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]

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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 that contains just two mo dalities an op erator
read as at al l future

states together with its dual op erator read as at some future state For

most of this pap er we will be working with various hybrid extension of this

simple language which we will call ML The purp ose of the present section

is to x notation and terminology to remind the reader of various standard

concepts in particular generated submodels and bisimulations and to present

a wishlist of prop erties for hybrid temp oral languages

Given a countable set of propositional symbols PROP fp q rg the

wellformed formulas of ML are dened as follows

WFF p j j j

Other Bo olean op erators and so on are dened in the usual

way and we dene to b e

ML is interpreted on models AmodelM is a triple S R V suchthat S is a

nonempty set of states and R is a binary relation on S the temporal precedence

relation the pair S R is called the frame underlying M The valuation V is a

function with domain PROP and range Pow S this tells us at which states if

any each prop ositional symb ol is true Dep ending on the application additional

prop erties may b e demanded of R in temp oral logic various combinations of

such prop erties as transitivity irreexivity density discreteness trichotomy

nobranchingtotheright and many others are common We shall deal with

such demands later

Blackburn and Tzakova a an extended version of the present pap er examines two

other lo cal solutions in detail adding a universal quantier over accessible states and

changing the underlying language from mo dal logic to tense logic This version will b e

made available at httpwwwcoliunisbde p atri ck An earlier version whichcontains

solution is already available

The satisfaction denition for ML is dened as follows Let M S R V

and s S Then

Ms j p i s V p where p PROP

Ms j i Ms j

Ms j i Ms j Ms j

Ms j
i s sR s Ms j

If Ms j wesaythat is satised in M at s If is true at all states in a

mo del M wesayitisvalid in M and write Mj

Note the lo cality of the satisfaction denition formulas are evaluated inside

mo dels at some particular state called the current state and the
and

op erators scan the states accessible from the current state via the precedence

relation R This lo calityintuition is arguably the central intuition underlying

mo dal approaches to temp oral logic it is certainly the intuition which prompted

Arthur Prior to pioneer the mo dal logic of time which he called tense logic

As he observed we are situated inside the temp oral ow and many asp ects of

language for example the use of tense and temp oral indexicals suchasnow

reect this internal p ersp ective Accordinglyhebelieved that mo dal analyses



of temp oral logic were likelytobemostrevealing

The lo calityofMLhasanobvious mathematical consequence satisfaction

of ML formulas is preserved under the formation of generate d submodels Tobe

more precise given a mo del M S R V and a state s of S the submo del

of M that is generated by s contains just those states of M that are accessible

from s by a nite numb er of transitions along R It follows by an easy induction

that for all formulas

s

Ms j i M s j

In what follows we use preservation under generated submo dels as a key crite

rion for judging hybrid temporal languages We are interested in local temp oral

languages and will reject hybrid extensions which lead to a loss of the generated

submo del preservation results

Now for a key question do es ML have the expressivity needed for temp oral

reasoning There is no absolute answer it dep ends on the application For some

applications ML will often be too strong For example if one is interested in

using mo dal languages to characterize various typ es of bisimulation invariance it

maybenecessarytowork with sublanguages of ML containing no prop ositional

symbols ws would be built using the constant or to shed some Bo olean

expressivity



But for many other applications ML is to o weak For a start as has

already b een mentioned no formula of ML is capable of distinguishing irreexive

from reexive states in all mo dels this means that a fundamental constrainton

temp oral precedence simply isnt reected Moreover consider the denition of



The b est intro duction to Priors views is Prior



A very obvious weakness is that ML oers us no way of lo oking backwards along R

for that we need Priors language of tense logic However while useful in natural language

semantics in many applications in AI and theoretical computer science backward lo oking

op erators dont play a prominent role Apart from o ccasional remarks wewont discuss tense

logic here but Blackburn and Tzakova a the extended version of the present pap er

contains a full treatment

the Until op erator

Ms j Until i s sR s Ms j tsR t tR s Mt j

This is an extremely natural local op erator note that formulas built using Until

