This paper presents the encoding of the hybridisation method into the HETS platform.FC

A. Madeira

C. Areces

G. Hoffmann

J.A. Goguen

M.A. Martins

P. Blackburn

T. Mossakowski

Renato Neves

Alexandre Madeira

Manuel A. Martins

Luís S. Barbosa

Crossref

Hybridisation at Work

Neves, Renato

Madeira, Alexandre

Martins, Manuel A.

Barbosa, L. S.

Universidade do Minho: RepositoriUM

Hybridisation at work
Renato Neves1, Alexandre Madeira2, Manuel A. Martins3, and Luı́s S. Barbosa1
1 HASLab - INESC TEC & Univ. Minho
{nevrenato@gmail.com, lsb@di.uminho.pt}
2 HASLab - INESC TEC & Dep. Mathematics, Univ. Aveiro & Critical Software S.A.
madeira@ua.pt
3 CIDMA - Dep. Mathematics, Univ. Aveiro
martins@ua.pt
Abstract. This paper presents the encoding of the hybridisation method pro-
posed in [MMDB11,DM13] into the HETS platform.
Keywords: Hybrid logics; institutions; reconfigurable systems.
1 Introduction and purpose
Hybrid logics [Bla00] are a brand of modal logics that provides appropriate syntax
for the possible worlds semantics through nominals. In particular, it adds to the modal
description of transition structures the ability to refer to specific states. This paves the
way to an expressive framework for specifying complex software able to evolve through
different execution configurations. In a number of papers, starting with [MFMB11], the
foundations and methodological aspects of such a framework for reconfigurability have
been developed, leading to a two-stage method:
– globally the system’s dynamics is represented by a transition structure described in
a hybrid language, whose states correspond to possible configurations;
– locally each state is endowed with a structure modeling the respective configuration
specification.
The logic used locally depends on the application requirements. Typical candidates
are equational, partial algebra or first-order logic (FOL), but one may equally resort
to multivalued logics or even to hybrid logic itself equipping, in the last case, each
state with another (local) transition system. Instead of fixing a particular hybrid logic,
a systematic method to develop on top of each local logic, the characteristic features of
hybrid logic was proposed. This process is called hybridisation and was characterised
in [MMDB11,DM13], framed in the context of the theory of institutions [GB92] to
achieve greater generality.
The hybridisation process abstracts away the syntactic and semantic details, that
are independent of the very essence of the hybrid logic idea. This has a number of
benefits, even if paying the price of a heavy notational burden. One is the focus on the
essential, the theoretical development not being hindered by irrelevant details. Another
one concerns its applicability to a wide number of concrete instances, since all of them
could be regarded as combinations between concrete versions of hybrid logics with
other logical systems. In this sense, the hybridisation method can be seen as a source of
logics for the specification of reconfigurable systems [MFMB11].
Institutions provide a systematic way to relate logics and transport results from one
to another, which means that a theorem prover for the latter can be used to reason about
specifications written in the former. This is achieved through a special class of maps
between institutions, referred to conservative comorphisms.
This paper reports on the implementation of the method introduced in [MMDB11]
along two different directions. Firstly the general hybridisation method is incorporated
