on single states, corresponding to the natural locality of states in object oriented programming; it is needed for some results in Section 3.1, as well as for final models to exist =-=[5]-=-; however, many other results of hidden algebra generalize to multiple hidden arguments [15, 58, 38, 57, 25]. Definition 8: Given hidden signatures \Sigma and \Sigma 0 , a hidden signature morphism \P... |

24 |
Burstall and Razvan Diaconescu. Hiding and behaviour: an institutional approach
- Rod
- 1994
(Show Context)
Citation Context ...ements some even more advanced techniques. A useful direction for future research is to extend our results to include the kind of subtyping given by order sorted algebra [37]. Burstall and Diaconescu =-=[4]-=- have extended the hiding process to many other institutions, and in particular, to order sorted algebra (in the sense of [37, 23]). Malcolm and Goguen [48] show that hidden order sorted logic forms a... |

24 | Category-based semantics for equational and constraint logic programming
- Diaconescu
- 1994
(Show Context)
Citation Context ...ery hidden (\Sigma; \Pi; C)-model. 2 Example 94: To illustrate this result, we add a relation, ascending, to the STACK theory of Example 13, defined by two axioms, using a notation like that of Eqlog =-=[34, 11]-=-, rel ascending : Stack . var N : Nat . var S : Stack . ax ascending(empty) . ax ascending(push(N,S)) if N ? top(S), ascending(S) . where ? : Nat Nat is a built in predicate symbol. To make things mor... |

24 | Hidden coinduction: behavioral correctness proofs for objects
- Goguen, Malcolm
- 1999
(Show Context)
Citation Context ...ACK-algebra. A detailed mechanical correctness proof for this implementation of stack can be found at http://www.cse.ucsd.edu/groups/tatami/kumo/exs/stack/ It is similar to the hand proof sketched in =-=[29]-=-. 2 See [31, 29] for a more general and comprehensive introduction to hidden algebra; the first paper on this subject [19] showed that equations as sentences, hidden algebras as models, and behavioral... |

22 |
On the Structure of Abstract Algebras
- Birkho
- 1935
(Show Context)
Citation Context ...){(4). 2 The following completeness result for conditional many-sorted algebra was rst proved by Goguen and Meseguer [33], though the unconditional one-sorted case is well known, going back to Birkho =-=[3]-=- in 1935: Theorem 4: Given a setE of (possibly conditional) -equations, an unconditional -equation is satis ed by every (;E)-algebra i it is derivable fromE using the rules (0){(5). 2 We use the stand... |

21 | Hyperprogramming: A formal approach to software environments
- Goguen
- 1990
(Show Context)
Citation Context ... on signature maps forms an institution (in the sense of [22], i.e., a logical system) and used this to model objects; combined with results in this paper, this implies that parameterized programming =-=[17, 18, 47]-=-, with its powerful generic module facilities, is available for our new paradigm. Two restrictions are that operations should have at most one argument of hidden sort, and that a fixed visible sorted ... |

19 | Coalgebra semantics for hidden algebra: Parameterised objects and inheritance
- Cîrstea
- 1998
(Show Context)
Citation Context ...idden model theory in Sections 3 and 4 further develops the research programme of [31], and the coinductive proof technique of Section 4 is useful in hidden algebra and related coalgebraic approaches =-=[56, 42, 46, 8]-=-. The consistency results in Section 3.3.1 are novel and useful. The hidden approach differs from classical algebraic approaches in using behavioral satisfaction except for a fixed interpretation of v... |

17 |
The logic of Horn clauses is equational
- Diaconescu
- 1993
(Show Context)
Citation Context ... this paper is a Herbrand theorem for hidden Horn clause logic, allowing solutions of queries to be constructed in a term algebra; it is obtained by applying the extended lifting result of Diaconescu =-=[10]-=- to the Hidden equational case. All this provides a semantic foundation for a novel programming paradigm, in which posing a query can activate methods that change the world so that a solution actually... |

13 | A category-based equational logic semantics to constraint programming
- Diaconescu
- 1996
(Show Context)
Citation Context ...isible sorts. This loose semantics allows hidden algebra to capture nondeterminism by underspecification [31]. Hidden algebra differs from Diaconescu's categorical approach to the constraint paradigm =-=[11, 12]-=-, which has loose ordinary satisfaction even for its built in data types. We have shown that a hidden theory has initial and final models iff it is consistent and lexic. The final model consists of ab... |

11 |
Behavioural equivalence, bisimilarity, and minimal realisation
- Malcolm
- 1996
(Show Context)
Citation Context ...ributes for all states that can be reached from the given state. It can also be shown that two elements of a hidden algebra are behaviorally equivalent iff their images in the final algebra are equal =-=[46]-=-. This prepares us for Theorem 49: If the equations in a hidden theory P = (H; \Sigma; E) have at most one variable of hidden sort, then P has a final model, denoted FP , iff it is consistent and lexi... |

8 | Interconnection of object specifications
- Malcolm
- 1996
(Show Context)
Citation Context ... on signature maps forms an institution (in the sense of [22], i.e., a logical system) and used this to model objects; combined with results in this paper, this implies that parameterized programming =-=[17, 18, 47]-=-, with its powerful generic module facilities, is available for our new paradigm. Two restrictions are that operations should have at most one argument of hidden sort, and that a fixed visible sorted ... |

6 | Behavioral coinductive rewriting
- Rosu
- 1999
(Show Context)
Citation Context ...ates in a more natural way, and also supports behavioral correctness and refinement proofs for systems with objects, inheritance, nondeterminism and concurrency, in addition to functional programming =-=[15, 38, 57, 58, 25]-=-. The main result of this paper is a Herbrand theorem for hidden Horn clause logic, allowing solutions of queries to be constructed in a term algebra; it is obtained by applying the extended lifting r... |

5 |
Coalgebra semantics for hidden algebra: parameterized objects and inheritance
- Crstea
- 1997
(Show Context)
Citation Context ...idden model theory in Sections 3 and 4 further develops the research programme of [31], and the coinductive proof technique of Section 4 is useful in hidden algebra and related coalgebraic approaches =-=[56, 42, 46, 8]-=-. The consistency results in Section 3.3.1 are novel and useful. The hidden approach di ers from classical algebraic approaches in using behavioral satisfaction except for a xed interpretation of visi... |

4 | Hidden order sorted algebra: subtypes for objects - Cirstea, Malcolm, et al. - 1999 |