## Kleene under a Modal Demonic Star (2004)

### Cached

### Download Links

- [www.informatik.uni-augsburg.de]
- [www.informatik.uni-augsburg.de]
- DBLP

### Other Repositories/Bibliography

Venue: | JOURNAL ON LOGIC AND ALGEBRAIC PROGRAMMING, SPECIAL ISSUE ON RELATION ALGEBRA AND KLEENE ALGEBRA |

Citations: | 7 - 5 self |

### BibTeX

@INPROCEEDINGS{Desharnais04kleeneunder,

author = {Jules Desharnais and Bernhard Möller and Fairouz Tchier},

title = {Kleene under a Modal Demonic Star},

booktitle = {JOURNAL ON LOGIC AND ALGEBRAIC PROGRAMMING, SPECIAL ISSUE ON RELATION ALGEBRA AND KLEENE ALGEBRA},

year = {2004},

pages = {2006},

publisher = {}

}

### Years of Citing Articles

### OpenURL

### Abstract

In relational semantics, the input-output semantics of a program is a relation on its set of states. We generalize this in considering elements of Kleene algebras as semantical values. In a nondeterministic context, the demonic semantics is calculated by considering the worst behavior of the program. In this paper, we concentrate on while loops. Calculating the semantics of a loop is difficult, but showing the correctness of any candidate abstraction is much easier. For deterministic programs, Mills has described a checking method known as the while statement verification rule. A

### Citations

821 | Dynamic logic
- Harel, Kozen, et al.
- 1984
(Show Context)
Citation Context ...x needs to be a test. By the correspondence with the modal box operator mentioned in Section 2.4, T � (a) = µ(|a]). In the propositional µ-calculus, this is known as the halting predicate (see, e.=-=g., [17]). It is easy to chec-=-k that ¬�(a ω ) is a fixed point of (x :: a → x). Hence, Corollary 4.11 Assume an omega MKA such that ν(|a〉) exists for all elements a. (a) T � (a) ≤ ¬�(a ω ). (b) T � (a) · a ω =... |

275 |
Regular Algebra and Finite Machines
- Conway
- 1971
(Show Context)
Citation Context ...f the abstract counterparts of the weakest liberal precondition and strongest liberal postcondition operators. A first treatment of this topic in the more restricted class of Standard Kleene Algebras =-=[8]-=- appeared in [14]; in the present paper we show that we can do without the assumption that the underlying lattice forms a complete Boolean algebra and that sequential composition is universally disjun... |

257 |
Refinement Calculus: A Systematic Introduction
- Back, Wright
- 1998
(Show Context)
Citation Context ...e the specification R of a while loop under the condition that R is strongly bounded, which guarantees termination after a finite amount of time. Further refinement theorems for loops can be found in =-=[1], -=-presented in the framework of predicate transformers. The main novelties in the present paper are the following. First, we fully generalize Mills’s approach to the nondeterministic case. This was al... |

211 |
On the caldus of relations
- Tarski
- 1941
(Show Context)
Citation Context ... , ∅, I), the algebra of homogeneous binary relations over some set M under relational composition ; . More generally than the concrete relation algebra REL, every abstract relation algebra (see e.g=-=. [6,30,32]-=-) is a KA. (c) A less abstract semiring than REL is the semiring PAT of path sets in a directed graph under union as addition and the extension of path concatenation to path sets (also known as fusion... |

186 | A completeness theorem for Kleene algebras and the algebra of regular events
- Kozen
- 1994
(Show Context)
Citation Context ...e modal semirings suffice for some applications, others require an explicit notion of iteration. This is achieved by extending idempotent semirings to Kleene algebras. Definition 4.5 A Kleene algebra =-=[19] is a structure (K, ∗ )-=- such that K is an idempotent semiring and the star ∗ satisfies, for a, b, c ∈ K, the unfold and 19sinduction laws 1 + a · a ∗ ≤ a ∗ , (∗-1) 1 + a ∗ · a ≤ a ∗ , (∗-2) b + a · c ... |