in the HETS platform [MML07] — parsing and static analysis for the hybridisation of
any base institution already supported in HETS being provided. Secondly, the comor-
phism HCASL → CASL is implemented, offering effective tool support for proofs on
a number of HCASL-sub-institutions, namely HFOL and hybrid propositional logic4.
This provides for free the proof support environment of a particularly well established
logic. Naturally, the final goal is to have in HETS for free a comorphism from a hy-
bridised logic HI to FOL given that a comorphism exists from the base logic I to
FOL.
HETS has been described as a “motherboard” where different “expansion cards”
can be plugged. These pieces are individual logics (with their particular analysers and
proof tools) as well as logic translations. To make them compatible, logics are for-
malised as institutions and translations as comorphisms. Therefore, the integration of
the hybrid specifications on the HETS platform is legitimate, since all formal require-
ments (e.g., that institutions exist, that a comorphism can be defined, etc.) are already
guaranteed by the hybridisation process itself.
The code for this extension to HETS, as well as a set of hybrid specification exam-
ples, is available from GITHUB (https://github.com/nevrenato/Hets Fork). Additionally
a ready–to–use HETS system is provided in a virtual machine available at SUGARSYNC
(https://www.sugarsync.com/pf/D7620475 67336482 6511440).
2 Hybridisation as a plug-in to HETS
2.1 Hybridisation of CASL
The hybridisation process was first incorporated into HETS through its direct applica-
tion to what is the platform lingua franca: CASL[MHST03]. A comorphism from the
outcome HCASL to CASL was also defined. Thus, assisted proof support for HCASL
becomes available for free. HCASL specifications add to the usual ones in CASL a dec-
laration of nominals and modalities. Sentences include the typical hybrid machinery
and quantification over nominals. Thus, the respective grammar is extended as follows:
CFor’ = HFor | . . . ;
HFor = @ n CFor’ | < m > CFor’ | [ m ] CFor’ | Here n | ! n CFor’ | ? n CFor’;
4 HCASL consists of the hybridisation of the institution CASL with the models restricted to
those with sorts commonly realised in all the states and with common realisation of the quan-
tified variables.
where n is a nominal, m a modality, and the last two alternatives the universal and
existential quantifiers over nominals. The grammar above besides extending sentences
with the typical hybrid machinery also includes the quantification over nominals. The
latter makes possible to define complex operators such as the ↓ binder, or the notion of
a rigid designator
Notice also the need to use the keyword Here, when having a nominal for a sen-
tence. Such is needed so that the parser for CASL does not take nominals as a common
proposition. In the following section a more sophisticated mechanism to prevent this
kind of ambiguities is discussed.
Fig. 1. Kripke frame for the swinging calculator.
Figure 1 depicts a toy example of a calculator which commutes, through a modality
shift, between two modes of operation: a binary operation is interpreted as arithmetic
addition in mode Sum and multiplication in Mult. Its HCASL encoding is reproduced
below.
logic HYBRID
spec RECONFCALC =
HNAT
then modalities Shift
nominals Sum,Mult
ops # : Nat × Nat→ Nat
%% global axioms
∀ n, m, p : Nat • n # m = m # n ∧ (n # m) # p = n # (m # p)
%% axioms specific to Sum and Mult
∀ n, m : Nat • @Sum n # 0 = n ∧ @Sum n # suc(m) = suc(n # m) ∧ @Mult n # 0 = 0
