## Borrowing Interpolation (2011)

### BibTeX

@MISC{Diaconescu11borrowinginterpolation,

author = {Răzvan Diaconescu},

title = {Borrowing Interpolation},

year = {2011}

}

### OpenURL

### Abstract

We present a generic method for establishing interpolation properties by ‘borrowing ’ across logical systems. The framework used is that of the so-caled ‘institution theory’ which is a categorical abstract model theory providing a formal definition for the informal concept of ‘logical system’ and a mathematical concept of ‘homomorphism’ between logical systems. We develop three different styles or patterns to apply the proposed borrowing interpolation method. These three ways are illustrated by the development of a series of concrete interpolation results for logical systems that are used in mathematical logic or in computing science, most of these interpolation properties apparently being new results. These logical systems include fragments of (classical many sorted) first order logic with equality, preordered algebra and its Horn fragment, partial algebra, higher order logic. Applications are also expected for many other logical systems, including membership algebra, various types of order sorted algebra, the logic of predefined types, etc., and various combinations of the logical systems discussed here.

### Citations

923 |
Categories for the Working Mathematician
- Lane, S
- 1971
(Show Context)
Citation Context ...tion, construction which is used for the encodings between the institutions. 2.1. Categories We assume the reader is familiar with basic notions and standard notations from category theory; e.g., see =-=[45]-=- for an introduction to this subject. The categorical notions required for this paper are rather basic, the reader is expected to have knowledge only of the concepts of category, functor, natural tran... |

854 |
A formulation of a simple theory of types
- Church
- 1940
(Show Context)
Citation Context ... realization of preordered algebra is based upon this institution. Example 2.7. (Higher order logic with Henkin semantics). Higher order logic with Henkin semantics has been introduced and studied in =-=[12, 40]-=-, a recent book on the topic being [3]. Here, in order to simplify the presentation and the illustration of our interpolation borrowing method, we consider an simplified variant close to the ‘higher o... |

479 |
Conditional rewriting logic as a unified model of concurrency
- Meseguer
- 1992
(Show Context)
Citation Context ...(¬def(t) ∧ ¬def(t ′ )). 5�� � �� generation of case analysis [25], and in general about reasoning about transitions between states of systems. They constitute an unlabeled form of rewriting logic of =-=[49]-=-. Let POA denote the institution of preordered algebras. The signatures are just ordinary algebraic signatures (S, F), i.e. FOL signatures without the relation component. The POA models are preordered... |

476 |
Institutions: Abstract model theory for specification and programming
- Goguen, Burstall
- 1992
(Show Context)
Citation Context ...fication) in response to the population explosion among logics in use there, with the ambition of doing as much as possible at a level of abstraction independent of commitment to any particular logic =-=[27, 34, 61]-=-. Besides its extensive use in specification theory (it has become the most fundamental mathematical structure in algebraic specification theory), there have been several substantial developments towa... |

452 |
Model Theory
- Chang, Keisler
- 1973
(Show Context)
Citation Context ...reprint submitted to Jouranl of Logic and Computation February 24, 2011Our work extends the borrowing idea to interpolation. Interpolation is one of the most studied properties in mathematical logic =-=[11, 63]-=-, a recent monography dedicated to interpolation in modal and intuitionistic being [33]. One of the reasons for the great interest in interpolation is perhaps the fact that it is the source for many o... |

429 |
Higher Order Categorical Logic
- Lambek, Scott
- 1986
(Show Context)
Citation Context ...ent model theory. A monography [23] dedicated to this topic has been recently published. Recently institutions have also been extended towards proof theory [22, 53] in the spirit of categorical logic =-=[43]-=-. One of the most important methods in institution-independent model theory is that of ‘borrowing’ logical structure or properties across logical systems formalized as institutions. This method owes t... |

