## Kleene Algebra with Domain (2003)

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@MISC{Desharnais03kleenealgebra,

author = {Jules Desharnais and Bernhard Möller and Georg Struth},

title = { Kleene Algebra with Domain},

year = {2003}

}

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### Abstract

We propose Kleene algebra with domain (KAD), an extension of Kleene algebra with two equational axioms for a domain and a codomain operation, respectively. KAD considerably augments the expressibility of Kleene algebra, in particular for the specification and analysis of state transition systems. We develop the basic calculus, discuss some related theories and present the most important models of KAD. We demonstrate applicability by two examples: First, an algebraic reconstruction of Noethericity and well-foundedness. Second, an algebraic reconstruction of propositional Hoare logic.

### Citations

1404 |
A Discipline of Programming
- Dijkstra
- 1976
(Show Context)
Citation Context ..., formalisms based on algebra such as KAT, set-based formalisms such as B [Abrial 1996] and Z [Spivey 1988], where domain is extensively used, and semantic approaches based on predicate transformers [=-=Dijkstra 1976-=-]. As in KAT, we embed propositions into actions. As in PDL, we also provide a mapping from actions to propositions: the domain operation. Adding domain to KAT is only natural. Relations are the stand... |

823 | First-Order Dynamic Logic
- Harel
- 1979
(Show Context)
Citation Context ...oeller,struth}@informatik.uni-augsburg.de. c○ 2003 J. Desharnais, B. Möller, G. Struth2 · J. Desharnais et al. ture: Propositional dynamic logic (PDL) and its algebraic variants (see, among others, [=-=Harel et al. 2000-=-; Kozen 1979b; Németi 1981; Pratt 1988; 1991; Trnkova and Reiterman 1987]) and Kleene algebra with tests (KAT) [Kozen 1997]. In PDL, only propositions are first-class citizens. This gives the approach... |

635 |
Modal Logic: An Introduction
- Chellas
- 1980
(Show Context)
Citation Context ...apting a result by Goldblatt [Goldblatt 1985], we show that for transitive relations our concept is also equivalent to an algebraic variant of Löb’s formula from modal logic [Bull and Segerberg 1984; =-=Chellas 1980-=-]. Finally, we show that some simple and well-known properties of well-founded relations can be calculated in KAD in a simple and elegant way. Intuitively, a set-theoretic relation R ⊆ A × A is well-f... |

360 |
Propositional dynamic logic of regular programs
- Fischer, Ladner
- 1979
(Show Context)
Citation Context ... Establishing this kind of subsumption relation is a popular exercise for many logics and algebras for imperative programming languages. Hoare logic has, for instance, already been embedded into PDL [=-=Fischer and Ladner 1979-=-] and KAT [Kozen 2001]. Since KAD is an extension of KAT, our subsumption result is no surprise. However we believe that it is interesting for at least two reasons. First, in KAD, an encoding of the i... |

186 | A completeness theorem for Kleene algebras and the algebra of regular events
- Kozen
- 1994
(Show Context)
Citation Context ... c ≤ b or ∀ c . b ≤ c ⇒ a ≤ c. Likewise, a = b can be proved by showing ∀ c . c ≤ a ⇔ c ≤ b or ∀ c . b ≤ c ⇔ a ≤ c.Kleene Algebra with Domain · 5 2.2 Kozen Semirings A Kozen semiring (a K-semiring) [=-=Kozen 1994-=-a] is a structure (A, +, ·, ∗ , 0, 1), such that (A, +, ·, 0, 1) is an i-semiring, a ∗ b is the least pre-fixed point of the function λx.b + ax and ba ∗ is the least pre-fixed point of λx.b + xa. Form... |

157 |
The B-Book
- Abrial
- 1996
(Show Context)
Citation Context ...uniform view of hitherto separate approaches to program analysis and development: formalisms based on modal logic such as PDL, formalisms based on algebra such as KAT, set-based formalisms such as B [=-=Abrial 1996-=-] and Z [Spivey 1988], where domain is extensively used, and semantic approaches based on predicate transformers [Dijkstra 1976]. As in KAT, we embed propositions into actions. As in PDL, we also prov... |

123 | Two-Dimensional Modal Logic - Segerberg - 1973 |

109 | Kleene algebra with tests
- Kozen
- 1997
(Show Context)
Citation Context ...al dynamic logic (PDL) and its algebraic variants (see, among others, [Harel et al. 2000; Kozen 1979b; Németi 1981; Pratt 1988; 1991; Trnkova and Reiterman 1987]) and Kleene algebra with tests (KAT) [=-=Kozen 1997-=-]. In PDL, only propositions are first-class citizens. This gives the approach a logical flavor. While equivalence of propositions is directly expressible, actions can only be observed indirectly thro... |