∃ p, q : Nat • @Mult n # suc(m) = p ∧ @Sum n # q = p ∧ @Mult n # m = q
%% axioms specific to the Kripke frame
•Here Sum ∨Here Mult
• @Sum (< Shift > Here Mult ∧ [Shift] Here Mult)
• @Mult (< Shift > Here Sum ∧ [Shift] Here Sum)
%% lt relation definition, using # op
∀ n, m, r : Nat • n <= m⇒ n # r <= m # r
The encoding from HCASL to CASL provides the expected proof support for this
sort of hybrid specifications, i.e. the set of proof tools available for CASL is brought
to HCASL. Below there are some examples of properties of the swinging calculator
specification verified in this way.
Figure 2 registers the corresponding HETS session, showing the proof window, part
of the model theory, and the specification graph.
• @Sum n # m >= n %(lemma2)%
• @Mult (m = 0 ∨ m = suc(0)⇒ n # m <= n) %(lemma6)%
• @Sum [Shift] [Shift] Here Sum %(Cyclicity1)%
∀ n, m, r : Nat
• n # 0 = 0⇒ < Shift > n # 0 = n %(StateExclusion)%
• ∃ p : Nat • @Sum n # n = p⇒ @Mult n # suc(suc(0)) = p %(DoubleDef)%
• ∃ p, q : Nat • m <= suc(0)⇒ @Sum n # m = p ∧ @Mult n # m = q⇒ p >= q
%(CasesSumBiggerMult)%
Fig. 2. A HETS session for the swinging calculator.
2.2 Generic hybridisation
The current integration of the hybridisation method into HETS offers the user hybridised
versions of logics already “plugged in” HETS . Examples include propositional logic,
CASL, COCASL, and any other logic previously hibridised. The implementation of the
hybridisation framework can also be used to expand this list: given an identifier, a sen-
tences’ parser and an analyser, a new logic can be taken into the picture.
In each case the resulting grammar for writing hybridised specifications is the com-
position of a specification in the base logic, the declaration of nominals and modalities,
and the sentences enriched with hybrid properties.
Note, finally, that hybridisation as described here may introduce ambiguities. In the
example below, for instance, it is not clear which nominals belong to the base or the
hybridised layer.
To clear out possible ambiguities resulting from double or multiple applications of
the hybridisation method, formulas associated to the base layer are wrapped into curly
brackets (i.e. they are transported to the hybridised level through an injection).
Follows an example specified in HETS using the double hybridisation of propositional
logic:
logic HYBRIDIZE
spec GEOGRAPHY =
baselogic Hybridize
Basic Spec{ baselogic Propositional
Basic Spec { props p }
Nominals Portugal, England, Canada
Modalities Car
•@ Portugal <Car> England }
Nominals Europe, America
Modalities Plane
• America => { not ( Portugal ∨ England ) } ∧ Europe => { not Canada };
•@ Europe <Plane> ( America ∧ { Canada } )
The specification above exemplifies a double hybridisation. It describes routes in
a map linked by some means of transport (the modalities) between different places
(identified by nominals). One level of hybridisation corresponds to countries; the second
one to continents. Clearly, in this case nominals can be ordered respecting the order used
to build an hierarchy of countries and continents. Note how this hierarchy is brought
back into sentences. For instance the last one in the above specification states : from
Europe one can travel by plane to America; and, in particular, to Canada.
3 Discussion
The hybridisation process and its implementation on HETS proved an effective and
flexible way to prove properties of hybrid specifications and thus to support the design
method in [MFMB11]. The implementation compares well with respect to dedicated
provers for (specific) hybrid logics, although a systematic comparison is still being