are preserved under the formation of generated submo dels and has proved a use

ful to ol for temp oral reasoning in computer science indeed computer scientists

usually regard Until as the fundamental mo dality However the Until op erator

is not denable in ML As the nondenabilityofboth Until and irreexivity

follows from the fact that ML formulas are preserved under bisimulations and

as we will later make use of sp ecial bisimulations called quasiinjective bisimu

lations it will b e useful to prove these nondenability results here

A bisimulation between twomodels M S R V andM S R V

is a nonempty binary relation Z between S and S such that

For all states s in S and s in S if s Zs then s and s satisfy the

same prop ositional symb ols

For all states s s in S and s in S ifs R s and s Zs then there

is a state s in S suchthats R s and s Zs

For all states s s in S ands in S if s R s and s Zs then there

is a state s in S suchthats R s and s Zs

The fundamental result concerning bisimulations which follows straightfor

wardly by induction on the structure of ML formulas is that if Z is a bisimula

satisfy exactly the tion b etween mo dels M and M and s Zs then s and s

same ML formulas

It follows that neither Until nor irreexivity is denable indeed the follow

ing counterexample whichwe b elieve is due to Johan van Benthem establishes

b oth p oints simultaneously Let M b e an irreexive mo del containing just two

states s and s let s Rs and s Rs and supp ose all prop ositional symb ols

are true at b oth states Let M b e an reexive mo del containing just one state

s and supp ose all prop ositional symb ols are true at s Clearly the relation Z

which links b oth s and s to s and viceversa is a bisimulation hence all states

in b oth mo dels satisfy exactly the same ML formulas So as M is irreexive

and M reexive it follows that no ML formula succeeds in distinguishing ir

reexive and reexive states Moreover observethat Until is false in M

at b oth s and s but true in M It follows that the Until op erator cannot

b e expressed in ML

Thus ML has expressiveweaknesses that are relevant to temp oral reasoning

and one of the key goals of this pap er will b e to repair them byhybridization

But what should a hybrid temp oral language lo ok like It is time to drawupa

wishlist

First wewould like our hybrid language to b e local Second wewould like

our hybrid language to b e expressive enough to detect irreexivityand dene

Until Third wewould liketondhybrid languages in which the central ideas

of mo dal and classical pro of systems can b e clearly combined Indeed wewould

like to exhibit a synergy b etween mo dal and classical ideas wewant the whole

so to sp eak to oer more than the sum of its parts Lets now examine the two

hybrid binders that have previously b een studied and see how they measure up

against these demands

Two hybrid binders

Syntacticallyhybridizing ML involves making twochanges First we sort the

atomic symb ols instead of having just one kind of atom namely the symb ols

in PROP we add a second sort called state symbols For reasons we shall so on

explain it is convenientto divide state symbols into two sub categories state

variables and nominals Second we add binders The binders will b e used to

bind state variables but not nominals or prop ositional symbols

Let PROP be as describ ed b efore Assume we have denumerably innite

set SVAR of state variables whose elements wetypically write as u v w x y

e and z and a denumerably innite set NOM of nominals whose elements w

typically write as i j k and l We assume that PROPSVAR and NOM are

pairwise disjoint We call SVAR NOM the set of state symbols andPROP

SVAR NOM the set of atoms Cho ose B to b e one of or We build the

wellformed formulas of the hybrid language over PROPSVAR NOM and B

as follows

WFF a j j j
j Bx

Here a ATOM and x SVAR If B was chosen to be we obtain the

language ML and if B was we get ML Strictly sp eaking dierent

choices of PROPSVAR and NOM give rise to dierent languages but we ignore

this whenever p ossible

A full discussion of the syntax of these languages would need to dene such

concepts as free b ound substitutable for and so on But exp erience with

classical logic is a reliable guide and anyway the relevant denitions may be

found in Blackburn and Tzakova so well simply remark that a sentence

is a formula containing no free variables or nominals and that we use the nota

tion sv to denote the formula obtained by substituting the state symbol s

for all free o ccurrences of the state variable v in

As promised in the intro duction our hybrid languages use formulas as lab els

in the semantics presented b elow b oth state variables and nominals will be

satised at exactly one state in any mo del Now the role of the state variables

the point of having nominals Simply this it is should be clear but what is

convenient to have a supply of lab els that cannot be bound by the binders

this simplies some of the technicalities for it saves us having to worry ab out

accidental binding In short nominals are reminiscent of the parameters used

in classical pro of theory

Now for the semantics The key idea is straightforward we are going to

insist that state symb ols are interpreted by singleton subsets of mo dels Well

also need a smo oth way to handle the fact that state variables may b ecome

b ound whereas this is not p ossible for nominals or prop ositional symb ols But

there is an obvious waytodothis well let the state variables b e handled bya

separate assignment function in the manner familiar from classical logic

Denition Standard mo dels and assignments L et L be a hybrid lan

guage over PROP SVAR and NOMAmodel M for L is a triple S R V such

that S is a nonempty set R a binary relation on S andV PROP NOM

Pow S A model is cal led standard i for al l nominals i NOM V i is a

singleton subset of S

An assignment for L on M is a mapping g SVAR Pow S An assign

ment is cal led standard i for al l state variables x SVAR g x is a singleton

x

subset of S The notation g g g is an xvariant of g means that g and g

are standard assignments on some model M such that g agrees with g on al l

arguments save possibly x

Let M S R V b e a standard mo del and g a standard assignment For

any atom a let V ga g a if a is a state variable and V a otherwise

Then interpretation of our hybrid languages is carried out using the following

denition

Mgs j a i s V ga where a ATOM

Mgs j i M g s j

Mgs j i Mgs j Mgs j

Mgs j
i s sR s Mgs j

x

g Mg s j Mgs j x i g g

x

xfsg Mgs j x i Mg s j where g g and g

Let M b e a standard mo del Wesay that is valid on M i for all standard

assignments g on M and all states s in M Mgs j and if this is the case

we write Mj Wesay that a formula is valid on a frame S R written

S R j i for all standard valuations V and standard assignments g on

S R and all s S S R V gs j

Lemma Substitution lemma Let M be a standard model let g be an

assignment on M and let be a formula of any of the hybrid languages dened

above Then for every state s in M if y is a variable that is substitutable for

x in and i is a nominal then

x

g and g xg y M gs j yx i M g s j where g

x

M gs j ix i M g s j where g g and g xV i

Proof By induction on the complexityof a

This concludes the preliminaries its time to take a closer lo ok at the binders

The binder

The binder is the stronger more classical of our binders indeed its just the

familiar universal quantier in a mo dal setting Note that if we dene x to

b e the dual binder x then

x

Mgs j x i g g g Mg s j

ML is a powerful language We saw in the intro duction that it can

distinguish irreexive from reexive states Moreover it can dene the Until

op erator

y Until y y

This denition says it is p ossible to bind the variable y to a successor state

in such a way that holds at the state lab eled y and holds at all

successors of the current state that precede this y lab eled state In addition

the minimal temp oral logic of ML has a simple axiomatization that can b e

proved complete reasonably straightforwardly All in all its a lovely language

But theres a snag it isnt lo cal To see that satisfaction of ML sentences

need not b e preserved under the formation of generated submo dels consider the

following counterexample taken from Blackburn and Seligman Let M

b e the following twoelement mo del where S fs tgandR fs sg

xx

s t

Then xx is true at s in M for we can assign the state t to x and s t R

s s

However it is not true at s in the submo del M generated by s for as M

contains only the state s all assignments assign s to x As s is reexive x

will always be false In short detects t from s even though t and s are

completely disconnected

If you want a strong hybrid language and are not interested in maintaining

lo calitythenML is probably an excellentchoice Indeed you maywishto

consider working with a hybrid language even less lo cal namely ML en



riched with the universal modality A The universal mo dalityhasthefollowing

satisfaction denition Ms j A i Ms j for all states s M It is

not hard to see that adding the universal mo dality yields a hybrid language

with rstorder expressivepower Prior knew this result and formulated it in

anumber of ways Moreover the A and work together extremely smo othly

making elegant axiomatizations p ossible see Bull But while suchrich

systems are interesting they are far removed from the lo cal temp oral languages

elop wewishtodev

The
binder

If one is interested in lo cal hybrid languages the binder is the most natural

starting p oint Quite simply binds a variable to the current state it creates



a lab el for the hereandnow Lets lo ok at it more closely

First note that is selfdual that is at any state in any standard mo del

under any standard assignment x is satised if and only if x is satised

to o To put it another way we are free to regard as either a universal

quantier over the current state or as an existential quantier over the current

state as there is exactly one current state these amount to the same thing

Next note that x is denable in ML we can dene it either as xx

orxx thus ML is a fragmentofML Its quite an in teresting



Virtually the entire literature on hybrid languages is devoted to such systems For exam

ple b oth Bull and Passy and Tinchev makeuseof and A



Incidentally while is a relativenewcomer to hybrid languages Goranko seems to

b e the rst published account essentially the same binder has b een intro duced to a number

of dierentnonhybrid languages for a wide variety of purp oses see for example Richards et

