## Dynamic Algebras as a well-behaved fragment of Relation Algebras (1990)

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Venue: | In Algebraic Logic and Universal Algebra in Computer Science, LNCS 425 |

Citations: | 33 - 5 self |

### BibTeX

@INPROCEEDINGS{Pratt90dynamicalgebras,

author = {Vaughan Pratt},

title = {Dynamic Algebras as a well-behaved fragment of Relation Algebras},

booktitle = {In Algebraic Logic and Universal Algebra in Computer Science, LNCS 425},

year = {1990},

pages = {77--110},

publisher = {Springer-Verlag}

}

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### Abstract

The varieties RA of relation algebras and DA of dynamic algebras are similar with regard to definitional capacity, admitting essentially the same equational definitions of converse and star. They differ with regard to completeness and decidability. The RA definitions that are incomplete with respect to representable relation algebras, when expressed in their DA form are complete with respect to representable dynamic algebras. Moreover, whereas the theory of RA is undecidable, that of DA is decidable in exponential time. These results follow from representability of the free intensional dynamic algebras. Dept. of Computer Science, Stanford, CA 94305. This paper is based on a talk given at the conference Algebra and Computer Science, Ames, Iowa, June 2-4, 1988. It will appear in the proceedings of that conference, to be published by SpringerVerlag in the Lecture Notes in Computer Science series. This work was supported by the National Science Foundation under grant number CCR-8814921 ...

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Citation Context ...ation, while test enables the rational performance of choice and iteration, expressed deterministically with if p then a else b and while p do a respectively, and more generally with guarded commands =-=[Dij76]-=-. The logic-of-programs significance of dynamic algebra is as follows. The set X is viewed as the states of a computer. Binary relations are viewed as programs: the meaning of (x; y) as an element of ... |

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Citation Context ...ater or equal to a. (Cl) asserts that it is the least such. There can be at most one least such, whence a is uniquely defined. Star, ancestral, or reflexive transitive closure, is variously notated a =-=[Kle56]-=-, a, and a 0 [Ded01, Sch95], while transitive closure has been written a + , a ! [NT77], and a 00 [Sch95]. The firstmentioned in each of these lists is the notation universally used in computer scienc... |

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Citation Context ...ng defined externally to a dynamic algebra, as is customary with algebras of scalars. These equations are essentially the Segerberg axioms [Seg77] for Fischer and Ladner's propositional dynamic logic =-=[FL79]-=-, translated into equational form. The differences from Segerberg's system are the equations for converse, which are due to Parikh [Par78], and the weakening to monotonicity of the equations that woul... |

277 |
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Citation Context ... Kleenean algebra when complement is dropped. Redko [Red64] has shown that the equational theory of the representable Kleenean algebras without converse is not finitely axiomatizable. Moreover Conway =-=[Con71]-=- has enumerated several finite models of this theory which do not satisfy axiom (S3) (given in the section on star for Boolean monoids) expressing that a is the least reflexive transitive element domi... |

236 | Modeling concurrency with partial orders
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Citation Context ...bras to process algebras may be described as inverse abstraction (back to relations as sets of pairs), two generalizations (from pairs as labeled linearly ordered doubletons to labeled partial orders =-=[Pra86]-=-, and from partial orders to generalized metrics [CCMP89]), and abstraction back to algebra, with the resulting logic having models far removed from binary relations, yet remaining remarkably like RA.... |

212 |
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Citation Context ...as well aware of the involutary nature of converse, which as we have seen entails KR!RK and KL!LK. In nominating Peirce rather than De Morgan as the "creator of the modern theory of relations&quo=-=t; Tarski [Tar41]-=- appears not to have taken Theorem K into account. Peirce gave an equivalent equational characterization of converse in 1870, [Pei33], but was no better equipped than De Morgan to appreciate its compl... |