done. Typically, such tools, such as HTAB[HA09] or HYLORES[AH02], are faster to
prove a formula with low complexity, but HETS achieves similar or even better perfor-
mance in more complex ones. In some cases, formulas hard to deal with in HTAB are
straightforward in HETS. A typical example is A(↓ x〈m〉¬x).
Moreover, the genericity of the approach reported in this paper seems highly attrac-
tive in practice.
The results of [MMDB11,DM13] have yet a great potential to be explored on top
of their integration in the HETS integration. The first reference shows that a comor-
phism from an arbitrary institution I to FOL gives rise to another comorphism from
its hybridisation HI to FOL. Reference [DM13] refines this by characterising the con-
servativeness of such maps. Conservativeness is sometimes achieved not for the “free
hybridisation” but for a restrict semantics of the hybridised institution satisfying a set
of properties. Those restrictions are axiomatised on the FOL “side” as suitable presen-
tations. The HETS rich support for FOL justifies the pertinence of the “hybridisation
of comorphisms” method, since it extends tool support for a wide class of hybridised
logics. Those includes not only hybrid equational logic and hybrid first-order logic (al-
ready supported by the comorphism presented above) but also hybridised modal logic
and even hybridised hybrid propositional logic, among others.
Acknowledgements: This work is funded by ERDF - European Regional Development Fund
through the COMPETE Programme (operational programme for competitiveness) and by Na-
tional Funds through the FCT, Portuguese Foundation for Science and Technology within project
FCOMP-01-0124-FEDER-028923, and by CIDMA-Centro de Investigação e Desenvolvimento
em Matemática e Aplicações of Universidade de Aveiro within the project FCOMP-01-0124-
FEDER-022690.
References
[AH02] C. Areces and J. Heguiabehere. Hylores: A hybrid logic prover based on direct
resolution. In CADE-18, volume 2392 of LNCS. Springer, 2002.
[Bla00] Patrick Blackburn. Representation, reasoning, and relational structures: A hybrid
logic manifesto. Logic Journal of IGPL, 8(3):339–365, 2000.
[DM13] R. Diaconescu and A. Madeira. Encoding hybridized institutions into first order
logic. Submitted to a journal, 2013.
[GB92] Joseph A. Goguen and Rod M. Burstall. Institutions: abstract model theory for spec-
ification and programming. J. ACM, 39:95–146, 1992.
[HA09] G. Hoffmann and C. Areces. Htab: a terminating tableaux system for hybrid logic.
Electr. Notes Theor. Comput. Sci., 231:3–19, 2009.
[MFMB11] A. Madeira, J. M. Faria, M. A. Martins, and L. S. Barbosa. Hybrid specification of
reactive systems: An institutional approach. In G. Barthe, A. Pardo, and G. Schnei-
der, editors, SEFM 2011: Software Engineering and Formal Methods, volume 7041
of LNCS, pages 269–285. Springer, 2011.
[MHST03] Till Mossakowski, Anne Haxthausen, Donald Sannella, and Andrzej Tarlecki.
CASL: The common algebraic specification language: Semantics and proof theory.
Computing and Informatics, 22:285–321, 2003.
[MMDB11] M. A. Martins, A. Madeira, R. Diaconescu, and L. S. Barbosa. Hybridization of
institutions. In A. Corradini, B. Klin, and C. Cı̂rstea, editors, CALCO 2011: Algebra
and Coalgebra in Computer Science, volume 6859 of LNCS. Springer, 2011.
[MML07] Till Mossakowski, Christian Maeder, and K. Lüttich. The heterogeneous tool set,
hets. In TACAS’07: Tools and Algorithms for Construction and Analysis of Systems,
volume 4424 of LNCS, pages 519–522. Springer, 2007.