al Cresswell and Sellink Lab eling the hereandnowseemstobean

imp ortant op eration

fragment For a start sentences of ML are preserved under the formation of

generated submo dels We leave the simple pro of to the reader Essentially it

boils down to the observation that the only states that canbindtovariables in

the course of evaluation must b e states in the generated submo del For example

in the previous diagram if weevaluate a sentence at s the only state that we

can bind to any variable is s itself ML cannot detect t which is what we

want Moreover adding the binder boosts the expressive power of ML in

temp orally interesting ways In particular note that the sentence

x
x

is true in a mo del at a state s i s is irreexive

Unfortunately ML has twodrawbacks First there is no obvious way

to provide a complete axiomatization without resorting to a fairly complex rule



of pro of Second for many purp oses it simply isnt expressive enough Lets

examine this second problem more closely

Although adding increases the expressive power Until still isnt den

able Toseewhy wemake use of the quasiinjective bisimulations intro duced

in Blackburn and Seligman Letussay that states s and s in a mo del

M S R V are mutual ly inaccessible i s is not in the submo del generated

by s and s is not in the submo del generated by s We then dene

Denition Quasiinjectivebisimulations Let Z be a bisimulation be

tween M and M Z is a quasiinjective bisimulation i

Zs and in M and s in M if s Zs and s For al l states s s

s s then s and s are mutual ly inaccessible and

For al l states s s in M and s in M if s Zs and s Zs and

s s s and s then are mutual ly inaccessible

Now ML sentences need not b e preserved under arbitrary bisimulations

the fact that x
x picks out irreexive states shows this but Blackburn

and Seligman showthatthey are preserved under quasiinjective bisimulations

That is

Prop osition Let Z be a quasiinjective bisimulation between models M and

M and let s and s be states in M and M respectively such that s Zs

i M s j Then for al l sentences of ML M s j

We can use this result to show that no sentence of ML denes the Until

op erator Tobemorespecicletp and q b e prop ositional symbols Then even

U pq 

over strictly partially ordered mo dels there is no sentence of ML



Blackburn and Tzakova axiomatize the set of valid ML by making use of the

COV rule see Gargov Passy and Tinchev Passy and Tinchev Gargovand

Goranko UnfortunatelytheCOV rule is rather complex it employs arbitrarily deep

nestings of mo dalities

The only other work on axiomatic systems for we know of are Goranko and

Goranko a However Gorankos investigations have little b earing on the concerns of the

present pap er for Gorankoinvestigates a language containing b oth the universal mo dality

and Note that the binder is denable in this language by x yAxAy thus

Gorankos language has full rstorder expressivepower

that is satised in a mo del M atastate s i Untilp q is satised in M at s

To see this consider the following two mo dels

p p p

q q q q

Untilp q Untilp q

In b oth mo dels the relation we are interested in is the transitive closure of

the relation indicated bythearrows thus b oth mo dels are strict partial orders

Note that Untilp q is false in the lefthand mo del at the ro ot no de and true

U pq 

in the righthand mo del at the ro ot Hence if some sentence of ML

expressed Untilp q it would b e false at the ro ot of the lefthand mo del and

true at the ro ot of the righthand side one But this is imp ossible for the obvious

unraveling relation b etween the two mo dels is a quasiinjective bisimulation

Summing up previously studied hybrid systems dont meet our three wish

list criteria The binder is interesting and elegant but to adopt it is to

abandon lo cality The binder is far more promising binding to the current

state is suchanintrinsically mo dal idea that it deserves further attention But

can weovercome its expressiveweakness And are there natural ways to avoid

e shall now dep endence on complex rules of pro of The answer is Yes As w

show we can do this by adding a retrieval op erator to match the action of

for two further solutions consult the extended version of this pap er

The op erator

Supp ose we were given a brand new webbrowser to test and we discovered

it had the following limitation although it allowed us to b o okmark URLs it

didnt allow us to jump to these lo cations byclicking on the stored b o okmark

Franklywewouldnt dream of working with sucha browser wed demand that

this shortcoming b e xed rightaway

ML is rather like this hop efully nonexistent browser pushes us

through cyb erspace and allows us to lab el the states we visit on our travels

but ML do esnt oer us a general mechanism for jumping to the states we

t We shall allow ourselves to construct formulas of the lab el Lets put this righ

form Toevaluate such a formula we will jump to the state s lab els and

s

see whether holds there in eect will enable us to use the values has so

carefully stored for us

Lets make this precise If s is a state symb ol and is a formula then is

s

a formula It is p ossible to think of as a binary mo dality whose rst argument

is a state symb ol and whose second argument is a formula but as will so on

b ecome clear it is more natural to view the comp osite symbol as a unary

s

mo dal op erator If we add all these statesymb olindexed unary mo dalities to

MLwe obtain ML Most syntactic asp ects of ML are obvious

though the following p ointisworth stressing does not bind variables Only

the binder do es that

Now for the semantics Let M S R V b e a standard mo del let g be a

standard assignmenton M and let Dens b e the denotation of the state symbol

s that is Dens is g s if s is a state variable and V s if s is a nominal

Then

Mgt j i MgDens j

s

As promised jumps to the denotation of s and evaluates its argumentthere

s

Sentences of ML are preserved under generated submo dels After all

in a sentence the only o ccurrences of will be of the form where y is a

y

state variable b ound by some o ccurrence of and as binds lo cally the result

follows Second can dene Until As we have already seen Until is not

denable in ML but it certainly is in ML

Until xy y
y

x

Note howthisworks we lab el the current state with x use to movetoan

accessible state which we lab el y and then use to jump us backto x We

then use the mo dalities to insist that holds at the state lab eled y and

holds at all successors of the current state that precede this y lab eled

state Note the similarities and dierences with our earlier based denition

of Until

As this example shows and make a great team they communicate

smo othly and their co op eration gives rise to an axiomatization called H K

This axiomatization is an extension of the minimal mo dal logic K Recall that

K is the smallest set of formulas containing all prop ositional tautologies and all

instances of


that is closed under modus ponens if

and are b oth provable then so is andnec essitation if is provable

then so is
To the axioms and rules of pro of of K we add axioms and rules

governing b oth and Lets deal with rst First wehave all instances of

the following schemas



A lot more could b e said ab out and we cant say it all here But two things should b e

said First the reader has almost certainly seen something like in nonhybrid languages for

example its Priors T s construct in thirdgrade tense logic its the Holdss op erator

intro duced by Allen for temp oral representation in AI and it is the characteristic

op erator of the Topological Logic of Rescher and Urquhart Note that the op erator

supp orts a variety of natural interpretations for example computationally it can b e viewed

as a goto instruction

But one p ersp ective is particularly relevant here can be viewedasarestricted version of

the universal modality First note that can b e dened as either As orE s

s

where E is the dual if A In short allows limited access to the p ower of A and the limitation

results in a generated submo del for sen tences But as we shall see b elow has enough p ower

to supp ort elegant pro of theories



Note that the prenex blo ck xy denes an existential quantier over states reachable

x

in Rstep xy this binder is discussed in detail in the extended version

x

y

of the pap er Similarlywe can dene an existential quantier over states accessible in R

steps xy Indeed for any natural number n we can dene an existential

x

y

quantier over states accessible in nRsteps Note that we also have simple denitions of the

universal quantiers over states reachable in nRsteps for example x

y

x

y

n

n

It is easy to see that and are dual binders for any natural number n

These axioms were used as part of the COV based axiomatization of Blackburn and

Tzakova In Blackburn and Tzakova a the extended version of the present

pap er these axioms are discussed further and analogs of QQ are given for the binder

mentioned in the previous fo otnote

Q v v

Q v s sv

Q v v v

SelfDual v v

Here v is a metavariable over state variables s a metavariables over state

symbols and and metavariables over arbitrary ws In Q cannot

contain free o ccurrences of v inQ s must b e substitutable for v in

Q and Q are obvious analogs of familiar rstorder axiom schemas The

ma jor dierence is that the presentversion of Q only lets us substitute state

symb ols for binders when the obvious lo cality condition is fullled s must b e

true in the current state This restriction motivates the intro duction of Q

which allows us to eliminate b ound o ccurrences of state variables in antecedent