108 | Kleene algebra with tests
- Kozen
- 1997
(Show Context)
Citation Context ...ng binary homogeneous relations. Second, at the same time we abstract from relational semantics to the more general setting of modal Kleene algebras, an extension of Kozen’s Kleene algebra with test=-=s [20]-=- that allows the internalization of the abstract counterparts of the weakest liberal precondition and strongest liberal postcondition operators. A first treatment of this topic in the more restricted ... |

57 |
First Steps in Modal Logic
- Popkorn
- 1994
(Show Context)
Citation Context ...efinition 3.11 We call a deterministic iff MD(a) holds, where MD(a) ∆ ⇔ |a〉 ≤ |a]. (9) This reflects a well-known characterization of determinacy that is used in modal correspondence theory (s=-=ee e.g. [29]). We q-=-uote from [15]: Lemma 3.12 All tests are deterministic. If a is deterministic and b ≤ a, then b is deterministic as well. Now we can show Theorem 3.13 a deterministic ⇒ a ✷ b = a · b. PROOF. a ... |

42 |
Ströhlein: Relations and Graphs
- Schmidt, T
- 1993
(Show Context)
Citation Context ... , ∅, I), the algebra of homogeneous binary relations over some set M under relational composition ; . More generally than the concrete relation algebra REL, every abstract relation algebra (see e.g=-=. [6,30,32]-=-) is a KA. (c) A less abstract semiring than REL is the semiring PAT of path sets in a directed graph under union as addition and the extension of path concatenation to path sets (also known as fusion... |

41 | Kleene algebra with domain
- Desharnais, Möller, et al.
(Show Context)
Citation Context ...er is organized as follows. In Section 2 we first introduce test semirings; they admit a direct abstract angelic semantics of loop-free programs. Next, we axiomatize a domain and a codomain operation =-=[11]-=- that assign to an abstract program a representation of its initial and final states, respectively. Based on that, forward and backward diamond and box operators can be defined, leading to modal semir... |

39 |
Separation and Reduction
- Cohen
- 2000
(Show Context)
Citation Context ...loop-free programs and show a number of basic properties such as associativity of demonic composition. In Section 4, we introduce finite and infinite iteration, leading to modal Kleene [13] and omega =-=[7,26]-=- algebras and show a number of auxiliary properties. Following that, we present in Section 5 a generalization of the while statement verification rule of Mills. This is followed by an example of appli... |

35 | Relational algebraic semantics of deterministic and nondeterministic programs - Berghammer, Zierer - 1986 |

33 | Mili: On The Lattice of Specifications: Applications to a Specification Methodology. Formal Aspects of Computing 4 - Boudriga, Elloumi, et al. - 1992 |

26 |
der Woude. Demonic Operators and Monotype Factors
- Backhouse, van
- 1993
(Show Context)
Citation Context ...rsa. Definition 2.11 |a]p ∆ = ¬|a〉¬p, [a|p ∆ = ¬〈a|¬p. Example 2.12 In the modal semiring REL, the forward box operator coincides with the monotype factor as defined by Backhouse and van d=-=er Woude in [3]. ⊓⊔-=- In the sequel, when the direction of diamonds and boxes does not matter, we will use the notation 〈a〉 and [a]. 7sFrom (1) it follows that diamonds (boxes) are lower (upper) adjoints of Galois con... |

20 |
The New Math of Computer Programming
- Mills
- 1975
(Show Context)
Citation Context ...matic way to calculate the relational abstraction of a while loop directly from the definition, it is possible to check the correctness of any candidate abstraction. For deterministic programs, Mills =-=[22,23]-=- has described a checking method known as the while statement verification rule. We generalize this rule to nondeterministic loops. We note here that half of the generalized theorem has been shown by ... |

