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Hybrid Languages and Temporal Logic University of Pennsylvania ScholarlyCommons IRCS Technical Reports Series Institute for Research in Cognitive Science January 1998 Hybrid Languages and Temporal Logic Patrick Blackburn Universität des Saarlandes, [email protected] Miroslava Tzakova Max-Planck-Institut für Informatik, [email protected] Follow this and additional works at: https://repository.upenn.edu/ircs_reports Blackburn, Patrick and Tzakova, Miroslava, "Hybrid Languages and Temporal Logic" (1998). IRCS Technical Reports Series. 61. https://repository.upenn.edu/ircs_reports/61 University of Pennsylvania Institute for Research in Cognitive Science Technical Report No. IRCS-98-16. This paper is posted at ScholarlyCommons. https://repository.upenn.edu/ircs_reports/61 For more information, please contact [email protected]. Hybrid Languages and Temporal Logic Abstract Hybridization is a method invented by Arthur Prior for extending the expressive power of modal languages. Although developed in interesting ways by Robert Bull, and by the Sofia school (notably, George Gargov, Valentin Goranko, Solomon Passy and Tinko Tinchev) the method remains little known. In our view this has deprived temporal logic of a valuable tool. The aim of the paper is to explain why hybridization is useful in temporal logic. We make two major points, the first technical, the second conceptual. First, we show that hybridization gives rise to well- behaved logics that exhibit an interesting synergy between modal and classical ideas. This synergy, obvious for hybrid languages with full first-order expressive strength, is demonstrated for a weaker local language capable of defining the Until operator, we provide a minimal axiomatization, and show that in a wide range of temporally interesting cases extended completeness results can be obtained automatically. Second, we argue that the idea of sorted atomic symbols which underpins the hybrid enterprise can be developed further. To illustrate this, we discuss the advantages and disadvantages of a simple hybrid language which can quantify over paths. Comments University of Pennsylvania Institute for Research in Cognitive Science Technical Report No. IRCS-98-16. This technical report is available at ScholarlyCommons: https://repository.upenn.edu/ircs_reports/61 Hybrid Languages and Temp oral Logic Patrick Blackburn Miroslava Tzakova Computerlinguistik MaxPlanckInstitut fur Informatik Universitat des Saarlandes Im Stadtwald Saarbruc ken Saarbruc ken Germany Germany patrickcoliunisbde tzakovampisbmpgde In memory of George Gargov Abstract Hybridization is a metho d invented by Arthur Prior for extending the expressivepower of mo dal languages Although develop ed in interesting ways by Rob ert Bull and by the Soa scho ol notably George Gargov Valentin Goranko Solomon Passy and Tinko Tinchev the metho d re mains little known In our view this has deprived temp oral logic of a valuable to ol The aim of the pap er is to explain whyhybridization is useful in tem poral logic We maketwo ma jor points the rst technical the second conceptual First weshowthathybridization gives rise to wellb ehaved logics that exhibit an interesting synergy between mo dal and classical ideas This synergyobvious for hybrid languages with full rstorder ex pressive strength is demonstrated for a weaker lo cal language capable of dening the Until op erator we provide a minimal axiomatization and show that in a wide range of temp orally interesting cases extended com pleteness results can b e obtained automatically Second we argue that the idea of sorted atomic symb ols which underpins the hybrid enterprise can b e develop ed further To illustrate this we discuss the advantages and disadvantages of a simple hybrid language which can quantify over paths Intro duction Arthur Prior prop osed using mo dal languages for temp oral reasoning more than years ago and since then the approach has b ecome widespread in a variety of disciplines Over this p erio d a wide range of often very p owerful mo dalities has b een used to reason ab out time This is unsurprising After all dierent choices of temp oral ontology such as instants intervals and events are rel evant for dierent purp oses and dep ending on the application considerable expressivepower may b e needed to cop e with the way information can b e dis tributed across such structures But inventing new mo dalities is not the only wayof b o osting mo dal expressivity There is a largely overlo oked alternative called hybridization and this pap er explores its relevance for temp oral logic Hybridization is best intro duced by example Consider the following sen tence from the language wecallML xx x The x in this expression is a state variable and all its o ccurrences are b