p osition In addition to these axioms wehavethe rule of state variable local

ization that is if is provable then so x Summing up supp orts a local

form of classical reasoning But in spite of the lo cality restriction the axioms

just intro duced are strong enough to supp ort many classical principles suchas

conversion As an illustration for full details see the extended version we

show

Lemma Normality For al l formulas and we have x

x x

Proof Note that x x is an instance of Qasisx

x Hence x x x Use lo calization to prex

this formula with x and then Q to distribute x over the main implication

to get x x xx Note that xx x is

an instance of Qsowe can simplify the consequent and so obtain the result

Using Q in this way to simplify the conditionals pro duced by applications of

Q is typical of H K pro ofs a

Lets turn to For every state symbol swehave the rule of necessitation

s

if is provable then so is In addition we have the rules Paste and

s

Paste these will be intro duced below In addition wehave all instances of

the following schemas These fall naturally into three groups The rst identies

the basic logic of

K

s s s

SelfDual

s s

Introduction s

s

Note that K is simply the familiar mo dal distribution schema hence as we

have the rule of necessitation is a normal mo dal op erator Obviously

s s

SelfDual states that is selfdual but note that viewed in more traditional

s

mo dal terms it tells us that is a mo dality whose transition relation is a

s

function one direction is the mo dal determinism axiom while the other is

tic logic Given the jumptothelab eledstate the characteristic axiom of deon

interpretation of this is exactly what we would exp ect Introduction tells

us howtointro duce information under the scop e of the op erator Actually

it also tells us how to get hold of such information for if we replace by

contrap ose and makeuseof SelfDual we obtain s we call this

s

is Elimination schema

The next group is a modal theory of labeling or to put it another way a

modal theory of state equality

Label s

s

Nom t

s t s

Swap t s

s t

Scope

t s s

The nal group tells us how and interact

Back

s s

Bridge s

s

And apart from the Paste rules thats H K Weleave the soundness

pro of to the reader and turn straight to the issue of completeness Essentially

were going to adapt the mo dal canonical mo del metho d to our new language

we assume the usual notions of consistency Maximal Consistent Sets MCSs

and so on see the extended version for further details

Denition Canonical Mo dels For a countable language Lthecanonical

c c c c c c

mo del M is S R V where S is the set of al l LMCSs R is the binary

c c

relation on S dened by R i
implies for al l Lformulas

c c c

and V is the valuation dened by V af S j a g where a is a

proposition symbol or nominal

We b egin by proving a key lemma without the help of the yettob eintro duced

Paste rules Let us say that an MCS is labeled if and only if it contains a state

symb ol if a state symb ol b elongs to an MCS wecallitalabel for that MCS

Lemma Let be a labeled MCS and for al l state symbols s let be

s

f j g Then

s

For every state symbol s is a labeled MCS that contains s

s

For al l state symbols s and t i

s t s

There is a state symbol s such that

s

For al l state symbols s f j g

s s s

For al l state symbols s and tif s then

t t s

Proof Clause First for every state symbol s wehavethe Label axiom s

s

hence s Next is consistent For assume for the sake of a contradiction

s s

that it is not Then there are suchthat By

n s n

necessitation hence isinandthus by

s s n s n

SelfDual isintoo On the other hand as

s n n s

wehave By simple mo dal argumentation all weneedisthe

s s

n

fact that is a normal mo dality it follows that as w ell

s s n

contradicting the consistency of We conclude that must be consistent

s

after all

It remains to show that is maximal So assume it is not Then there

s

is a formula such that neither nor is in But then b oth and

s s

b elong to and this is imp ossible if then by self duality

s s

as w ell and wecontradict the consistency of So is maximal

s s

Clause We have i By Scope i

s t t s t s

We call this the agreement property though simple it plays an

s

imp ortant role in our completeness pro of

Clause By assumption con tains at least one state symbol let us call

it s If we can show that we will have the result But this is easy

s

Supp ose Then as s by Introduction and hence

s s

Converselyif then Hence as s by Elimination wehave

s s

Clause Use Introduction and Elimination much as in the previous clause

Clause Let b e such that s we shall showthat First

t t t s

we have that s Hence by Swap t observe that since s

t t s

to o But now the result is moreorless immediate First For if

t s

then Hence as t it follows by Nom that

t t s s

and hence that as required A similar Nom based argumentshows that

s

a

s t

This lemma gives us a lot in essence it says that the subscripted op

erators in any lab eled MCS index a wellb ehav ed collection of lab eled MCSs

Now thinking ahead to the Truth Lemma wewillhavetoprove it should b e

clear why we want to work with labeled MCSs with the help of Q we can

use these lab els to instantiate state variables b ound by and hence establish

the inductive step for Thus the are plausible mo delbuilding material

s

nonetheless they dont yet have all the prop erties wewant

First theres a small wrinkle wewould like the MCSs we use to b e lab eled

byanominal notjustafreevariable this isnt crucial but it saves having to

worry ab out ab out accidental binding But note that even if itself con tains

a nominal say i wehave no guarantee that all the do to o for example

s

may contain j for all nominals j in which case wont contain any

x x

nominals at all though of course it will contain x

And theres a second far more serious problem Supp ose wetake the collec

tion of yielded by a lab eled MCS as the building blo cks of our mo del Doing

s

this means wehavethrown away MCSs wewillbeworking in a submodel of the

canonical mo del Howdoweknowthatamodalstyle Existence Lemma holds

for this submo del That is how can prove the clause of the Truth Lemma for

the mo dalities Bluntly there is no obvious way to do this

The Paste rules enable us to x b oth problems Here they are

t t

s s

s s

The rule on the left is called Paste the rule on the right Paste In b oth t

mustbeastatesymb ol distinct from s that do es not o ccur in or

The key rule is Paste Read contrap ositively that is read from b ottom

to top it tells is that pasting a brand new state symb ol under the scop e of

is a consistency preserving op eration for if we cant derivea contradiction

that is without the new nominal then we cant derive the contradiction

after wehave pasted We shall leave the reader to p onder the simpler Paste

rule essentially it says that giving a brand new name to a lab eled state isnt

going to cause any problems and prove the Extended Lindenbaums Lemma we

need

Denition Pasted MCSs An MCS is pasted i implies

s

that for some nominal i i It is pasted i implies

s s

that for some nominal i i We say that is pasted i it is both

s

pasted and pasted

Lemma Extended Lindenbaums Lemma Let L and L becountable

languages such that L is L enriched with a countably innite set of new nom

d inals Then every consistent set of Lformulas can be extended to a paste

L MCS that is labeled by a nominal

Proof Enumerate the new nominals Given a consistent set of Lformulas

dene to be fj g where j is the rst new nominal is consistent

j j

For supp ose not Then for some conjunction of formulas from we have

that j as j is from the newnominal enumeration it do es not o ccur

in Let P be a pro of of j and let x be any state variable that

do es not app ear in this pro of Then replacing every o ccurrence of j in P by x

yields a pro of of x Lo calization then yields xx By Q

x Nowvacuous o ccurrences of the binder are eliminable in H K

for so for anyvariable x not o ccurring in lo calization and Q

yield x whereup on contrap osition and the self dualityof yield

the result Hence which contradicts the consistency of Thus is

j

consistent after all

W enow paste Enumerate all the formulas of L dene to b e and

j

m

supp ose wehave dened where m Let b e the m th formula in

m

The extended version of this pap er discusses the admissibility of these rules Asemantic

argument is given which strongly suggests that Paste isnt a genuine enrichment of the

system though at the time of writing this hadnt b een backed up byasyntactic pro of The

admissibilityof Paste is p osed as an op en problem

But while interesting to fo cus exclusively on the admissibilityof Paste over an axiomatic

basis is to miss the true signicance of this rule Paste is actually the most natural part

of H K its the other comp onents that should b e eliminated This is the strategy

adopted in Blackburn and Seligman Drawing on ideas from Seligman an

based sequent system is presented and the idea underlying Paste nds its true home