210 | Semantical considerations on Floyd-Hoare logic, in
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Citation Context ...matization), and vice versa. In this account of RA, converse and star become siblings, as they are in DA. In fact, although converse appeared in the first dynamic logic paper four pages ahead of star =-=[Pra76]-=-, the relative importance of star to programmers had caused converse to temporarily disappear from dynamic logic by the time of Segerberg's axiomatization of propositional dynamic logic [Seg77]. It wa... |

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Citation Context ...sses. One of the classes consisted of the models of Segerberg's axioms, which I then called Hoare algebras. That this semantic proof method is complete is an immediate corollary of Birkhoff's theorem =-=[Bir35]-=- that every class closed under homomorphisms, subalgebras, and direct products forms a variety. The one step in this proof that I did not supply was the inclusion \Theta FKRI ` \Theta DA , FKRI denoti... |

155 |
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Citation Context ...quations and thus making the variety BA of Boolean algebras finitely axiomatized. Evidently RBA ` BA. It is one of nature's little pranks that RBA is a variety, but it is a bigger prank that RBA = BA =-=[Sto36]-=-. The class RRA of representable relation algebras is the quasivariety generated by algebras Rel X . The variety RA of relation algebras is finitely based, with RRA ` RA. We again have the little pran... |

130 | Partially Ordered Algebraic Systems - Fuchs - 1962 |

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Citation Context ... relation algebra [TG87] and dynamic algebra [Koz79b, Pra79a]. The nonalgebraic origins of the former lie in the subject of foundations of mathematics, and of the latter in that of logics of programs =-=[KT89]-=-. When the organizers of this conference very kindly asked me to talk on a subject of my choice it seemed a foregone conclusion that a conference organized by relation algebraists would expect a talk ... |

122 |
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Citation Context ...der grant number CCR-8814921 1 Introduction 1.1 Overview Binary relations have proved a fruitful framework in both logic and computer science. In logic they have served as the eliminator of variables =-=[TG87]-=-, and in computer science as the illuminator of software [dBdR72, Pra76]. One finds the algebraic versions of these topics today under the respective rubrics of relation algebra [TG87] and dynamic alg... |

84 |
V.: 1979, Models of program logics
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(Show Context)
Citation Context ...sivariety and HSP(RDA)=HSP(IDA). 4.6 Computational Complexity Theorem 22 There exist 1 ! c ! d such that \Theta DA and its complement �� \Theta DA are not in DTIME(c n ) [FL79] but are in DTIME(d =-=n ) [Pra79b]-=-. That is, the time required to deterministically test either satisfiability or validity of dynamic algebra equations is one exponential in the number n of occurrences of variables in the formula, a b... |

65 |
The representation of relational algebras
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- 1950
(Show Context)
Citation Context ... The variety RA of relation algebras is finitely based, with RRA ` RA. We again have the little prank, that RRA is a variety [Tar55]. The difference is that we no longer have the big prank: RA 6= RRA =-=[Lyn50]-=-, and although there are infinitely many finitely axiomatizable varieties between these two, RRA itself is not finitely axiomatizable [Mon64]. With dynamic algebras the situation is in between these t... |

57 |
Tarski: Distributive and modular laws in the arithmetic of relation algebras
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- 1951
(Show Context)
Citation Context ...a unary operation called converse, notated a, satisfying anb = (a; b \Gamma ) \Gamma and a=b = (a \Gamma ; b) \Gamma . This definition of the class RA is the content of Theorem 2.2 of Chin and Tarski =-=[CT51]-=-, which together with the axioms for a Boolean monoid suffice to axiomatize RA. 2 It is customary in giving equational axiomatizations of RA to mention converse in at least four equations, which invar... |

54 | Residuated lattices - Ward, Dilworth - 1939 |

52 |
A completeness theorem in the modal logic of programs
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- 1977
(Show Context)
Citation Context ...f star [Pra76], the relative importance of star to programmers had caused converse to temporarily disappear from dynamic logic by the time of Segerberg's axiomatization of propositional dynamic logic =-=[Seg77]-=-. It was restored and straightforwardly axiomatized the following year by Parikh [Par78], whose axioms have here become our single-equation definition of RA converse. The surviving advantages of DA re... |