16 | Typed Kleene algebras
- Möller
- 1999
(Show Context)
Citation Context ...hat acts like a when the initial state satisfies p and aborts otherwise. Symmetrically, a · p describes a restriction of a in its possible final states. We now introduce an abstract domain operator �=-=�� [24] tha-=-t assigns to a the test that describes precisely its enabling states. Definition 2.5 A semiring with domain [11] (a �-semiring) is a structure (K, �), where K is an idempotent semiring and the dom... |

16 |
Derivation of graph and pointer algorithms
- Möller
- 1993
(Show Context)
Citation Context ...g than REL is the semiring PAT of path sets in a directed graph under union as addition and the extension of path concatenation to path sets (also known as fusion product) as multiplication (see e.g. =-=[25] for-=- details). Whereas REL only gives information about ex3sistence of a path between a pair of nodes, PAT gives the possibility to talk about different paths between that same pair. ⊓⊔ Programs and s... |

14 | Embedding a Demonic Semilattice in a Relation Algebra - Desharnais, Belkhiter, et al. - 1995 |

12 | A Calculus for Predicative Programming
- Sekerinski
- 1993
(Show Context)
Citation Context ...bed a checking method known as the while statement verification rule. We generalize this rule to nondeterministic loops. We note here that half of the generalized theorem has been shown by Sekerinski =-=[31], -=-who uses an approach based on predicative programming [18]. A related theorem has been given by Norvell [28] in the framework of predicative programming with time bounds. Norvell’s theorem shows how... |

11 | Characterizing determinacy in Kleene algebras
- Desharnais, Möller
(Show Context)
Citation Context ... ✷ c) = (b → c) · �a · (a → c) = �a · (a → c) = �(a ✷ c), so that (b → c) · (a ✷ c) = a ✷ c by (llp). Symmetrically, (a → c) · (b ✷ c) = b ✷ c. ⊓⊔ For the next the=-=orem we need a notion of determinacy [15]. Defini-=-tion 3.11 We call a deterministic iff MD(a) holds, where MD(a) ∆ ⇔ |a〉 ≤ |a]. (9) This reflects a well-known characterization of determinacy that is used in modal correspondence theory (see e.... |

10 |
Refinement and demonic semantics
- Desharnais, Mili, et al.
- 1997
(Show Context)
Citation Context ... The following example is rather contrived, but it is simple and fully illustrates the various cases that may happen. Consider the following loop, where the variable n ranges over the set of integers =-=[10,34]: Example 6.2 Consid-=-er the program do n > 0 → if n = 1 → n := 1 n = 1 → n := −3 n = 3 → n := 2 n = 3 → n := −1 n ≥ 4 → n := n−4 fi od Notice that all n > 0 such that n mod 4 = 1 may lead to terminatio... |

10 | Kleene under a demonic star
- Desharnais, Möller, et al.
- 2000
(Show Context)
Citation Context ...ounterparts of the weakest liberal precondition and strongest liberal postcondition operators. A first treatment of this topic in the more restricted class of Standard Kleene Algebras [8] appeared in =-=[14]-=-; in the present paper we show that we can do without the assumption that the underlying lattice forms a complete Boolean algebra and that sequential composition is universally disjunctive. In doing s... |

9 | Termination in modal Kleene algebra - Desharnais, Möller, et al. |

8 |
Relational heuristics for the design of deterministic programs
- Mili, Desharnais, et al.
- 1987
(Show Context)
Citation Context ...the case of relation algebra to arbitrary KAs. For more details on relational demonic semantics and demonic operators, see [3– 5,9,10,33]. Definition 3.1 We say that an element a refines an element =-=b [21], denoted by a ��-=-� b, iff �b ≤ �a ∧ �b · a ≤ b. It is easy to show that ⊑ is indeed a partial ordering. Since the following theorem employs meets, we first quote the following properties [26]. Lemma 3.2... |

