## Quasi-Boolean Encodings and Conditionals in Algebraic Specification

Citations: | 3 - 3 self |

### BibTeX

@MISC{Diaconescu_quasi-booleanencodings,

author = {Răzvan Diaconescu},

title = {Quasi-Boolean Encodings and Conditionals in Algebraic Specification},

year = {}

}

### OpenURL

### Abstract

We develop a general study of the algebraic specification practice, originating from the OBJ tradition, which encodes atomic sentences in logical specification languages as Boolean terms. This practice originally motivated by operational aspects, but also leading to significant increase in expressivity power, has recently become important within the context of some formal verification methodologies mainly because it allows the use of simple equational reasoning for frameworks based on logics that do not have an equational nature. Our development includes a generic rigorous definition of the logics underlying the above mentioned practice, based on the novel concept of ‘quasi-Boolean encoding’, a general result on existence of initial semantics for these logics, and presents a general method for employing Birkhoff calculus of conditional equations as a sound calculus for these logics. The high level of generality of our study means that the concepts are introduced and the results are obtained at the level of abstract institutions (in the sense of Goguen and Burstall [12]) and are therefore applicable to a multitude of logical systems and environments.

### Citations

921 |
Categories for the working mathematician
- Lane
- 1998
(Show Context)
Citation Context ... be used to illustrate instances of the general developments of our paper. Category theory.. We assume the reader is familiar with basic notions and standard notations from category theory; e.g., see =-=[19]-=- for an introduction to this subject. Here we recall very briefly some of them. By way of notation, |C| denotes the class of objects of a category C, C(A, B) the set of arrows with domain A and codoma... |

476 |
Institutions: Abstract model theory for specication and programming
- Goguen, Burstall
- 1992
(Show Context)
Citation Context ... these logics. The high level of generality of our study means that the concepts are introduced and the results are obtained at the level of abstract institutions (in the sense of Goguen and Burstall =-=[12]-=-) and are therefore applicable to a multitude of logical systems and environments. Key words: algebraic specification, institution theory, Boolean algebra, encodings, OBJ, CafeOBJ 1. Introduction Equa... |

168 |
General logics
- Meseguer
- 1989
(Show Context)
Citation Context ... sentences. The following property, needed by our work, plays a crucial role for the semantics studies of formal specifications and comes up in very many works in the area, a few early examples being =-=[9, 20, 26, 27]-=-. It is a necessary condition in many model theoretic results using institutions (see [4]), thus being one of the most desirable properties for an institution. It is also not to be confused with a muc... |

161 |
The semantics of Clear, a specification language
- Burstall, Goguen
- 1980
(Show Context)
Citation Context ...former concept because it is technically enough and in the applications it has the advantage of having to check less conditions. Institutions. Institutions have been defined by Goguen and Burstall in =-=[2]-=-, the journal seminal paper [12] 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, semant... |

100 |
EQLOG: equality, types and generic modules for logic programming
- Goguen, Meseguer
(Show Context)
Citation Context ... verification aspects of formal methods based upon conditional equational logic. Moreover, equational logic in conditional form provides the framework for the so-called ‘equational logic programming’ =-=[14, 15]-=-, a rather powerful logic programming paradigm. In some cases, including OBJ and CafeOBJ, the execution mechanism of conditional equational logic specifications by rewriting requires the following tri... |

85 | Logical support for modularisation
- Diaconescu, Goguen, et al.
- 1991
(Show Context)
Citation Context ... sentences. The following property, needed by our work, plays a crucial role for the semantics studies of formal specifications and comes up in very many works in the area, a few early examples being =-=[9, 20, 26, 27]-=-. It is a necessary condition in many model theoretic results using institutions (see [4]), thus being one of the most desirable properties for an institution. It is also not to be confused with a muc... |

85 | Towards an algebraic semantics for the object paradigm
- GOGUEN, DIACONESCU
- 1994
(Show Context)
Citation Context ...rtial algebras is extended to all sentences like in ALG; note the role played by the assumption that the quantifications are total. Example 2.5 (Hidden algebra). Hidden algebra has been introduced in =-=[11, 13]-=- as an algebraic formalism underlying the behavioural specification paradigm and further developed by works such as [7, 18, 25]. In an essential form it can be presented as the following institution, ... |