49 |
Vorlesungen über die Algebra der Logik
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(Show Context)
Citation Context ... observed their interdefinability, notating them ab, a b , a b and adjoining a fourth connective for relative sum, which he subsequently notated ayb; the notation a + b we use here is due to Schroder =-=[Sch95], subseque-=-ntly (1897) adopted in modified form (the "scorpion tail") by Peirce. For 0; 1; 0'; 1' Peirce writes 0; 1;n; 1 respectively, and calls relations as1', as0', as1', and as0' respectively concu... |

45 |
On representable relation algebras
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(Show Context)
Citation Context ...ference is that we no longer have the big prank: RA 6= RRA [Lyn50], and although there are infinitely many finitely axiomatizable varieties between these two, RRA itself is not finitely axiomatizable =-=[Mon64]-=-. With dynamic algebras the situation is in between these two. The class RDA of representable dynamic algebras is the quasivariety generated by algebras Kri X . The variety DA of dynamic algebras is f... |

45 |
A near-optimal method for reasoning about action
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- 1980
(Show Context)
Citation Context ...f deciding PDL theoremhood, I undertook to find a structure that would work for me both as an understandable completeness proof and a decision method. This resulted in [Pra78], extensively revised as =-=[Pra80b]. The form-=-er axiomatized dynamic logic in the language of Gentzen sequents, the latter extended this approach to a theory whose atomic formulas were u j= p meaning "state u satisfies proposition p" an... |

38 |
Contributions to the theory of models
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(Show Context)
Citation Context ...RRA) is a representable Boolean monoid equipped with converse having its usual meaning for binary relations. Equivalently it is a subalgebra of an algebra Eqv E. The class RRA of such forms a variety =-=[Tar55]-=-. 3.1 Star The operation star, or ancestral, or reflexive transitive closure, which we shall notate a , resembles converse in some respects. It is a unary operation definable in any ordered monoid. Wh... |

34 |
A practical decision method for propositional dynamic logic: Preliminary report
- Pratt
- 1978
(Show Context)
Citation Context ...he proof theory to the problem of deciding PDL theoremhood, I undertook to find a structure that would work for me both as an understandable completeness proof and a decision method. This resulted in =-=[Pra78], ext-=-ensively revised as [Pra80b]. The former axiomatized dynamic logic in the language of Gentzen sequents, the latter extended this approach to a theory whose atomic formulas were u j= p meaning "st... |

30 | A calculus for recursive programs schemes - Bakker, Roever - 1972 |

29 | Temporal structures - Casley, Crew, et al. - 1991 |

29 |
On defining relations for the algebra of regular events
- Redko
- 1964
(Show Context)
Citation Context ... not come up here since we no longer have either intersection or a top element. Any representable relation algebra with star becomes a representable Kleenean algebra when complement is dropped. Redko =-=[Red64]-=- has shown that the equational theory of the representable Kleenean algebras without converse is not finitely axiomatizable. Moreover Conway [Con71] has enumerated several finite models of this theory... |

27 |
Relation algebras with transitive closure
- Ng
- 1984
(Show Context)
Citation Context ...b ! asb (Cl) as(a + b) & (a=a) = a=a (Ta) as(a + b) & (a(a=a)) + = a(a=a) (Ng) where a + abbreviates a; a . Proof: (Ta) appears in [NT77] (in the equivalent form a; (ana) + = a), and is attributed in =-=[Ng84]-=- to Tarski. (Ng) appears in [Ng84], where the equivalence of (Cl), (Ta), and (Ng) are treated. We showed the equivalence of (S3) and (Ind) above, and that (S3) implied (Cl), assuming (S1) and (S2). It... |