Incidentally Paste is closely related to a rule intro duced byGabbay and Ho dkinson

for Until Since logic The Gabbay and Ho dkinson metho d is discussed in detail in the ex

tended version of the pap er and Paste is intro duced as so to sp eak an based imple

mentation of their idea that bypasses the need to work with arbitrary sequences of tense

op erators

m m

our enumeration We dene as follows If f g is inconsistent

m

m m

then Otherwise

m m

f g if is not of the form v or Here s

m m v s

is a state symbol and v is a state variable

m m

f g f k v gif is of the form v Here k

m v m v

m

is the next new nominal that do es not o ccur in

m m

f gf k gif is of the form Here

m s m s

m

k is the next new nominal that do es not o ccur in or

s

S

n

Let It is clear that this set is lab eled bya nominal maxi

n

mal and pasted Furthermore it must b e consistent for the only nontrivial

asp ects of the expansion are those dened by items and and Paste and

Paste resp ectively guarantee that these are consistency preserving

So it only remains to check that is pasted b ecause of the rather limited

way item uses Paste this may not b e entirely obvious First note that by

basic mo dal reasoning So supp ose If s

s s s s

is a nominal say i then b ecause i is an axiom i as required

i i

On the other hand if s is a variable say x then b ecause of the pasting pro cess

carried out in item for some nominal i wehavethat i x As is

x s

a normal mo dal op erator i so i We conclude that is

x x

the required L MCS a

Were now ready to prove the completeness of H K in fact wehave

everything we need to prove the completeness of many of its extensions as well

Denition Lab eled mo dels and natural assignments Let bea pasted

ols s let be f j g MCS labeled by a nominal For al l state symb

s s

and dene S to be f j s is a state symbolg Then we dene M the labeled

s

c

model yielded by to be S R V where R and V are the restrictions of R

c

the canonical relation and V the canonical valuation to S We dene the

natural assignment g SVAR Sby g xfs S j x sg

Such lab eled mo dels have all the structure wewant For a start by Clause

of Lemma S and by Clause V is a standard valuation and g is a

standard assignment Further all states in the mo del contain nominals b ecause

is pasted and hence are w ellb ehaved as far as is concerned Moreover

we know from Lemma that M is extremely wellb ehaved with resp ect to

So it only remains to ensure that such mo dels are wellb ehaved with resp ect to

the mo dalities that is wewant an Existence Lemma This of course is where

pasting comes in

Lemma Existence Lemma Let M S R V be the labeled model

yielded by a pasted set that is labeled by some nominal Suppose S

and Then thereisa M such that R and

Proof As S for some nominal i wehave that hence as

i

But is pasted and hence pasted so for some nominal k

i

R k and so k If we could show that

i k i i

and then would b e a suitable choice of And in fact Bridge

k k

and Back aided by the agreementproperty of our mo del that is item of

Lemma will let us establish this

For we need to show that for any wehavethat So

k i

supp ose This means that By agreement But

k k k i

k Hence by Bridge as required

i i

For we know that k But k this is an

i k

instance of Introduction hence But then by Back

k i k i

By agreement Hence as required a

k k

Lemma Truth Lemma Let be an MCS in M For al l formulas

i M j

Proof By induction the atomic b o olean and mo dal steps are standard we

use the Existence Lemma just proved for the latter

So supp ose x Since contains a nominal say i by Q ix

By the inductivehyp othesis M g j ix Thus M g j i ix and

by the contrap ositiveofthe Q axiom M g j x For the other direction

x

assume M g j x That is M g j where g g such that

g xfg Now contains a nominal say isoby the Substitution Lemma

M g j ix hence by the inductive hyp othesis ix So by the

contrap ositive of the Q axiom x is in as required

The argumentforrunsas follows M j i M j for by

s s

Clause of Lemma is the only MCS containing s and hence by the the

s

atomic case of the present lemma the only state in M where s is true i

s

inductivehyp othesis i using the fact that s together with

s s s

Introduction for the lefttoright direction and Elimination for the righttoleft

direction i by the agreement prop erty for the MCSs in S Thus

s

all cases have b een proved and the Truth Lemma follows by induction a

K consistent set of formulas Theorem Completeness Every H

in a countable language L is satisable in a countable standardmodel with respect

to a standard assignment function Moreover every H K consistent set

of sentences in L is satisable in a countable connected standardmodel

Proof Therstisproved in the exp ected way given a H K consistentset

of formulas use the Extended Lindenbaum Lemma to expand it to a pasted

set lab eled by some nominal in a countable language L By the Truth

Lemma just proved the lab eled mo del and natural assignmentthat gives

rise to satisfy at This mo del need not b e connected but the submo del

generated by is and all sentences in are true in this submo del a

But theres no need to stop here one of the nicest things ab out hybrid lan

guages is the ease with which general completeness results for richer logics can

be proved Moreover such results typically link completeness and frame

denability in a very straightforward way

Historically this has b een a ma jor motivation for exploring hybrid languages Bull

points out see page that all statesymb olbased extensions of the basic logic are com

plete and a neat argument to the same eect is given at the end of Gargov Passy and

Tinchev Passy and Tinchev push matters further like the earlier Passy and

Tinchev this pap er takes PDL as the underlying mo dal language and explores what

happens beyond the rstorder barrier The present pap er applies similar arguments to weaker

lo cal languages

A formula is said to dene some prop erty of frames for example transi

tivity i it is valid on precisely the frames with that prop erty recall from

Section that a formula is valid on a frame i it is imp ossible to falsify it at

any state in that frame no matter whichvaluation or assignment is used The

sort of results we are after have roughly the following form for anyformula

from some sp ecied syntactic class if denes a prop erty P then using it as

an additional axiom guarantees completeness with resp ect to the class of frames

with prop erty P For ordinary mo dal languages the Sahlqvist Theorems are the

b est known result of this typ e see Sahlqvist as we shall see analogous

results for hybrid languages come far more easily We shall givetwo The idea

underlying b oth is the same stop thinking in terms of prop ositional variables

and start thinking in terms of state symbols

Wesay that a formula of ML is pure i it contains no prop ositional

variables our rst result concerns pure sentences As the following examples

show pure sentences are remarkably expressive eachsentence denes the prop

erty listed to its right All these prop erties are relevant to temp oral reasoning