72 |
Types as theories
- Goguen
- 1991
(Show Context)
Citation Context ...rtial algebras is extended to all sentences like in ALG; note the role played by the assumption that the quantifications are total. Example 2.5 (Hidden algebra). Hidden algebra has been introduced in =-=[11, 13]-=- as an algebraic formalism underlying the behavioural specification paradigm and further developed by works such as [7, 18, 25]. In an essential form it can be presented as the following institution, ... |

59 |
Rewriting as a unified model of concurrency
- Meseguer
- 1990
(Show Context)
Citation Context ...tended to full first order sentences formed from the atoms of PDL as in the case of ALG. Example 2.3 (Preordered algebra). This institution, denoted POA, represents a dilluted form of rewriting logic =-=[21]-=- in that it considers only unlabelled transitions. It is directly realized as a paradigm for specifying transitions by the language CafeOBJ [6]. 2 If w = s1 . . . sn and a = (a1, . . . , an) then by h... |

58 | Institution morphisms
- Goguen, Ros¸u
- 2002
(Show Context)
Citation Context ...the left adjoints to the model translation functors of this comorphism. More details about this can be found in [5]. More on the concept of persistently liberal theoroidal comorphisms can be found in =-=[4, 16, 22, 23]-=-. In fact all quasi-Boolean encodings of Examples 3.1, 3.2, 3.3, 3.4, and 3.5 (but not those of Ex. 3.6) can be presented as persistently liberal simple theoroidal comorphisms, and this gives a genera... |

52 | Observational logic
- Hennicker, Bidoit
- 1999
(Show Context)
Citation Context ... total. Example 2.5 (Hidden algebra). Hidden algebra has been introduced in [11, 13] as an algebraic formalism underlying the behavioural specification paradigm and further developed by works such as =-=[7, 18, 25]-=-. In an essential form it can be presented as the following institution, denoted HA. The signatures of HA are triples (H, V, F) where V and H are sets of visible and hidden sort symbols, respectively,... |

49 |
A Model Theoretic Oriented Approach to Partial Algebras (Introduction to Theory and Application of Partial Algebras
- Burmeister
- 1986
(Show Context)
Citation Context ...). This institution is denoted PA. Here we refer to the partial algebra as used in CASL [22] which represents a slight refinement of the concept of partial algebra as defined in the standard textbook =-=[1]-=-. A partial algebraic signature is a tuple (S, TF, PF), where both (S, TF) and (S, PF) are algebraic signatures such that TFw→s and PFw→s are always disjoint. TF stands for ‘total’ function symbols wh... |

42 |
Models and equality for logical programming
- Goguen, Meseguer
- 1987
(Show Context)
Citation Context ... verification aspects of formal methods based upon conditional equational logic. Moreover, equational logic in conditional form provides the framework for the so-called ‘equational logic programming’ =-=[14, 15]-=-, a rather powerful logic programming paradigm. In some cases, including OBJ and CafeOBJ, the execution mechanism of conditional equational logic specifications by rewriting requires the following tri... |

41 |
Institution-independent Model Theory
- Diaconescu
- 2008
(Show Context)
Citation Context ... of Σ-sentences in an arbitrary institution by E |=Σ E ′ we denote that for all Σ-models M, if M |=Σ E then M |=Σ E ′ . There are myriads examples of institutions from logic or computing science (see =-=[4]-=- for some of these). The examples presented below will be used as concrete benchmarks for our general results. Example 2.1 (Total algebra). This institution is denoted ALG. Its signatures are called a... |

34 | Relating Casl with other specification languages: the institution level
- Mossakowski
(Show Context)
Citation Context ... if and only if Mt ≤ Mt ′ for any preordered algebra M of the respective signature. Example 2.4 (Partial algebra). This institution is denoted PA. Here we refer to the partial algebra as used in CASL =-=[22]-=- which represents a slight refinement of the concept of partial algebra as defined in the standard textbook [1]. A partial algebraic signature is a tuple (S, TF, PF), where both (S, TF) and (S, PF) ar... |