97 | Verification of sequential and concurrent programs - Apt, Olderog - 1997 |

83 | Quantales and (noncommutative) linear logic - Yetter - 1990 |

75 | Methods and applications of (max,+) linear algebra
- Gaubert, Plus
- 1997
(Show Context)
Citation Context .... Example 2.14. Set N−∞ = N ∪ {−∞} and consider the structure (max, +) = (N−∞, max, +, −∞, 0) with operations defined in the obvious way. Then (max, +) is an i-semiring, called the max-plus semiring [=-=Gaubert and Plus 1997-=-]. Its natural ordering coincides with the standard ordering on N−∞. Unlike the tropical semiring, the max-plus semiring cannot be extended to a K-semiring. The reason is that for a > 0 the set {a n |... |

70 | The Foundations of Program Verification - Loeckx, Sieber - 1984 |

50 | Action logic and pure induction
- Pratt
- 1990
(Show Context)
Citation Context ...t can be axiomatized by a finite set of equations. The following lemma is immediate. Lemma 8.1. TSD is a finitely based variety. The next lemma is not so immediate. It has been shown in [Kozen 1994b; =-=Pratt 1990-=-] that KA with a residuation operation is a finitely based variety. The same phenomenon might occur when adding a domain or codomain operation. The following lemma shows that this is not the case. A s... |

49 |
Understanding Z
- Spivey
- 1988
(Show Context)
Citation Context ...erto separate approaches to program analysis and development: formalisms based on modal logic such as PDL, formalisms based on algebra such as KAT, set-based formalisms such as B [Abrial 1996] and Z [=-=Spivey 1988-=-], where domain is extensively used, and semantic approaches based on predicate transformers [Dijkstra 1976]. As in KAT, we embed propositions into actions. As in PDL, we also provide a mapping from a... |

41 | On Hoare logic and Kleene algebra with tests - Kozen - 2000 |

39 | Separation and reduction - Cohen - 2000 |

36 |
Semirings and Formal Power Series: Their Relevance to Formal Languages and Automata
- Kuich
- 1997
(Show Context)
Citation Context ...in Example 2.9. Example 2.13. Set N∞ = N ∪ {∞} and define the operations min and + in the obvious way. Then the structure (min, +) = (N∞, min, +, ∞, 0) is an i-semiring, called the tropical semiring [=-=Kuich 1997-=-]. Its natural ordering is the converse of the standard ordering on N∞. Hence 0 — the semiring multiplicative unit — is the largest element, so that by (10) (min, +) can uniquely be extended to a K-se... |

22 | F.: Kleene algebra with tests: Completeness and decidability
- Kozen, Smith
- 1996
(Show Context)
Citation Context ...expressed and analyzed. Each approach has its particular advantages and merits. Note, however, that PDL is EXPTIME-complete [Harel et al. 2000], while the equational theory of KAT is PSPACE-complete [=-=Kozen and Smith 1996-=-]. We propose Kleene algebra with domain (KAD) as an extension of KAT and as a reconciliation of KAT and PDL with equal opportunities for propositions and actions. We believe that KAD not only combine... |

17 | Dynamic Algebras: Examples, Constructions, Applications - Pratt - 1979 |

16 | Galois connections presented calculationally
- Aarts
(Show Context)
Citation Context ...operator” for relations and KAD supports abstract algebraic reasoning with it. Domain has already been defined algebraically in extensions of Kleene algebra like quantales and relation algebras (cf. [=-=Aarts 1992-=-; Desharnais and Möller 2001; Desharnais et al. 2000; Schmidt and Ströhlein 1993]). But there is no straightforward transfer to KAT. Again, KAD offers several benefits. In opposition to relation algeb... |

16 | Typed Kleene algebras
- Möller
- 1999
(Show Context)
Citation Context ... δ(aδ(b)) ≤ δ(a) ≤ p. In the second case, 4.6 Domain in b-Monoids δ(aδ(b)) = δ(a0) = δ(0) = 0 ≤ p. Definitions for predomain have originally been given for b-monoids and b-quantales (cf. [Aarts 1992; =-=Möller 1999-=-; Desharnais and Möller 2001]). There, the situation is considerably simpler. First, as we have pointed out in Section 3, the entire set of subidentities of a b-monoid forms a Boolean sublattice and t... |

