Composition nominative modal and temporal logics

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Composition nominative modal and temporal logics
Taras Shevchenko National University of Kyiv

• Oksana
  Shkilnyak,
• Assistant professor,
• Department of Information Systems,
• Faculty of Cybernetics
• (e-mail
  me.oksana_at_gmail.com)

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Quasi-ary predicates

• Restrictions of classical logic
• syntax prevails over semantics
• functions and predicates are considered as total
  mappings of fixed arity
• Quasi-ary predicates
• partial mappings on nominative data

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Composition nominative approach

• Common for logic and programming
• Principle of development from abstract to
  concrete
• Principle of subordination of syntax to semantics
• Compositionality principle (logical connectives
  and quantifiers are reduced to compositions of
  predicates)
• Nominativity principle (nomination and
  denomination relations are used)

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Spectrum of composition nominative logics
PROPOSITIONAL LEVEL
NOMINATIVE LEVEL
Renominative level
First-order level
Quantifier with level
Functional level
Quantifier level
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Composition nominative logics of quasi-ary
predicates
Modal logics

• Composition nominative modal logics

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Composition nominative modal systems

• We define CNMS as an object
• M ((S, R, Pr, C), F?, I), where
• S is a set of states of the universe
• R is a relation R ? S ? Sn
• Pr is a set of predicates on states
• C is a set of compositions on Pr
• Fm is a set of formulas of the language
• I is an interpretation mapping

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Composition nominative modal systems

• For CNMS of nominative levels, S is a set of
  structures ?  (A?, Pr?), where Pr? is a set of
  quasi-ary predicates VA? ?T, F
• , is a
  set of basic data of the universe
• I  ?s ? S ?Pr (Ps is a set of propositional
  symbols)
• J?  F? ? S ?Pr is a continuation of I
• J?(?, ?) ? Pr?.
• ? ? J?(?, ?) is true
• M ? ? ?, for all ??S
• ? is logically valid M ? holds for all CNMS
  M of the same type
• Propositional CNMS. Abstract elements are states
  of the universe.
• I  ?s ? S ?T, F continues to J  F?m ? S?T,
  F

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Composition nominative modal logics
CNML
Transition CNML
General CNML
Temporal CNML
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Composition nominative modal systems

• For modal transition systems (MTS),
• R  is a transition relation R ? S ? S

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Composition nominative modal systems

• General MTS
• (GMTS) transition MTS with modal compositions ?
  and ?

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Composition nominative modal systems

• Temporal MTS (TMTS) transition MTS with
  temporal compositions
• ??, ??, ??, ?? 

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General CNML

• Modal compositions ? and ?.
• Basic composition ? (? can be taken as
  well).
• Lets define J? for ?? and ?? more precisely.
• For each ??S and d?VA?
• J?(??, ?)(d)
• J?(??, ?)(d)
• Types of general MTS
• R-GMTS, T-GMTS, S-GMTS,
• RT-GMTS, RS-GMTS, TS-GMTS, RTS-GMTS.

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General CNML

• Theorem 1. Basic compositions of general MTS
  preserve equitonicity.
• Theorem 2. For any ?, d ??(d) ? ?(d)
• and ??(d) ? ?(d).
• Corollary. ???? ? and ???? ? are
  logically valid.
• Theorem 3. ??x???x??, ?x?????x?,
• ?x?????x? and ??x???x?? are logically valid.
• Example. The Barcans formula ?x?????x? isnt
  logically valid.

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CNML of quantifier-equational level

• Equality predicate xy, for each d?VA
• Interpretation of equality as identity
• for the same data d it is impossible that
  d(x)? d(y)? on one state and d(x)? ? d(y)? on
  the other.
• Theorem 4. xy ? ?xy is logically valid.
• Theorem 5. 1) It is impossible to have a xy
  and
• a ? ?xy at the same time.
• 2) It can be a ?xy and a ? xy at the same
  time.

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Temporal CNML

• Temporal compositions ??, ??, ??, ??
• Basic compositions ??, ?? (or ??, ??)
• For each ??S and d?VA?
• J?(???, ?)(d)
• J?(???, ?)(d)
• Types of temporal MTS
• R-TMTS, T-TMTS, S-TMTS, RT-TMTS, RS-TMTS,
  TS-TMTS, RTS-TMTS.