33 |
CafeOBJ Report: The Language, Proof Techniques, and Methodologies for Object-Oriented Algebraic Specification
- Diaconescu, Futatsugi
- 1998
(Show Context)
Citation Context ...tn = t ′ n). For example this is assumed to be the case for the famous pioneering language OBJ [17] or for the functional part of many of the modern algebraic specification languages, such as CafeOBJ =-=[6, 8]-=-. Conditional equational logics have a series of properties that make them rather suitable for formal specification. In particular they admit initial semantics, which is the main way to specify data t... |

31 |
On the Existence of Free Models in Abstract Algebraic Institutions
- Tarlecki
- 1985
(Show Context)
Citation Context ... sentences. The following property, needed by our work, plays a crucial role for the semantics studies of formal specifications and comes up in very many works in the area, a few early examples being =-=[9, 20, 26, 27]-=-. It is a necessary condition in many model theoretic results using institutions (see [4]), thus being one of the most desirable properties for an institution. It is also not to be confused with a muc... |

29 |
Heterogeneous Specification and the Heterogeneous Tool Set, Habilitation thesis
- Mossakowski
- 2005
(Show Context)
Citation Context ...the left adjoints to the model translation functors of this comorphism. More details about this can be found in [5]. More on the concept of persistently liberal theoroidal comorphisms can be found in =-=[4, 16, 22, 23]-=-. In fact all quasi-Boolean encodings of Examples 3.1, 3.2, 3.3, 3.4, and 3.5 (but not those of Ex. 3.6) can be presented as persistently liberal simple theoroidal comorphisms, and this gives a genera... |

24 |
and Andrzej Tarlecki. Specifications in an arbitrary institution
- Sannella
- 1988
(Show Context)
Citation Context |

22 |
Quasi-Varieties in Abstract Algebraic Institutions
- Tarlecki
- 1986
(Show Context)
Citation Context ...s paper we assume that all abstract quasi-Boolean encodings are truth injective. The main idea of the following treatment of quantifiers at the level of abstract institutions originates probably from =-=[28]-=- (see also [4]). Proposition 3.1. Let I be any institution with pushout of signatures and weak model amalgamation and let D be a class of its signature morphisms such that for any signature morphisms ... |

17 |
The logic of Horn clauses is equational
- Diaconescu
- 1993
(Show Context)
Citation Context ...a, b). For all terms t1 and t2 of sort s, – α(S,F)(t1 = t2) = (t1 =○ st2). Example 3.2 (Predicate logic). This quasi-Boolean encoding of A(PDL), i.e. the atomic part of PDL, has been first defined in =-=[3]-=-. For each PDL signature (S, C, P), Φ(S, C, P) = (S ⊎ {B}, C⊕P) where – (C⊕P)→s = C→s when s � B, – (C⊕P)w→B = { π○ | π ∈ Pw}, when w is non-empty, – (C⊕P)→B = {true}, and – (C⊕P)w→s = ∅ when w is non... |

3 | An encoding of partial algebras as total algebras
- Diaconescu
- 2009
(Show Context)
Citation Context ...of a ‘persistently liberal’ simple theoroidal comorphism PA → ALG with γ(S,TF,PF) being the left adjoints to the model translation functors of this comorphism. More details about this can be found in =-=[5]-=-. More on the concept of persistently liberal theoroidal comorphisms can be found in [4, 16, 22, 23]. In fact all quasi-Boolean encodings of Examples 3.1, 3.2, 3.3, 3.4, and 3.5 (but not those of Ex. ... |

2 | Verifying design with proof scores
- Futatsugi, Goguen, et al.
- 2005
(Show Context)
Citation Context .... The encoding of equations as Boolean terms via the encoding of the syntactic equality = as a Boolean valued operation == plays a central role in the recent so-called OTS/CafeOBJ verification method =-=[10, 24]-=-. The work reported in this paper may provide the necessary foundations for at least some aspects of the above mentioned verification method. 2. Preliminaries In this section we introduce some institu... |

1 |
and Kokichi Futatsugi. Simulation-based verification for invariant properties in the OTS/CafeOBJ method
- Ogata
(Show Context)
Citation Context .... The encoding of equations as Boolean terms via the encoding of the syntactic equality = as a Boolean valued operation == plays a central role in the recent so-called OTS/CafeOBJ verification method =-=[10, 24]-=-. The work reported in this paper may provide the necessary foundations for at least some aspects of the above mentioned verification method. 2. Preliminaries In this section we introduce some institu... |