English

Hybridisation at work

Hybridisation at workRenato Neves1, Alexandre Madeira2, Manuel A. Martins3, and Luı´s S. Barbosa11 HASLab - INESC TEC & Univ. Minho{nevrenato@gmail.com, lsb@di.uminho.pt}2 HASLab - INESC TEC & Dep. Mathematics, Univ. Aveiro & Critical Software S.A.madeira@ua.pt3 CIDMA - Dep. Mathematics, Univ. Aveiromartins@ua.ptAbstract. This paper presents the encoding of the hybridisation method pro-posed in [MMDB11,DM13] into the HETS platform.Keywords: Hybrid logics; institutions; reconfigurable systems.1 Introduction and purposeHybrid logics [Bla00] are a brand of modal logics that provides appropriate syntaxfor the possible worlds semantics through nominals. In particular, it adds to the modaldescription of transition structures the ability to refer to specific states. This paves theway to an expressive framework for specifying complex software able to evolve throughdifferent execution configurations. In a number of papers, starting with [MFMB11], thefoundations and methodological aspects of such a framework for reconfigurability havebeen developed, leading to a two-stage method:– globally the system’s dynamics is represented by a transition structure described ina hybrid language, whose states correspond to possible configurations;– locally each state is endowed with a structure modeling the respective configurationspecification.The logic used locally depends on the application requirements. Typical candidatesare equational, partial algebra or first-order logic (FOL), but one may equally resortto multivalued logics or even to hybrid logic itself equipping, in the last case, eachstate with another (local) transition system. Instead of fixing a particular hybrid logic,a systematic method to develop on top of each local logic, the characteristic features ofhybrid logic was proposed. This process is called hybridisation and was characterisedin [MMDB11,DM13], framed in the context of the theory of institutions [GB92] toachieve greater generality.The hybridisation process abstracts away the syntactic and semantic details, thatare independent of the very essence of the hybrid logic idea. This has a number ofbenefits, even if paying the price of a heavy notational burden. One is the focus on theessential, the theoretical development not being hindered by irrelevant details. Anotherone concerns its applicability to a wide number of concrete instances, since all of themcould be regarded as combinations between concrete versions of hybrid logics withother logical systems. In this sense, the hybridisation method can be seen as a source oflogics for the specification of reconfigurable systems [MFMB11].Institutions provide a systematic way to relate logics and transport results from oneto another, which means that a theorem prover for the latter can be used to reason aboutspecifications written in the former. This is achieved through a special class of mapsbetween institutions, referred to conservative comorphisms.This paper reports on the implementation of the method introduced in [MMDB11]along two different directions. Firstly the general hybridisation method is incorporatedin the HETS platform [MML07] — parsing and static analysis for the hybridisation ofany base institution already supported in HETS being provided. Secondly, the comor-phism HCASL → CASL is implemented, offering effective tool support for proofs ona number of HCASL-sub-institutions, namely HFOL and hybrid propositional logic4.This provides for free the proof support environment of a particularly well establishedlogic. Naturally, the final goal is to have in HETS for free a comorphism from a hy-bridised logic HI to FOL given that a comorphism exists from the base logic I toFOL.HETS has been described as a “motherboard” where different “expansion cards”can be plugged. These pieces are individual logics (with their particular analysers andproof tools) as well as logic translations. To make them compatible, logics are for-malised as institutions and translations as comorphisms. Therefore, the integration ofthe hybrid specifications on the HETS platform is legitimate, since all formal require-ments (e.g., that institutions exist, that a comorphism can be defined, etc.) are alreadyguaranteed by the hybridisation process itself.The code for this extension to HETS, as well as a set of hybrid specification exam-ples, is available from GITHUB (https://github.com/nevrenato/Hets Fork). Additionallya ready–to–use HETS system is provided in a virtual machine available at SUGARSYNC(https://www.sugarsync.com/pf/D7620475 67336482 6511440).2 