ound by the binder Syntactically state variables are formulas after all the ex pression x x is built using and in the same way that p p is Semantically however state variables are b est thought of as terms Our semantics will stipulate that state variables are satised at exactly one state in In eect state variables act as names they label the unique state anymodel they are true at The use of formulas as terms gives hybrid languages their unique avor they are formalisms which blend the op eratorbased p ersp ective of mo dal logic with the classical idea of explicitly binding variables to states Unsurprisingly this combination oers increased expressive power The ab ove sentence for example is true at any irreexive state in any mo del and false at all reexive ones No ordinary mo dal formula has this prop erty Now the language ML is not the only hybrid language and for many purp oses it is not the most natural one Oneofthekey intuitions underlying mo dal semantics is locality and it is intuitively clear we shall b e precise later that is not lo cal as our notation suggests quanties across al l states So if wewantalocalhybrid language ML is not a suitable choice But what are the alternatives To the b est of our knowledge only one has been considered namely the binder we here call Now do es something simple and natural it binds a variable to the current state Unfortunately while ML is a lo cal language it has twodrawbacks First it is not expressive enough for many applications for example we shall show that it is not strong The literature on hybrid languages consists of a handful of pap ers published over the last thirtyyears by researchers with very dierentinterests Conning ourselves to the main line of development the idea can b e traced back Prior and the p osthumously published Prior and Fine contains some of Priors unnished pap ers on the sub ject together with an app endix by Kit Fine Priors concerns were largely philosophical technical development seems to have started with Bull Bull investigated a hybrid temp oral language con taining the binder and the universal mo dality A and intro duced the idea of quantication over paths In addition he initiated the algebraic study of such systems The pap er never attracted the attention it deserved in fact apart from citations in the hybrid literature the only mention we know of is from Burgesss survey of tense logic Other hybrids of a dierent sort not easytodescribe briey aretreated in an interesting paper of Bul l Burgess page This is probably the rst use of hybrid in connection with such languages The idea was indep endently invented by the Soa Scho ol as a spino of their investigation of mo dal logic with names The b est guide to the Bulgarian tradition is the b eautiful and ambitious Passy and Tinchev drafts of whichwere in circulation in the late s Hybridization is discussed in Chapter I I I and deals with Prop ositional Dynamic Logic enriched with b oth and the univ ersal mo dality see also Passy and Tinchev and the brief remarks at the end of Gargov Passy and Tinchev Recent pap ers on the sub ject include Goranko probably the rst published account of hybrid languages containing the binder Blackburn and Seligman and Selig man whichinvestigates hybrid natural deduction and sequent calculi for applications in Situation Theory and Blackburn and Tzakova ab Also relevant are Gar gov and Goranko Blackburn these lo ok at mo dal and tense logics enriched with nominals in eect the free variable fragments of hybrid languages enough to dene the Until op erator Second in stark contrast to ML which has an elegant axiomatization axiomatizing ML seems to require complex pro of rules What are we to do Here we showthat intro ducing an op erator which retrieves the value stored by solves these problems it oers the expressivity weneed the minimal logic is elegant and we automatically get completeness results for a wide class of interesting frame classes many of which are not mo dally denable All this without sacricing lo cality These results are the technical core of the pap er but to close our discussion we change gears there is an imp ortant conceptual point to be made ab out not simply hybridization and its relevance to temp oral logic hybridization is ab out quantifying over states Rather hybridization is about hand ling dierent types of information in a uniform way We illustrate this idea by discussing a simple hybrid language for quantifying over paths But we are jumping ahead There is muchtobedonebeforewe can usefully discuss such ideas so lets call a halt to our intro ductory remarks and start developing the idea of hybridization systematically The basic mo dal language One of the simplest languages for temp oral reasoning is the prop ositional mo dal language
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