27 |
Description of a notation for the logic of relatives, resulting from an amplification of the conceptions of Boole’s calculus of logic, Memoirs of the American Academy of Sciences 9
- Peirce
- 1870
(Show Context)
Citation Context ...e Morgan as the "creator of the modern theory of relations" Tarski [Tar41] appears not to have taken Theorem K into account. Peirce gave an equivalent equational characterization of converse=-= in 1870, [Pei33]-=-, but was no better equipped than De Morgan to appreciate its completeness. a; bsb ! a ; bsb (S3) We say that a converges at b when a; bsb. Thus (S2) asserts that a converges at a , and (S3) asserts t... |

25 | Varieties of relation algebras - Jónsson - 1982 |

24 | Representation problems for relation algebras - Jónsson, Tarski - 1192 |

22 |
On the syllogism, no. iv, and on the logic of relations
- Morgan
(Show Context)
Citation Context ... by the binary relation a, an element of R. We require that if a 6= b then there exist x; y such that exactly one of (x; a; y) and (x; b; y) hold; this ensures that each element of R acts 1 De Morgan =-=[DM64] writes ab, ab 0 , and a 0 b f-=-or a; b, a/-b, and a,!b respectively, construing them as "an a of a b of", "an a of every b of", and "an a of none but b's of," and asserting their sufficiency. Peirce [P... |

20 | On induction vs. *-continuity
- Kozen
- 1981
(Show Context)
Citation Context ...s of elements from the same X i . Proposition 12 RDA ` IDA. Proof: Every Kri X is separable. RDA is the ISP closure of the Kri X while IDA is the ISP closure of SDA. It follows from a result of Kozen =-=[Koz81]-=- that the converse does not hold. In the next example the Boolean and Kleenean elements are languages as sets of strings over a common alphabet X . For languages we must omit converse and test, but se... |

18 |
Boolean modules
- Brink
- 1981
(Show Context)
Citation Context ...uld be a nice idea to organize this paper as a comparison of the merits of relation and dynamic algebras. It also seemed a good idea to build up these notions from Boolean monoids and Boolean modules =-=[Bri81]-=- respectively, with the former mingling logical and relative notions in a single sort and the latter keeping them segregated. My initial impression was that modules improved on monoids in the areas of... |

17 | Dynamic Algebras: Examples, Constructions, Applications
- Pratt
- 1979
(Show Context)
Citation Context ...of representable dynamic algebras is the quasivariety generated by algebras Kri X . The variety DA of dynamic algebras is finitely based, with RDA ` IDA ` DA. Now the free IDA's are residually finite =-=[Pra79a]-=- and moreover are representable [N'em82], whence the equational theory of IDA completely axiomatizes RDA, i.e. RDA and IDA have the same equational theory. Also RDA, IDA, and DA have the same Boolean ... |

15 | A satisfiability tester for non-clausal propositional calculus
- Gelder
- 1988
(Show Context)
Citation Context ...mproved by more than by a polynomial of degree log c d. For comparison, the best deterministic procedure known for pure Boolean equations, i.e. propositional calculus, requires time 2 n=4 or 1:1892 n =-=[VG88]-=-, down to 1:093 n for equations t = 0 when t is in conjunctive normal form. Fischer and Ladner do not supply a specific value for c, but their proof is constructive and if pushed hard might conceivabl... |

13 |
On the duality of dynamic algebras and Kripke models
- Kozen
- 1981
(Show Context)
Citation Context ...e if and only if ha i = hai. (An operator f is reflexive when psf(p) and transitive when f(f(p))sf(p).) A preKleenean algebra is the Kleenean algebra of a dynamic algebra. A separable dynamic algebra =-=[Koz79a]-=- is one for which 3 is injective, satisfying the \Pi 0 2 sentence 8p[haip = hbip] ! a = b. An intensional dynamic algebra is an algebra isomorphic to a subalgebra of a separable dynamic algebra. A Kle... |

13 |
A decidable mu-calculus
- Pratt
- 1981
(Show Context)
Citation Context ...grip on why Segerberg's axioms were complete. Further reflection on the meaning of Segerberg's induction axiom led me to propose a formulation of the least-fixpoint or ��-calculus for Boolean modu=-=les [Pra81]-=-. This would appear to be the first time that the notions of least fixpoint and Boolean module were brought together. This juxtaposition has since enjoyed considerable attention from the computer scie... |