399 | Simplification by cooperating decision procedures
- Nelson, Oppen
- 1979
(Show Context)
Citation Context ...inability. Interpolation has numerous applications in computing science especially in formal specification theory [5, 8, 27, 29, 30, 69]. but also in data bases (ontologies) [42], automated reasoning =-=[54, 55]-=-, type checking [41], model checking [47], and structured theorem proving [1, 46]. Consequently it has also received special attention within institution-independent model theory [20, 21, 39]. In spit... |

309 |
An Introduction to Mathematical Logic and Type Theory: to Truth through Proof
- Andrews
- 1986
(Show Context)
Citation Context ... upon this institution. Example 2.7. (Higher order logic with Henkin semantics). Higher order logic with Henkin semantics has been introduced and studied in [12, 40], a recent book on the topic being =-=[3]-=-. Here, in order to simplify the presentation and the illustration of our interpolation borrowing method, we consider an simplified variant close to the ‘higher order algebra’ of [51] which does not c... |

183 |
Completeness in the theory of types
- Henkin
- 1950
(Show Context)
Citation Context ... realization of preordered algebra is based upon this institution. Example 2.7. (Higher order logic with Henkin semantics). Higher order logic with Henkin semantics has been introduced and studied in =-=[12, 40]-=-, a recent book on the topic being [3]. Here, in order to simplify the presentation and the illustration of our interpolation borrowing method, we consider an simplified variant close to the ‘higher o... |

167 |
General Logics
- Meseguer
- 1989
(Show Context)
Citation Context ...ucturally ‘simpler’ one. Howvever, the institution mapping which is appropriate for our task here of borrowing interpolation is that of institution comorphisms [37], previously know as ‘plain map’ in =-=[48]-=- or ‘representation’ in [67, 68]. Institution comorphisms realize the intuition of ‘embedding’ a ‘simpler’ institution into a ‘more complex’ one, which is dual to the intuition realized by the institu... |

161 |
The semantics of CLEAR, a specification language
- Burstall, Goguen
- 1980
(Show Context)
Citation Context ... The category of sets (as objects) and functions (as arrows) is denoted by Set, and CAT is the category of all categories. 1 2.2. Institutions Institutions have been defined by Goguen and Burstall in =-=[9]-=-, the seminal paper [34] being printed after a delay of many years. Below we recall the concept of institution which formalises the intuitive notion of logical system, including syntax, semantics, and... |

143 | Membership algebra as a logical framework for equational specification
- Meseguer
- 1997
(Show Context)
Citation Context ..., but most of them being rather unconventional from the viewpoint of the logic tradition. Some important institutions in use in computing science (other than those mentioned above) include membership =-=[50]-=-, polymorphic [62], various modal logics such as temporal [32], process [32], behavioural [6], coalgebraic [13], object-oriented [36], multi-algebraic (non-determinism) [44], and pre-defined types [17... |

113 |
Module algebra
- Bergstra, Heering, et al.
- 1990
(Show Context)
Citation Context ...or many other logical results and properties, a rather notorius case being that of definability. Interpolation has numerous applications in computing science especially in formal specification theory =-=[5, 8, 27, 29, 30, 69]-=-. but also in data bases (ontologies) [42], automated reasoning [54, 55], type checking [41], model checking [47], and structured theorem proving [1, 46]. Consequently it has also received special att... |

85 | Towards an algebraic semantics for the object paradigm
- Goguen, Diaconescu
- 1994
(Show Context)
Citation Context ...ting science (other than those mentioned above) include membership [50], polymorphic [62], various modal logics such as temporal [32], process [32], behavioural [6], coalgebraic [13], object-oriented =-=[36]-=-, multi-algebraic (non-determinism) [44], and pre-defined types [17] logics. The following definitions introduce some very basic model theoretic concepts at the level of abstract institutions. Definit... |

57 | Institution morphisms
- Goguen, Roşu
- 2002
(Show Context)
Citation Context ...l structure or properties across logical systems formalized as institutions. This method owes to the existence of a concept of homomorphism between institutions, in the literature called ‘comorphism’ =-=[37]-=-. Institution comorphisms are mappings which preserve the mathematical structure of institutions, but from a logic and model theoretic perspective their significance is that of either embeddings or en... |