and with the exception of transitivity and density none are denable in ordi

nary mo dal logic

x
x Irreexivity

x

x Asymmetry

x
x x Antisymmetry

x
y y Density

x

x

y y Transitivity

x

xy

y
z z z Discr eteness

x y

n 

The last three expressions can b e simplied using notation

Let us saythata pure sentential axiomatic extension of H K isany

system obtained by adding as axioms a set of pure sentences of ML

Theorem Extended Completeness I Let Pure be a set of pure sen

tences of ML and let P be the pure sentential axiomatic extension of

H K obtained by adding al l sentences in Pure as axioms Then every

P consistent set of formulas in a countable language L is satisable in a count

able standardmodel basedonaframe that validates every axiom in Pure with

function Moreover every consistent set of respect to a standard assignment

sentences in L is satisable in a countable connected standard model based on

aframe that validates Pure



This notation was intro duced in Fo otnote The denition of density can b e rewritten

as y every state y that can b e reached in one step can b e reached in two steps

y

the denition of transitivityis y every state y that can b e reached in two steps can b e

y

reached in one step while discreteness simplies to

y z z there is a

y

y z

successor state y thatisnot step reachable from whichany successor state z is or step

reachable

Proof An easy corollary of Theorem given a P consistent set of formulas

build a satisfying mo del by expanding to a set in a countable language

L and forming the lab eled mo del M S R V and the natural assignment

g Now the lab eled mo del is built of MCSs and each axiom in Pure b elongs

to every P MCS thus by the Truth Lemma Mg j Pure But as Pure con

tains only sentencesthechoice of assignmentis irrelevant hence Mj Pure

Moreover as Pure contains only pure sentences the choice of valuation is also

irrelevant and S R j Pure This proves the rst claim Finallyifcontains

only sentences we obtain a connected mo del by restricting our attention to the

submo del generated by the underlying subframe validates Pure a

As a simple application note that we obtain the logic of strictly partial ly

ordered frames which many writers for example van Benthem would

regard as the minimal temporal logic by adding as axioms x
x and y

y

the previous theorem guarantees that the lab eled mo del validates these axioms

hence as they dene irreexivity and transitivity resp ectively the lab eled mo del

will have these prop erties

This is pleasant but lets push things further Theorem requires us to

use sentences as axioms However it can b e more natural to use pureschemas

Consider for example the schema s s Any instance of this schema

denes transitivity and it is easy to verify that including all instances as axioms

guarantees a transitive lab eled mo del Similarlyany instance of the schema

s t s t s t t s

denes the nobranchingtotheright prop erty and including all instances as

axioms guarantees a lab eled mo del with this prop erty Both transitivity and



nobranchingtotherightare denable using pure sentences but the use of

schemas can oer more A simple example is the schema s any instance of

this denes the class of frames S R such that R S S and its inclusion as



an axiom schema imp oses this prop erty on lab eled mo dels

A pure schematic extension of H K isany system obtained by adding

all ML instances of a set of pure schemas of ML as axioms to

H K

Theorem Extended Completeness I I Let Schemas be a set of pure

schemas of ML and let S be the pure schematic extension of H K

obtained by adding al l instances of the schemas in Schemas as axioms Then

in every S consistent set of sentences in a countable language L is satisable

a countable standard model based on a frame that validates al l these axioms

with respect to a standard assignment function Moreover every consistent set

of sentences in L is satisable in a countable connected standard model based

on a frame that validates al l these axioms

Proof See the extended version of this pap er a



The pure sentence y z y z y z z y denes no

y z

branchingtotheright



We dont know many temp orally relevant examples in ML that require the use

of schemas but examples are easy to nd in tense logic enriched with For example the

schema P s s F s guarantees trichotomy that is xy xRy x y yRx while PFs

that is xy z zRx zRy guarantees us leftdirectedness

What sort of coverage do Theorems and oer For a start note that

all our examples of frame prop erties denable bypure sentences or instances

of pure schemas were rstorder This is no accident a simple extension of

the Standard Translation for the basic mo dal language shows that every pure

formula of ML denes a rstorder condition on frames The Standard

Translation for the basic mo dal language is dened as follows

ST p Px for all prop ositional symbols p

x

ST ST

x x

ST ST ST

x x x

ST
y xR y ST

x y

In the rst clause P is a monadic secondorder predicate variable eachpropo

sitional symb ol corresp onds uniquely to suchasymb ol Following Blackburn

and Seligman we extend this translation to ML as follows we

assume that the rstorder variables wehaveavailable consist of all the usual

state variables plus a distinct variable x for each nominal i and dene

i

ST y x y for all state variables y

x

ST i x x for all nominals i

x i

ST x y x y ST

x x

ST ST

x y y

Supp ose is a formulaofML we supp ose that has b een converted

so that it contains no o ccurrences of the variable x we reservethisvariable to

denote the current state It is easy to see that ST will contain at least

x

one free variable namely x It is also easy to see that this extended version of

ST preserves satisfaction That is for anyML formula any standard

mo del M S R V any standard assignment g and any s S

z V i V p Msg j i Mj ST s g

x

The notation on the right means assign s to the free variable x assign the

unique elementof g z toz if z o ccurs free in the translation assign the unique

elementofV itox if x o ccurs free in the translation and assign V ptoP

i i

if P is a monadic predicate variable that o ccurs free in the translation Nowwe

can see whyit pays to b e pure if contains no prop ositional variables then

the previous expression simplies to

Mgs j i Mj ST s g z V i

x

Wearenow rmly in the world of rst order logic But lets carry on Wehave

Mg j i Mj xS T g z V i

x

and hence

S R j i S R j z z xS T

n x

On the righthand side wehave simply universally quantied over all the free

variables in xS T In short the frame prop ertyanypureformula denes can

x

be calculated by applying the standard translation and forming the universal

bear a certain family resemblance to the closure Thus Theorem and

Sahlqvist Theorems all these results cover rstorder prop erties whichcanbe

eectively calculated from the relevant axioms

There are a host of related questions worth pursuing For example wehave

seen many examples of rstorder prop erties which are not mo dally denable but

which are denable using pure formulas can al l mo dally denable rstorder

conditions be captured in this way And if not can all Sahlqvist denable



prop erties b e so captured

Working with other sorts

Our technical work is done but our conceptual work is not The reader mayhave

gained the impression that hybridization is simply the business of quantifying

over states in a mo dal setting But while thats part of the story and an

imp ortant part to o we believe that a more general idea deserves to be made

explicit

Our preceding work rested on a simple idea combining two forms of infor

mation in a uniform way Our languages dealt with arbitrary information via

the prop ositional symb ols and lab eling information via the state symb ols and

yet we drew no distinction b etween terms and formulas b oth typ es of informa

tion were handled propositional ly Now the natural question is if this works for

statelab el information why shouldnt it work for other typ es of information as

well For example in some applications we mightwanttowork with intervals

or events or paths or some combination of these entitiessowhynotintro

duce sp ecial atomic symb ols that lab el suchentities and allowourselves to bind



them In short why not attempt hybridization in more ambitious ways

Intriguingly there are at least twoways of doing this The rst involves little

change to the work of previous sections For example working with intervals

in a mo dal logic standardly means working with richer frames p erhaps frames

Here S is thought of as a set of intervals as the of the form S v

precedence relation on intervals and v as the inclusion relation on intervals



There are rstorder prop erties which are mo dally denable but not Sahlqvist denable

which can b e dened by pure sentences For example transitivity atomicityxy xRy

z yRz z y is denable by the conjunction of the mo dal transitivity axiom p

p and the McKinsey formula
p
p but no Sahlqvist formula denes this condition

Incidentally McKinsey do es not dene atomicity and in fact no ordinary mo dal formula do es

so only transitivity atomicity is mo dally denable But the following pure sentence denes

atomicity y
y Wehave already seen that transitivity is denable by a pure sentence