Hybridisation as a plug-in to HETS2.1 Hybridisation of CASLThe hybridisation process was first incorporated into HETS through its direct applica-tion to what is the platform lingua franca: CASL[MHST03]. A comorphism from theoutcome HCASL to CASL was also defined. Thus, assisted proof support for HCASLbecomes available for free.HCASL specifications add to the usual ones in CASL a dec-laration of nominals and modalities. Sentences include the typical hybrid machineryand quantification over nominals. Thus, the respective grammar is extended as follows:CFor’ = HFor | . . . ;HFor = @ n CFor’ | < m > CFor’ | [ m ] CFor’ | Here n | ! n CFor’ | ? n CFor’;4 HCASL consists of the hybridisation of the institution CASL with the models restricted tothose with sorts commonly realised in all the states and with common realisation of the quan-tified variables.where n is a nominal, m a modality, and the last two alternatives the universal andexistential quantifiers over nominals. The grammar above besides extending sentenceswith the typical hybrid machinery also includes the quantification over nominals. Thelatter makes possible to define complex operators such as the ↓ binder, or the notion ofa rigid designatorNotice also the need to use the keyword Here, when having a nominal for a sen-tence. Such is needed so that the parser for CASL does not take nominals as a commonproposition. In the following section a more sophisticated mechanism to prevent thiskind of ambiguities is discussed.Fig. 1. Kripke frame for the swinging calculator.Figure 1 depicts a toy example of a calculator which commutes, through a modalityshift, between two modes of operation: a binary operation is interpreted as arithmeticaddition in mode Sum and multiplication in Mult. ItsHCASL encoding is reproducedbelow.logic HYBRIDspec RECONFCALC =HNATthen modalities Shiftnominals Sum,Multops # : Nat × Nat→ Nat%% global axioms∀ n, m, p : Nat • n # m = m # n ∧ (n # m) # p = n # (m # p)%% axioms specific to Sum and Mult∀ n, m : Nat • @Sum n # 0 = n ∧ @Sum n # suc(m) = suc(n # m) ∧ @Mult n # 0 = 0∃ p, q : Nat • @Mult n # suc(m) = p ∧ @Sum n # q = p ∧ @Mult n # m = q%% axioms specific to the Kripke frame•Here Sum ∨Here Mult• @Sum (< Shift > Here Mult ∧ [Shift] Here Mult)• @Mult (< Shift > Here Sum ∧ [Shift] Here Sum)%% lt relation definition, using # op∀ n, m, r : Nat • n <= m⇒ n # r <= m # rThe encoding from HCASL to CASL provides the expected proof support for thissort of hybrid specifications, i.e. the set of proof tools available for CASL is broughtto HCASL. Below there are some examples of properties of the swinging calculatorspecification verified in this way.Figure 2 registers the corresponding HETS session, showing the proof window, partof the model theory, and the specification graph.• @Sum n # m >= n %(lemma2)%• @Mult (m = 0 ∨ m = suc(0)⇒ n # m <= n) %(lemma6)%• @Sum [Shift] [Shift] Here Sum %(Cyclicity1)%∀ n, m, r : Nat• n # 0 = 0⇒ < Shift > n # 0 = n %(StateExclusion)%• ∃ p : Nat • @Sum n # n = p⇒ @Mult n # suc(suc(0)) = p %(DoubleDef)%• ∃ p, q : Nat • m <= suc(0)⇒ @Sum n # m = p ∧ @Mult n # m = q⇒ p >= q%(CasesSumBiggerMult)%Fig. 2. A HETS session for the swinging calculator.2.2 Generic hybridisationThe current integration of the hybridisation method into HETS offers the user hybridisedversions of logics already “plugged in” HETS . Examples include propositional logic,CASL, COCASL, and any other logic previously hibridised. The implementation of thehybridisation framework can also be used to expand this list: given an identifier, a sen-tences’ parser and an analyser, a new logic can be taken into the picture.In each case the resulting grammar for writing hybridised specifications is the com-position of a specification in the base logic, the declaration of nominals and modalities,and the sentences enriched with hybrid properties.Note, finally, that hybridisation as described here may introduce ambiguities. In theexample below, for instance, it is not clear which nominals belong to the base or thehybridised layer.To clear out possible ambiguities resulting from double or multiple applications ofthe hybridisation method, formulas associated to the base layer are wrapped into curlybrackets (i.e. they are transported to the hybridised level through an injection).Follows an example specified in HETS using the double hybridisation of propositionallogic:logic HYBRIDIZEspec GEOGRAPHY =baselogic HybridizeBasic Spec{ baselogic PropositionalBasic Spec { props p }Nominals Portugal, England, CanadaModalities Car•@ Portugal <Car> England }Nominals Europe, AmericaModalities