48 |
Bits and pieces of the theory of institutions
- Tarlecki
- 1986
(Show Context)
Citation Context ...tion theorem. 3.1. Craig interpolation In the algebraic specification literature there are several institution independent formulations of Craig interpolation (which we abbreviate by CI). For example =-=[64]-=- is one of the first work introducing the concept of interpolation at the level of abstract institutions. The common feature of these formulations is that they generalise the conventional intersection... |

46 |
I borrow your logic? (transporting logical structures along maps
- Cerioli, Meseguer
(Show Context)
Citation Context ...both the syntactic and the semantic components of comorphisms are coherent to each other. One of the early contributions in the area of borrowing of logical structure along institution comorphisms is =-=[10]-=-. Email address: Razvan.Diaconescu@imar.ro (Răzvan Diaconescu) Preprint submitted to Jouranl of Logic and Computation February 24, 2011Our work extends the borrowing idea to interpolation. Interpolat... |

40 |
Institution-independent Model Theory. Birkhäuser
- Diaconescu
- 2008
(Show Context)
Citation Context ...s an ‘institution-independent’ (abstract) model theory, [15, 18–20, 38, 39, 56, 57, 65, 66] being just few examples from a growing list of papers on institution-independent model theory. A monography =-=[23]-=- dedicated to this topic has been recently published. Recently institutions have also been extended towards proof theory [22, 53] in the spirit of categorical logic [43]. One of the most important met... |

36 | Logical systems for structured specifications
- Borzyszkowski
(Show Context)
Citation Context ...or many other logical results and properties, a rather notorius case being that of definability. Interpolation has numerous applications in computing science especially in formal specification theory =-=[5, 8, 27, 29, 30, 69]-=-. but also in data bases (ontologies) [42], automated reasoning [54, 55], type checking [41], model checking [47], and structured theorem proving [1, 46]. Consequently it has also received special att... |

33 | Relating CASL with Other Specification Languages: The Institution
- Mossakowski
- 2002
(Show Context)
Citation Context ...Γ2 |=Σ2 E2 But these just mean E1 |=(Σ1,Γ1) ϕ1(E) and ϕ2(E) ∪ K2 |=(Σ2,Γ2) E2 which shows the CRI property for presentations. ✷ 4.3. Interpolation in PA In the algebraic specification literature (see =-=[52]-=- for example) there are two major comorphisms PA → FOL pres . A third one has been recently developed in [24]. Two of them encodes the partial operations as total operations, and only one encodes them... |

32 | Applications of Craig interpolants in model checking
- McMillan
- 2005
(Show Context)
Citation Context ...tions in computing science especially in formal specification theory [5, 8, 27, 29, 30, 69]. but also in data bases (ontologies) [42], automated reasoning [54, 55], type checking [41], model checking =-=[47]-=-, and structured theorem proving [1, 46]. Consequently it has also received special attention within institution-independent model theory [20, 21, 39]. In spite of its naturalness and simplicity, just... |

31 |
and R*azvan Diaconescu. An Oxford survey of order sorted algebra
- Goguen
- 1994
(Show Context)
Citation Context ...gment. Based on the already existing encoding comorphisms in the literature, we strongly expect similar results in membership algebra [50] and in some of the types of order sorted algebra surveyed in =-=[35]-=-, as well as in their corresponding Horn clause fragments. Due to lack of space we have not included the development of these result here, but we encourage the interested reader to do this by herself ... |

26 | Theorem proving with structured theories
- Amir, McIlraith
(Show Context)
Citation Context ... in formal specification theory [5, 8, 27, 29, 30, 69]. but also in data bases (ontologies) [42], automated reasoning [54, 55], type checking [41], model checking [47], and structured theorem proving =-=[1, 46]-=-. Consequently it has also received special attention within institution-independent model theory [20, 21, 39]. In spite of its naturalness and simplicity, just saying that any deduction ρ1 ⊢ ρ2 can b... |