15 | Lattice Theory, volume 25 of Colloquium Publications - Birkhoff - 1984 |

13 | On the duality of dynamic algebras and Kripke models - Kozen - 1981 |

12 | From kleene algebra to refinement algebra - Wright - 2002 |

11 | Characterizing determinacy in Kleene algebras
- Desharnais, Möller
(Show Context)
Citation Context ...r relations and KAD supports abstract algebraic reasoning with it. Domain has already been defined algebraically in extensions of Kleene algebra like quantales and relation algebras (cf. [Aarts 1992; =-=Desharnais and Möller 2001-=-; Desharnais et al. 2000; Schmidt and Ströhlein 1993]). But there is no straightforward transfer to KAT. Again, KAD offers several benefits. In opposition to relation algebra, it focuses entirely on t... |

10 | Kleene under a demonic star
- Desharnais, Möller, et al.
- 2000
(Show Context)
Citation Context ... abstract algebraic reasoning with it. Domain has already been defined algebraically in extensions of Kleene algebra like quantales and relation algebras (cf. [Aarts 1992; Desharnais and Möller 2001; =-=Desharnais et al. 2000-=-; Schmidt and Ströhlein 1993]). But there is no straightforward transfer to KAT. Again, KAD offers several benefits. In opposition to relation algebra, it focuses entirely on the essential operations ... |

10 | On action algebras
- Kozen
- 1994
(Show Context)
Citation Context ... c ≤ b or ∀ c . b ≤ c ⇒ a ≤ c. Likewise, a = b can be proved by showing ∀ c . c ≤ a ⇔ c ≤ b or ∀ c . b ≤ c ⇔ a ≤ c.Kleene Algebra with Domain · 5 2.2 Kozen Semirings A Kozen semiring (a K-semiring) [=-=Kozen 1994-=-a] is a structure (A, +, ·, ∗ , 0, 1), such that (A, +, ·, 0, 1) is an i-semiring, a ∗ b is the least pre-fixed point of the function λx.b + ax and ba ∗ is the least pre-fixed point of λx.b + xa. Form... |

9 |
The variety of Kleene algebras with conversion is not finitely based. Theoretical Computer Science 230
- Crvenkovič, Dolinka, et al.
- 2000
(Show Context)
Citation Context ...a, (c1) (a + b) ◦ = a ◦ + b ◦ , (c2) (ab) ◦ = b ◦ a ◦ . (c3) An i-semiring with weak converse is an i-semiring with preconverse such that all p ≤ 1 satisfy p ◦ ≤ p. (iii) An i-semiring with converse [=-=Crvenkovič et al. 2000-=-] is an i-semiring with preconverse that satisfies the equation It is easy to show that the properties a ≤ aa ◦ a. hold in every i-semiring with preconverse. The equation (c4) (c5) 1 ◦ = 1, (39) 0 ◦ =... |

7 | 2003b. Modal Kleene algebra and partial correctness - Möller, Struth |

7 |
An algebraic study of well-foundedness
- Goldblatt
- 1985
(Show Context)
Citation Context ...a [Cohen 2000], an extension of KA with infinite iteration that is defined as a greatest fixed point by expressions similar to (∗-1), (∗-2), (∗-3) and (∗-4). Moreover, adapting a result by Goldblatt [=-=Goldblatt 1985-=-], we show that for transitive relations our concept is also equivalent to an algebraic variant of Löb’s formula from modal logic [Bull and Segerberg 1984; Chellas 1980]. Finally, we show that some si... |

6 |
Regular Algebra and Finite State Machines
- Conway
- 1971
(Show Context)
Citation Context ...died in [Birkhoff 1984]. An l-monoid is bounded if it has a least element 0 and a greatest element ⊤. It is complete if the underlying lattice is. A quantale [Mulvey 1986] or standard Kleene algebra [=-=Conway 1971-=-] is a complete l-monoid in which left and right multiplication is universally additive. Quantales have been investigated in contexts like the logic of quantum mechanics [Mulvey 1986] and algebraic mo... |

6 |
Kleene modules
- Ehm, Möller, et al.
(Show Context)
Citation Context ...ity, completeness or representability of KAD have not been investigated in this text. The same holds for the apparent relation to modal algebras and in particular algebraic variants of PDL (cf. e.g. [=-=Ehm et al. 2003-=-]). On the practical side, it might be interesting to continue our investigations of termination analysis and greedy algorithms [Möller and Struth 2003a]. Moreover, a combination of the two methods fo... |