In suggesting this we are merely echoing Arthur Prior for this idea was an imp ortant

p erhaps the dominant theme in his later work the key reference here is the p osthumous

Prior and Fine which consists of draft chapters of a b o ok together with pap ers and an

invaluable app endix by Kit Fine which attempts to systematically reconstruct Priors views

Prior attached immense philosophical weight to this pro ject in his view it showed that that

p ossible worlds were not needed to analyze mo dal notions and indeed that times were not

needed to analyze temp oral expressions Only suitably sorted prop ositions and prop erties

mattered

Priors philosophical p osition is in teresting it is strongly information oriented has natural

anities with frameworks such as Prop erty Theory and Situation Semantics and deserves

further exploration Nonetheless here we prefer to adopt a neutral p ersp ective on the philo

sophical signicance of hybrid languages for present purp oses they are simply an eleganttool

for talking ab out structures lo cally and adding further sorts is simply an interesting technical

idea



Various constraints would b e imp osed to makethisinterpretation plausible Typically we

would demand that S b e a strict partial order that S v b e partial order and that

Or p erhaps wed prefer working with frames b earing the relations demanded

in Allen Either way the fundamental point is that we are enriching

our notion of what a state is by lo cating it in aricher web of relations This

mo de of enrichmentisobviously compatible with the metho ds discussed earlier

for example it is straightforward to work with Allenstyle intervals using and

Such an approach naturally leads to multisorted systems For example if

wewanted to work with atomic interval structures it would b e natural to have

a sort which lab eled arbitrary intervals and a subsort which lab eled atomic

intervals see Blackburn

But there is another wayofdeveloping multisorted hybrid languages This

hinges on the following observation some entities can b e thought of as struc

tured sets of states For example an interval is the set of all states b etween two

end p oints Why not add atomic symb ols that range over such sets After all

we already have prop ositional symb ols ranging over arbitrary subsets and state

symb ols ranging over singleton subsets so whynotsymb ols that range over

convex sets to o This is arguably a useful idea see Blackburn

and it is certainly simple to handle logically But to illustrate the structured

set approach to sorting in more detail we want to discuss not intervals but

paths b ecause this example not only provides a nice illustration of the p oten

tial of sorting for temp oral logic it also makes clear that even simplelo oking

extensions can give rise to nontrivial problems

Many applications of temp oral logic demand the use of paths or courses

of history For example for philosophical purp oses it is natural to mo del the

idea that the future is unknown by using treelike mo dels of time that branch

into alternative futures and in computer science it is standard to reason ab out

unravelings of nondeterministic transition systems On the face of it these

applications only seem to demand that we work with new classes of treelike

mo dels and clearly we can do that with the to ols we already have But this is

only half the story As well as new mo dels we are faced with new expressive

demands and these will lead us to new territory

For example in natural language semantics wewould liketohave a future

tense op erator F suchthat F is true precisely when holds somewhere in every

p ossible future that is when holds at least once on every path through the

current state However we cant dene F in anyofourhybrid languages even

abandoning lo cality and working with MLA do esnt help As a second

example consider fairness In computer science applications we may want to

insist that a pro cess is activated innitely often along every p ossible computation

and v interacted appropriately for example wed want stt s v t t t s v t see

van Benthem for further discussion



The straightforward is justied many of the frame prop erties required are expressible

by pure sentences or schemas hence completeness will often b e automatic For example xv

Gy Fy regulates the interaction of and v here v means at all sup erintervals

x

As a second example wehave already noted that atomicity whichwemaywantforvis

enforceable using a pure sentence see Fo otnote It would b e interesting to compare an

and based treatment with Yde Venemas twodimensional analysis see Venema

Of course one mightwant to distinguish b etween various typ es of intervals suchasopen

and closed but wewont do so here

Readers familiar with the representation theorems for abstract interval structures in terms

of p ointbased structures proved in van Benthem will rightly susp ect that in many

cases this structuredset approachtohybrid interval logic will turn out to b e equivalentto

the additionalrelations approach Incidentally this duality b etween the additionalrelations

and the structuredset approaches may b e relevant for paths to o

path but our state symb ols wont help us dene a fairness op erator Thus we

have a genuine expressivity shortcoming on our hands Lets try to x it by

hybridization

The basic strategy for dealing with paths in hybrid languages should be

clear First we add a third sort the sort of path symbols presumably wewant

to keep the state symb ols though this of course is optional As with state

symb ols path symb ols should b e divided into two sub categories namely path

variables which will b e op en to binding and path nominals which will not

So we cho ose PVAR to be a countably innite set of path variables whose

and andPNOMtobeacountably innite elements wetypically write as

set of path nominals whose elements we typically write as and and of

course we cho ose these sets to be disjoint from each other and from PROP

SVAR and NOM We dene the set of atoms of our enriched language to be

PROP SVAR NOM PVAR PNOM

The second step is to add a binder We shall add a binder called thus

forming the language ML As the notation is meant to suggest is

auniversal quantier over paths through the current state that is lo cal paths

The ws of this language are dened in the exp ected way as are such concepts

as free and b ound path variables so lets pro ceed straight to the semantics

We shall work with strictly partially ordered trees S R and adopt Bulls

denition of a path a path in S R is a linearly ordered subset of S that

is maximal among the linearly ordered subsets of S That is paths are convex

subsets of S that contain the ro ot no de and are closed under R successorship

We denote the set of paths in S R byS R If S R and s then

wesay that passes through s Obviously S R isnever empty and at least

one path passes through every state

Denition Standard mo dels and assignments Let ML

be a hybrid language built over PROP SVAR NOM PVAR and PNOM A

model M for this language is a triple S R V such that S R is a strictly

partial ly ordered tree and V PROP NOM PNOM Pow S A model

is cal led standard i for al l nominals i NOM V i is a singleton subset of S

and for al l path nominals PNOM V S R

S An An assignment on M is a mapping g SVAR PVAR Pow

assignment is cal led standard i for al l state variables x SVAR g x is a

singleton subset of S and for al l path variables PNOM V S R

Now to interpret the language The atomic clause is automatically taken

care of byourV g notation and the clauses for the Bo oleans and mo dalities

are unchanged So it only remains to interpret

Mgs j i Mg s j for all g g suchthat s g

That is is a universal quantier over lo cal paths the dual binder is an

existential quantier over lo cal paths

We are not the rst to do this Motivated by Priors arguments Rob ert Bull added a

universal quantier over paths to TLA in his classic pap er thus far from b eing

the new kid on the blo ck hybridization is actually one of the oldest approaches to path