25 | Interpolation in Grothendieck institutions
- Diaconescu
(Show Context)
Citation Context ...2. The borrowing theorem The method of borrowing interpolation properties along institution comorphisms requires some special interpolation properties of the comorphism. These have been introduced in =-=[21]-=- and are recalled by the Definitions 3.3 and 3.4 below. Definition 3.3 (Left interpolation property for comorphisms). For a fixed class S ⊆ Sig of signature morphisms, we say that an institution comor... |

24 |
and Andrzej Tarlecki. Specifications in an arbitrary institution
- Sannella
- 1988
(Show Context)
Citation Context ...fication) in response to the population explosion among logics in use there, with the ambition of doing as much as possible at a level of abstraction independent of commitment to any particular logic =-=[27, 34, 61]-=-. Besides its extensive use in specification theory (it has become the most fundamental mathematical structure in algebraic specification theory), there have been several substantial developments towa... |

21 | On the integration of observability and reachability concepts
- Bidoit, Hennicker
(Show Context)
Citation Context ... important institutions in use in computing science (other than those mentioned above) include membership [50], polymorphic [62], various modal logics such as temporal [32], process [32], behavioural =-=[6]-=-, coalgebraic [13], object-oriented [36], multi-algebraic (non-determinism) [44], and pre-defined types [17] logics. The following definitions introduce some very basic model theoretic concepts at the... |

18 | Elementary diagrams in institutions - Diaconescu |

18 |
An Institution-independent Proof of Craig Interpolation Theorem
- Diaconescu
(Show Context)
Citation Context ...ed reasoning [54, 55], type checking [41], model checking [47], and structured theorem proving [1, 46]. Consequently it has also received special attention within institution-independent model theory =-=[20, 21, 39]-=-. In spite of its naturalness and simplicity, just saying that any deduction ρ1 ⊢ ρ2 can be split as ρ1 ⊢ ρ ⊢ ρ2 such that ρ contains only symbols ‘common’ to both ρ1 and ρ2, interpolation is notoriou... |

15 |
Andrzej Tarlecki. Casl: The Common Algebraic Specification Language
- Astesiano, Bidoit, et al.
(Show Context)
Citation Context ...notes the sub-institution of UNIV determined by the Horn clauses. Example 2.4. (Partial algebra). Here we consider the institution PA of partial algebra as employed by the specification language CASL =-=[4]-=-. A partial algebraic signature is a tuple (S, TF, PF), where TF is a family of sets of total function symbols and PF is a family of sets of partial function symbols such that TFw→s ∩ PFw→s = ∅ for ea... |

15 |
Institution-independent ultraproducts
- Diaconescu
(Show Context)
Citation Context ... the main technical task to be solved in this section. For this section we assume familiarity with the concept of ultraproduct of models in FOL (see [11] for the classic approach to ultraproducts and =-=[18, 23, 28]-=- for an institution-independent approach). Proposition 5.1. The institution (embedding) comorphism UNIV → FOL has the Craig (ie∗)-left interpolation property. Proof. Consider ϕ1 : Σ → Σ1 be a (ie∗) mo... |

14 |
A simple algebraic proof of the equational interpolation theorem. Algebra Universalis
- Rodenburg
- 1991
(Show Context)
Citation Context ... E2 of Σ2-sentences such that θ1(E1) |= θ2(E2) there exists a set E of Σ-sentences such that E1 |= ϕ1(E) and ϕ2(E) |= E2. The set E is called the interpolant of E1 and E2. 9�� �� � � �� �� The works =-=[60]-=- and [27] argue succesfully that the above formulation of CI in terms of sets of sentences is more natural than the more traditional formulations of CI in terms of single sentences. First, the applica... |