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Sequent calculi for GMTL and TMTL

• Lets introduce a relation of logical
  consequence for sets of formulas. It is a
  semantic basis for sequent calculi construction.
• ? and ? are sets of specified with state
  names formulas
• Propositional level
• ? M ? J?(?, ?)  T for all ???? implies the
  impossibility that J?(?, ?)  F for all ????.
• Quantifier level
• ? M ? for all d?VA, ??(d?)  T for all ????
  implies the impossibility that ??(d?)  F for
  all ????.
• Here d? means d ?VA?, where d?VA.
• ?  ? ? M ? for each CNMS M of the same
  type.

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Sequent calculi for GMTL and TMTL

• Sequent calculi for CNML are constructed basing
  on their relational semantics. Sequent forms are
  syntactic analogues of properties of .
• State specification ?? or ??
• ? is a state on which a specified formula is
  considered.
• Sequents are enriched with constructed on the
  current derivation stage sets S and R.
• Enriched sequent for propositional levels
  ? // M
• ? set of specified formulas,
• M model of the universe scheme (constructed on
  the current stage reachability relation, written
  for state names).
• Enriched sequent for nominative levels
  ? // St // M
• St constructed on the current stage set of
  state names with sets of their basic data.

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Sequent calculi for GMTL and TMTL of quantifier
level

• Sequent forms
• ? ?, ? ?, ??, ??, ?RT, ?RT, ?RR, ?RT,
  ?R?, ?R?, ?R?, ?R?, ?R?, ?R?, ??N, ??N,
  ?R?, ?R?, ?R??, ?R??, ??, ??
• forms for modal operators
• ? ?, ? ? for general MTL,
• ? ??, ? ??, ? ??, ? ?? for temporal MTL.

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Soundness and completeness of sequent calculi
for GMTL and TMTL

• Soundness theorem. Let ? ?? ? is derivable.
  Then ?  ?.
• Completeness theorem. Let ?  ?. Then sequent
  ? ?? ? is derivable.

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Sequent calculi for GMTL and TMTL of quantifier
level

• ??
• ? is totally nonsignificant and ??n?(?, ?). New
  element y is added to the carrier A? of the
  state ?.
• ??
• z1,, z? is a set of names of available
  formulas of the sequent ???x?, ? and its
  descendants.

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Forms for modal operators (general MTL)

• General case
• (no conditions on ? imposed)
• ? ?
• ? is any state connected with the current by
  reachability relation ?,
• ??? ? is an auxiliary formula, generating
  formulas ?1? ?, ..., ?n? ?   for all states
  ?1, , ?n, available at the moment, such that
  ???1, ..., ???n .
• The form doesnt affect the set of basic data of
  states and the model of the universe scheme.

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Forms for modal operators (general MTL)

• General case
• (no conditions on ? imposed)
• ? ?
• ? is a new state of the universe, B1, , Bm are
  the formulas obtained from ??? Bi , generated by
  ?? ?Bi.
• The form adds a new state ? such that ??? and
  A?  A?.

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Forms for modal operators (general MTL)

• ? is transitive and reflexive (RT-calculus)
• ? ?
• ??? ? generates ?1? ?, ..., ?n? ?  and
  ?1???, ..., ?n???  for all states ?1, , ?n,
  available at the moment, such that
  ???1, ..., ???n .
• ? ?
• ? is a new state of the universe, B1, , Bm are
  the formulas obtained from ???Bi, generated by
  ???Bi.

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Forms for temporal operators (temporal MTL)

• Forms are similar to those of general MTL. They
  are specified according to the direction of time
  ? ??, ? ?? and ? ??, ? ??.
• For example, forms ? ?? and ? ?? for transitive
  and reflexive ? (temporal RT-calculus)
• ? ??
• ? ??
• ? is a new state of the universe, B1, , Bm are
  the formulas obtained from ??? Bi after applying
  ? ?? / ??? Bi after applying ? ??.
• If ? is a symmetric relation, then operators ??
  and ?? work identically.

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Summary

• Modal and temporal logics are widely used in
  software systems specification and verification.
• We propose semantically based definitions of such
  logics.
• A special refinement of the notion of CNMS is
  introduced.
• We specify general and temporal modal transition
  systems and logics based on such systems.
• Semantic properties of transition and temporal
  MTL are investigated.
• Sequent calculi are constructed for the defined
  logics of nominative levels soundness and
  completeness theorems are proved.

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Taras Shevchenko National University of Kyiv