Plane• America => { not ( Portugal ∨ England ) } ∧ Europe => { not Canada };•@ Europe <Plane> ( America ∧ { Canada } )The specification above exemplifies a double hybridisation. It describes routes ina map linked by some means of transport (the modalities) between different places(identified by nominals). One level of hybridisation corresponds to countries; the secondone to continents. Clearly, in this case nominals can be ordered respecting the order usedto build an hierarchy of countries and continents. Note how this hierarchy is broughtback into sentences. For instance the last one in the above specification states : fromEurope one can travel by plane to America; and, in particular, to Canada.3 DiscussionThe hybridisation process and its implementation on HETS proved an effective andflexible way to prove properties of hybrid specifications and thus to support the designmethod in [MFMB11]. The implementation compares well with respect to dedicatedprovers for (specific) hybrid logics, although a systematic comparison is still beingdone. Typically, such tools, such as HTAB[HA09] or HYLORES[AH02], are faster toprove a formula with low complexity, but HETS achieves similar or even better perfor-mance in more complex ones. In some cases, formulas hard to deal with in HTAB arestraightforward in HETS. A typical example is A(↓ x〈m〉¬x).Moreover, the genericity of the approach reported in this paper seems highly attrac-tive in practice.The results of [MMDB11,DM13] have yet a great potential to be explored on topof their integration in the HETS integration. The first reference shows that a comor-phism from an arbitrary institution I to FOL gives rise to another comorphism fromits hybridisationHI to FOL. Reference [DM13] refines this by characterising the con-servativeness of such maps. Conservativeness is sometimes achieved not for the “freehybridisation” but for a restrict semantics of the hybridised institution satisfying a setof properties. Those restrictions are axiomatised on the FOL “side” as suitable presen-tations. The HETS rich support for FOL justifies the pertinence of the “hybridisationof comorphisms” method, since it extends tool support for a wide class of hybridisedlogics. Those includes not only hybrid equational logic and hybrid first-order logic (al-ready supported by the comorphism presented above) but also hybridised modal logicand even hybridised hybrid propositional logic, among others.Acknowledgements: This work is funded by ERDF - European Regional Development Fundthrough the COMPETE Programme (operational programme for competitiveness) and by Na-tional Funds through the FCT, Portuguese Foundation for Science and Technology within projectFCOMP-01-0124-FEDER-028923, and by CIDMA-Centro de Investigac¸a˜o e Desenvolvimentoem Matema´tica e Aplicac¸o˜es of Universidade de Aveiro within the project FCOMP-01-0124-FEDER-022690.References[AH02] C. Areces and J. Heguiabehere. Hylores: A hybrid logic prover based on directresolution. In CADE-18, volume 2392 of LNCS. Springer, 2002.[Bla00] Patrick Blackburn. Representation, reasoning, and relational structures: A hybridlogic manifesto. Logic Journal of IGPL, 8(3):339–365, 2000.[DM13] R. Diaconescu and A. Madeira. Encoding hybridized institutions into first orderlogic. Submitted to a journal, 2013.[GB92] Joseph A. Goguen and Rod M. Burstall. Institutions: abstract model theory for spec-ification and programming. J. ACM, 39:95–146, 1992.[HA09] G. Hoffmann and C. Areces. Htab: a terminating tableaux system for hybrid logic.Electr. Notes Theor. Comput. Sci., 231:3–19, 2009.[MFMB11] A. Madeira, J. M. Faria, M. A. Martins, and L. S. Barbosa. Hybrid specification ofreactive systems: An institutional approach. In G. Barthe, A. Pardo, and G. Schnei-der, editors, SEFM 2011: Software Engineering and Formal Methods, volume 7041of LNCS, pages 269–285. Springer, 2011.[MHST03] Till Mossakowski, Anne Haxthausen, Donald Sannella, and Andrzej Tarlecki.CASL: The common algebraic specification language: Semantics and proof theory.Computing and Informatics, 22:285–321, 2003.[MMDB11] M. A. Martins, A. Madeira, R. Diaconescu, and L. S. Barbosa. Hybridization ofinstitutions. In A. Corradini, B. Klin, and C. Cıˆrstea, editors, CALCO 2011: Algebraand Coalgebra in Computer Science, volume 6859 of LNCS. Springer, 2011.[MML07] Till Mossakowski, Christian Maeder, and K. Lu¨ttich. The heterogeneous tool set,hets. In TACAS’07: Tools and Algorithms for Construction and Analysis of Systems,volume 4424 of LNCS, pages 519–522. Springer, 2007.

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