12 |
Formal Support for Specification Design and Implementation
- Dimitrakos
- 1998
(Show Context)
Citation Context ...or many other logical results and properties, a rather notorius case being that of definability. Interpolation has numerous applications in computing science especially in formal specification theory =-=[5, 8, 27, 29, 30, 69]-=-. but also in data bases (ontologies) [42], automated reasoning [54, 55], type checking [41], model checking [47], and structured theorem proving [1, 46]. Consequently it has also received special att... |

12 | Type class polymorphism in an institutional framework
- Schröder, Mossakowski, et al.
- 2004
(Show Context)
Citation Context ... being rather unconventional from the viewpoint of the logic tradition. Some important institutions in use in computing science (other than those mentioned above) include membership [50], polymorphic =-=[62]-=-, various modal logics such as temporal [32], process [32], behavioural [6], coalgebraic [13], object-oriented [36], multi-algebraic (non-determinism) [44], and pre-defined types [17] logics. The foll... |

11 |
Algebraic Specifications of Reachable Higher-Order Algebras
- Möller, Tarlecki, et al.
- 1988
(Show Context)
Citation Context ...n the topic being [3]. Here, in order to simplify the presentation and the illustration of our interpolation borrowing method, we consider an simplified variant close to the ‘higher order algebra’ of =-=[51]-=- which does not consider λ-abstraction and choice functions. Let us first define an institution of ‘plain’ higher order logic, denoted HOL. For any set S of sorts, let −→ S be the set of S -types defi... |

10 |
Generalized interpolation
- Borzyszkowski
- 2001
(Show Context)
Citation Context ...ormulations is that they generalise the conventional intersection-union (of signatures) framework to commuting squares of signature morphisms. In most cases these are required to be pushouts (like in =-=[7, 8, 31, 65]-=-) and when this is not the case the signature morphisms are required to be (abstract) inclusions [27]. It has been noticed in [20] that the mere formulation of CI does not require any extra technical ... |

10 |
Răzvan Diaconescu, and Andrzej Tarlecki, What is a logic translation
- Mossakowski
(Show Context)
Citation Context ...owing list of papers on institution-independent model theory. A monography [23] dedicated to this topic has been recently published. Recently institutions have also been extended towards proof theory =-=[22, 53]-=- in the spirit of categorical logic [43]. One of the most important methods in institution-independent model theory is that of ‘borrowing’ logical structure or properties across logical systems formal... |

9 |
Interpolation and Definability: Modal and Intuitionistic Logic. Number 46 in Oxford Logic Guides
- Gabbay, Maksimova
- 2005
(Show Context)
Citation Context ...e borrowing idea to interpolation. Interpolation is one of the most studied properties in mathematical logic [11, 63], a recent monography dedicated to interpolation in modal and intuitionistic being =-=[33]-=-. One of the reasons for the great interest in interpolation is perhaps the fact that it is the source for many other logical results and properties, a rather notorius case being that of definability.... |

9 |
An Institution-Independant Proof of Robinson Consistency Theorem. Studia Logica
- Găină, Popescu
- 2006
(Show Context)
Citation Context ...ed reasoning [54, 55], type checking [41], model checking [47], and structured theorem proving [1, 46]. Consequently it has also received special attention within institution-independent model theory =-=[20, 21, 39]-=-. In spite of its naturalness and simplicity, just saying that any deduction ρ1 ⊢ ρ2 can be split as ρ1 ⊢ ρ ⊢ ρ2 such that ρ contains only symbols ‘common’ to both ρ1 and ρ2, interpolation is notoriou... |

9 | State of the union: Type inference via Craig interpolation
- Jhala, Majumdar, et al.
- 2007
(Show Context)
Citation Context ... has numerous applications in computing science especially in formal specification theory [5, 8, 27, 29, 30, 69]. but also in data bases (ontologies) [42], automated reasoning [54, 55], type checking =-=[41]-=-, model checking [47], and structured theorem proving [1, 46]. Consequently it has also received special attention within institution-independent model theory [20, 21, 39]. In spite of its naturalness... |