based reasoning weknow of A recent pap er by Gorankoonhybrid languages strong enough

to embed CTL see Goranko b is worth noting Gorankos language do esnt contain

path binders but it do es contain path nominals

It is easy to see that sentences of this language are preserved under generated

submo dels Moreover the expressivity has clearly b een b o osted For example

we can now dene the F op erator

F

It is also straightforward to dene a fairness op erator

Fair

At any state s in a standard mo del Fair is true at a state s i is true

innitely often along every path through s

Moreover familiarlo oking principles of hybrid reasoning extend to our new

alization if is provable then binder For example the rule of path variable loc

so is for any path variable preserves validity and all instances of the

following three schemas are valid

Q

Q p p

Q

LocalPath

Here and p are used as metavariables across path variables and path symb ols

resp ectively In Q must not b e free in andinQ p must b e substitutable

for in In short the basic quanticational p owers of describ ed by QQ

are analogous to those of and LocalPath is analogous to the validity xx



Moreover wehavea Barcan analog

Barcan

is p erhaps easier to grasp The contrap osed and dualised form

Essentially this says if we can select a suitable path at a successor state

then we can select a suitable path at the current state it is a path existence

principle

reect path geometry weuse p Our language also supp orts schemas that

as a metavariable over path symbols and s and t as metavariable over state

nominals

P p p

P p p

P s p t p s t s t t s



The signicance of this may not b e apparent to readers of this short version Roughly

sp eaking in hybrid languages the validity of Barcan analogs is often a sign that the logic will

be wellb ehaved For further discussion see Blackburn and Tzakova a the extended

version of the present pap er

Clearly P reects convexity P reects R maximality under successorship and

P reects linearity note the way the state and path symb ols co op erate here

Summing up in manyways ML is a pleasant language

Thats the go o d part lets turn to the bad It seems that proving com

pleteness results for will require new ideas the lab eled mo del metho d used

in the previous section do es not automatically give us completeness results for

the new binder or at least not with resp ect to the standard semantics dened

ab ove Whats the problem Its simple but deadly although the lab eled

mo del construction will guarantee that all states are lab eled we dont haveany



guarantee that all paths will b e lab eled by some path symbol

This is not easy to x What are we to do Rob ert Bull makes an interesting

remark He comments see his Fo otnote on page that although not every

path is the interpretation of some path symb ol his mo del

does provide enough paths Vu to give a reasonable interpreta

tion

With this remark Bull hints at a line of work that has subsequently b ecome

common in pathbased temp oral logic All reasonably expressive pathbased



logics we know of for example Ockhamist logic or CTL face similar dif

culties regarding completeness A standard resp onse to the problem is to

prove completeness with resp ect to some suitably lib eralized notion of mo del

for example mo dels containing bundles of paths see Zanardo such ap

proaches have anities with the use of generalized mo dels in secondorder logic

or general frames in mo dal logic We b elieveitwould b e interesting to explore

this landscap e using hybrid path languages and susp ect that the lab eled mo del

construction may b e useful in suchinvestigations

But what of the standard semantics dened ab ove This may call for a

the use of innitary rules Intuitively what is more brutal line of approach

needed is an innitary extension of the LocalPath schema From LocalPath

we can deduce that there is a path through the current state what we also

need is a principle that ensures that given a sequence of states one of which

is the current state that satises the convexity R maximality and linearity

principles then there is a path nominal that is true at all the states in this

sequence Innitary rules are unpalatable but a clean innitary approach

may provide a framework which can at least in some cases of interest be

suitably nitized however wemust admit that at presentwe dont knowhow

realistic the prosp ects of success here are

And thats a taste of the joys and sorrows of hybrid path languages Wehave

only scratched the surface of a vast topic but wehopewehave said enough to

indicate whywe nd this terrain worthy of further exploration Moreover we

hop e we have given the reader a taste of the variety of options hybridization

oers to the study of rich temp oral ontologies



Incidentally were not claiming that adding the axioms and rules mentioned ab ove to

H K yields a system complete with resp ect to the standard semantics its obvious

that it do esnt Rather the p ointisthateven after we plug up all the obvious gaps with

suitable axioms well still face a tough problem For further discussion see the extended

version

Concluding remarks

We haveargued that the hybridization technique intro duced by Arthur Prior

and develop ed by Rob ert Bull and the Soa Scho ol is a natural to ol for temp oral

logic Our argument had b oth a technical and conceptual side

Our technical results showed that hybridization is compatible with a temp o

rally natural locality assumption namely that temp oral op erators and binders

should only b e able to work with temp orally accessible states Weshowed that

ML a lo cal language in which Until was denable had an elegant min

imal logic and that many temp orally interesting extended completeness results

could be obtained automatically In our view this language meets the three

criteria listed at the end of Section in particular we feel it exhibits a genuine

synergy of mo dal and classical ideas

Its only fair to warn the reader that we pay a price for this synergy

H K lacks the nite mo del and is undecidable and the same is true



of the logic of strict partial orders Of course the logics of manyinteresting

frame classes are decidable for example the logics of various classes of trees

canbeproved decidable using Rabinstyle arguments see Blackburn and Selig

man nonetheless the fact remains that binding variables to states tilts

the underlying computational prop erties rmly in the classical direction

But we b elieve this is a price worth paying Lab eled deductive systems

Fitting Gabbay have proved an imp ortant technique for au

tomating mo dal inference but lab els are usually regarded as a convenientif

somewhat adhoc metalinguistic to ol Lab els are far more imp ortant than that

indeed if Prior is right they are fundamental to the entire mo dal enterprise

of lab el in the ob ject language and Hybrid languages internalize the notion

this internalization can b e motivated on grounds that are completely indep en

dent of the desire for deductive felicity Nonetheless as the use of the Paste

rule already indicates see Fo otnote deductive felicity isthereforthe tak

ing Seligman discusses natural deduction and sequentbased metho ds

for global hybrid languages containing b oth and and Blackburn and Selig

man a shows that these metho ds can b e adapted even to weak decidable

languages that contain no binders at all In our view the deductive and concep

tual clarityoeredbyinternalized lab els is more than ample comp ensation for

the undecidability results just noted

Our main conceptual argumentinfavor of hybridization is essentially a sec

ular version of Priors vision of abstract entities as prop ositions That is wefeel

that regardless of whether there is an interesting metaphysical sense in which

arbitrary information typ es should be thought of prop ositionally freely com

bining dierent sorts of information in one mo dal algebra is a natural wayof

mo deling temp oral reasoning over richontologies

Acknowledgments

Wewould like to thank the referees for their comments on the rst version of

this pap er and Natasha Kurtonina for her comments on later versions Patrick



For pro ofs of the latter see Blackburn and Seligman Pro ofs of the former are easy

to come by using the spy p oint technique intro duced in Blackburn and Seligman

Blackburn would liketothankAravind Joshi and the sta of the IRCS Univer

sityof Pennsylvania for their hospitality while the nal version of this pap er

was b eing prepared

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