7 |
An algebraic study of well-foundedness
- Goldblatt
- 1985
(Show Context)
Citation Context ...ogram admits infinite execution sequences. To this end we abstract a notion of termination for modal semirings from set-theoretic relations. A similar characterisation has been used, for instance, in =-=[16] for -=-related structures. A set-theoretic relation R ⊆ A×A on a set A is well-founded if there are no infinitely descending R-chains, that is, no infinite chains x0, x1, . . . such that (xi+1, xi) ∈ R.... |

7 | 2003b. Modal Kleene algebra and partial correctness
- Möller, Struth
(Show Context)
Citation Context ...K) is complete then a domain operation always exists. If test(K) is not complete, this need not be the case. Another important consequence of the axioms is that � preserves arbitrary existing suprem=-=a [27]. Exa-=-mple 2.6 (a) In LAN, the domain of a language L is ∅ if L = ∅ and {ε} otherwise; i.e., domain decides merely about being 0 or not. The same applies to all test semirings with discrete test algebr... |

5 |
Backhouse et al. Fixed point calculus
- C
- 1995
(Show Context)
Citation Context ... a is progressively finite. (b) In a modal omega algebra, ¬�(a ω ) · a is progressively finite. (c) Let f(x) ∆ = a · x + b. If a is progressively finite, then f has a unique fixed point, viz. =-=a ∗ · b [2]. (d) a ω = a ∗ · a ω-=- . PROOF. (a) Immediate from isotonicity of the omega operator. (b) Set b ∆ = ¬�(a ω ) · a. Since b ≤ a we get b ω ≤ a ω and hence �(b ω ) ≤ �(a ω ). 21sOn the other hand, by (d2)... |

3 |
Predicative programming: Parts i and ii
- Hehner
- 1984
(Show Context)
Citation Context ...ion rule. We generalize this rule to nondeterministic loops. We note here that half of the generalized theorem has been shown by Sekerinski [31], who uses an approach based on predicative programming =-=[18]. -=-A related theorem has been given by Norvell [28] in the framework of predicative programming with time bounds. Norvell’s theorem shows how to refine the specification R of a while loop under the con... |

3 | Predicative semantics of loops
- Norvell
- 1997
(Show Context)
Citation Context ...tic loops. We note here that half of the generalized theorem has been shown by Sekerinski [31], who uses an approach based on predicative programming [18]. A related theorem has been given by Norvell =-=[28] i-=-n the framework of predicative programming with time bounds. Norvell’s theorem shows how to refine the specification R of a while loop under the condition that R is strongly bounded, which guarantee... |

2 |
Sémantiques relationelles demoniaques et vérification de boucles nondéterministes
- Tchier
- 1996
(Show Context)
Citation Context ...e transformers. The main novelties in the present paper are the following. First, we fully generalize Mills’s approach to the nondeterministic case. This was already achieved by Desharnais and Tchie=-=r [33,34] u-=-sing binary homogeneous relations. Second, at the same time we abstract from relational semantics to the more general setting of modal Kleene algebras, an extension of Kozen’s Kleene algebra with te... |

2 |
Applying a generalization of a theorem of Mills to generalized looping structures. Colloquium Science and Engineering for Software Development, organized
- Tchier, Desharnais
- 1999
(Show Context)
Citation Context ...e transformers. The main novelties in the present paper are the following. First, we fully generalize Mills’s approach to the nondeterministic case. This was already achieved by Desharnais and Tchie=-=r [33,34] u-=-sing binary homogeneous relations. Second, at the same time we abstract from relational semantics to the more general setting of modal Kleene algebras, an extension of Kozen’s Kleene algebra with te... |

1 |
Applications of modal Kleene algebra — A survey. Technical report DIUL-RR-0401, Département d’informatique et de génie logiciel, Université
- Desharnais, Möller, et al.
- 2004
(Show Context)
Citation Context ...c semantics of loop-free programs and show a number of basic properties such as associativity of demonic composition. In Section 4, we introduce finite and infinite iteration, leading to modal Kleene =-=[13]-=- and omega [7,26] algebras and show a number of auxiliary properties. Following that, we present in Section 5 a generalization of the while statement verification rule of Mills. This is followed by an... |