7 | Category-based constraint logic
- Diaconescu
(Show Context)
Citation Context ...50], polymorphic [62], various modal logics such as temporal [32], process [32], behavioural [6], coalgebraic [13], object-oriented [36], multi-algebraic (non-determinism) [44], and pre-defined types =-=[17]-=- logics. The following definitions introduce some very basic model theoretic concepts at the level of abstract institutions. Definition 2.2. For any signature Σ in an institution I: – For each set E o... |

7 |
Petria and Răzvan Diaconescu. Abstract Beth definability in institutions
- Marius
- 2006
(Show Context)
Citation Context ...een recently developed in [24]. Two of them encodes the partial operations as total operations, and only one encodes them as relations. This latter comorphism, also used for borrowing definability in =-=[57]-=- is the one which is appropriate for our task here. We recall its definition below: • Each PA-signature (S, TF, PF) gets mapped to the FOL presentation ((S, TF, PF), Γ(S,TF,PF)) such that PFws = PFw→s... |

6 | The Institution of Multialgebras – a general framework for algebraic software development
- Lamo
- 2003
(Show Context)
Citation Context ... above) include membership [50], polymorphic [62], various modal logics such as temporal [32], process [32], behavioural [6], coalgebraic [13], object-oriented [36], multi-algebraic (non-determinism) =-=[44]-=-, and pre-defined types [17] logics. The following definitions introduce some very basic model theoretic concepts at the level of abstract institutions. Definition 2.2. For any signature Σ in an insti... |

5 |
Proof systems for institutional logic
- Diaconescu
(Show Context)
Citation Context ...owing list of papers on institution-independent model theory. A monography [23] dedicated to this topic has been recently published. Recently institutions have also been extended towards proof theory =-=[22, 53]-=- in the spirit of categorical logic [43]. One of the most important methods in institution-independent model theory is that of ‘borrowing’ logical structure or properties across logical systems formal... |

5 |
On the role of interpolation in stepwise refinement
- Dimitrakos, Maibaum
- 1997
(Show Context)
Citation Context |

4 | Modal logics are coalgebraic
- Cîrstea, Kurz, et al.
(Show Context)
Citation Context ...utions in use in computing science (other than those mentioned above) include membership [50], polymorphic [62], various modal logics such as temporal [32], process [32], behavioural [6], coalgebraic =-=[13]-=-, object-oriented [36], multi-algebraic (non-determinism) [44], and pre-defined types [17] logics. The following definitions introduce some very basic model theoretic concepts at the level of abstract... |

4 | Birkhoff completeness in institutions - Codescu, Găină |

4 | An institution-independent generalization of Tarski’s Elementary Chain Theorem - Găină, Popescu |

3 | Improving the Efficiency of Reasoning Through Structure-Based Reformulation
- Amir, McIlraith
- 2000
(Show Context)
Citation Context ... in formal specification theory [5, 8, 27, 29, 30, 69]. but also in data bases (ontologies) [42], automated reasoning [54, 55], type checking [41], model checking [47], and structured theorem proving =-=[1, 46]-=-. Consequently it has also received special attention within institution-independent model theory [20, 21, 39]. In spite of its naturalness and simplicity, just saying that any deduction ρ1 ⊢ ρ2 can b... |

3 | An encoding of partial algebras as total algebras - Diaconescu - 2009 |

3 | Modules in transition. Conservativity, Composition, and Colimits
- Kutz, Mossakowski
- 2007
(Show Context)
Citation Context ...ius case being that of definability. Interpolation has numerous applications in computing science especially in formal specification theory [5, 8, 27, 29, 30, 69]. but also in data bases (ontologies) =-=[42]-=-, automated reasoning [54, 55], type checking [41], model checking [47], and structured theorem proving [1, 46]. Consequently it has also received special attention within institution-independent mode... |