12 |
Relation algebras with transitive closure, abstract 742-02-09
- Ng, Tarski
- 1977
(Show Context)
Citation Context ...st such, whence a is uniquely defined. Star, ancestral, or reflexive transitive closure, is variously notated a [Kle56], a, and a 0 [Ded01, Sch95], while transitive closure has been written a + , a ! =-=[NT77]-=-, and a 00 [Sch95]. The firstmentioned in each of these lists is the notation universally used in computer science and is adopted here. In any ordered monoid a + is definable in terms of a via the equ... |

12 |
Dynamic algebras and the nature of induction
- Pratt
- 1980
(Show Context)
Citation Context ...eenean congruence. This was first observed for -continuous dynamic algebras by Kozen [Koz79b, Koz80] (see the history section). It was generalized to the weaker Segerberg notion of star by the author =-=[Pra80a]-=-. Proposition 15 Every dynamic algebra D has a unique quotient D 0 in IDA such that B = B 0 . Proof: By the preceding proposition, the unique factorization of the function 3 as the composition of an i... |

11 |
A decision procedure for the propositional -calculus
- Kozen, Parikh
- 1984
(Show Context)
Citation Context ...point and Boolean module were brought together. This juxtaposition has since enjoyed considerable attention from the computer science community, most notably in its expression as Kozen's L�� calcu=-=lus [KP83]-=-. Acknowledgments. I am very grateful to Dexter Kozen for his insights in 1979 which were most helpful to me in clarifying my thinking on this subject. Much email traffic and long phone calls to Georg... |

10 |
A representation theorem for models of *-free PDL
- Kozen
- 1979
(Show Context)
Citation Context ...g the proper HSP formulation of the completeness proof. This however I was unable to do in time for the FOCS-79 deadline. Shortly after submitting that paper I received a manuscript from Dexter Kozen =-=[Koz79b]-=-. It gave the full details of Kozen's dynamic algebra (a term I subsequently adopted for the proceedings version of my paper). Kozen's notion was the same as mine in most respects. The biggest differe... |

10 |
A decision procedure for the propositional μ-calculus
- Kozen, Parikh
- 1984
(Show Context)
Citation Context ...xpoint and Boolean module were brought together. This juxtaposition has since enjoyed considerable attention from the computer science community, most notably in its expression as Kozen’s Lµ calculus =-=[KP83]-=-. Acknowledgments. I am very grateful to Dexter Kozen for his insights in 1979 which were most helpful to me in clarifying my thinking on this subject. Much email traffic and long phone calls to Georg... |

7 | Lattice Theory, volume 25 of Am - Birkhoff - 1948 |

7 | Ultraproducts of algebraists - Eklof - 1977 |

4 |
Essays on the Theory of Numbers. The Open Court Publishing Company. Online at http://www.gutenberg.org/ etext/21016 [3
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Citation Context ...lattice monoid, a is definable via either of the equations a = a + + 1' or a = (a + 1') + . Star was first studied in detail by Schroder [Sch95], who notated it a 0 following Dedekind's 1888 notation =-=[Ded01] for the "chain-=-" of a "transformation" a. There Dedekind gave three axioms for chains, in paragraphs numbered respectively 45-47. Schroder translated Dedekind's axioms into the language of binary rela... |

4 | Every free algebra in the variety generated by the representable dynamic algebras is separable and representable - Németi - 1982 |

2 |
A completeness result for a propositional dynamic logic
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(Show Context)
Citation Context ...mporarily disappear from dynamic logic by the time of Segerberg's axiomatization of propositional dynamic logic [Seg77]. It was restored and straightforwardly axiomatized the following year by Parikh =-=[Par78]-=-, whose axioms have here become our single-equation definition of RA converse. The surviving advantages of DA remain those of decidability of the equational theory, and representability, as per Kozen ... |

1 | On PC \Delta classes in the theory of models - Makkai - 1964 |