6 |
Lattice theory, in: Colloquium Publications, vol
- Birkhoff
- 1967
(Show Context)
Citation Context ...ered monoid (an l-monoid) is a structure (A, +, ⊓, ·, 1), such that (A, +, ⊓) is a lattice, (A, ·, 1) is a monoid and left and right multiplication are additive. l-monoids are extensively studied in [=-=Birkhoff 1984-=-]. An l-monoid is bounded if it has a least element 0 and a greatest element ⊤. It is complete if the underlying lattice is. A quantale [Mulvey 1986] or standard Kleene algebra [Conway 1971] is a comp... |

5 | Sequential calculus - Hoare, Karger - 1995 |

4 | Equational axioms of test algebra
- Hollenberg
- 1996
(Show Context)
Citation Context ...tion is a finitely based variety. The same phenomenon might occur when adding a domain or codomain operation. The following lemma shows that this is not the case. A similar argument has been used in [=-=Hollenberg 1997-=-] for algebras related to PDL. Lemma 8.2. KAP and KAD are not finitely based varieties. Proof. In [Conway 1971], p. 106, Conway gives an algebra Ap for showing that the algebra of regular events (cf. ... |

4 |
Dynamic algebras of programs
- Németi
- 1981
(Show Context)
Citation Context ...gsburg.de. c○ 2003 J. Desharnais, B. Möller, G. Struth2 · J. Desharnais et al. ture: Propositional dynamic logic (PDL) and its algebraic variants (see, among others, [Harel et al. 2000; Kozen 1979b; =-=Németi 1981-=-; Pratt 1988; 1991; Trnkova and Reiterman 1987]) and Kleene algebra with tests (KAT) [Kozen 1997]. In PDL, only propositions are first-class citizens. This gives the approach a logical flavor. While e... |

4 |
The Foundations of Program Verification, 2nd ed
- Loeckx, Sieber
- 1987
(Show Context)
Citation Context ...n KAD, an encoding of the inference rules of the Hoare calculus is much more crisp and clear and so are their correctness proofs. Second, the properties of the standard partial correctness semantics [=-=Loeckx and Sieber 1987-=-;36 · J. Desharnais et al. Apt and Olderog 1997] for Hoare logic mirror precisely those of domain, so that KAD may be considered a natural abstract algebraic semantics for propositional Hoare logic. ... |

3 |
A representation theorem for ∗-free PDL
- Kozen
- 1979
(Show Context)
Citation Context ...rmatik.uni-augsburg.de. c○ 2003 J. Desharnais, B. Möller, G. Struth2 · J. Desharnais et al. ture: Propositional dynamic logic (PDL) and its algebraic variants (see, among others, [Harel et al. 2000; =-=Kozen 1979-=-b; Németi 1981; Pratt 1988; 1991; Trnkova and Reiterman 1987]) and Kleene algebra with tests (KAT) [Kozen 1997]. In PDL, only propositions are first-class citizens. This gives the approach a logical f... |

3 |
Dynamic algebras with tests
- Trnkova, Reiterman
- 1987
(Show Context)
Citation Context ...s, B. Möller, G. Struth2 · J. Desharnais et al. ture: Propositional dynamic logic (PDL) and its algebraic variants (see, among others, [Harel et al. 2000; Kozen 1979b; Németi 1981; Pratt 1988; 1991; =-=Trnkova and Reiterman 1987-=-]) and Kleene algebra with tests (KAT) [Kozen 1997]. In PDL, only propositions are first-class citizens. This gives the approach a logical flavor. While equivalence of propositions is directly express... |

2 | Hypotheses in Kleene algebra. Unpublished manuscript - Cohen - 1994 |

2 | Greedy-like algorithms in Kleene algebra
- Möller, Struth
- 2003
(Show Context)
Citation Context ...d KAT in this context is the possibility to express the weakest precondition operator as wlp(a, p) = (a : p ′ ) ′ . However, to keep matters short, we stay with Hoare logic in this text and refer to [=-=Möller and Struth 2003-=-b] for a full account of wlp in KAD. We start by encoding the relevant programming constructs in KA. a ; b = ab, (78) if p then a else b = pa + p ′ b, (79) while p do a = (pa) ∗ p ′ . (80) We now brie... |

1 |
Dynamic logic as a well-behaved fragment of relation algebras
- Pratt
- 1988
(Show Context)
Citation Context ... 2003 J. Desharnais, B. Möller, G. Struth2 · J. Desharnais et al. ture: Propositional dynamic logic (PDL) and its algebraic variants (see, among others, [Harel et al. 2000; Kozen 1979b; Németi 1981; =-=Pratt 1988-=-; 1991; Trnkova and Reiterman 1987]) and Kleene algebra with tests (KAT) [Kozen 1997]. In PDL, only propositions are first-class citizens. This gives the approach a logical flavor. While equivalence o... |