Logics for Graphs and Graph Transformations

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Logics for Graphs and Graph Transformations

• For course Logics for knowledge representation
  and reasoning by Luciano Serafini. Aliaksei
  Novikau.

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The plan

• What is Tropos
• Graph grammars String grammars
• String Grammars with more details
• Turing machines
• Graph Grammars different types
• MSO logic
• Transductions
• F-magma
• VR and HR grammars decidable
• Decidability of the task in common
• Conclusions

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Tropos Bresciani2003

• Tropos is based on two key ideas.
• First, the notion of agent and all related
  mentalistic notions (for instance goals and
  plans) are used in all phases of software
  development, from early analysis down to the
  actual implementation.
• Second, Tropos covers also the very early phases
  of requirements analysis, thus allowing for a
  deeper understanding of the environment where the
  software must operate, and of the kind of
  interactions that should occur between software
  and human agents.

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Tropos
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Tropos graph development rules
The rule for goal decomposition.
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Tropos - graph transformation approach advantages

• Requirement traceability history graph and saved
  sequence of applied rules
• Consistency of the developed diagram to a
  metamodel because the set of the rules is
  consistent with the metamodel - ???

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Graph Grammar

• Graph grammar Bardohl1999 G (L, T, S, P)
• L finit set of labels for nodes and edges
• T terminal symbols
• S an axiom, i.e., an initial graph
• P productions
• Each node, respectively edge, can be equipped
  with an element of the label set. These element
  can be used as types or as terminal or
  nonterminal symbols.
• The difference with a string grammar
  concatenation

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String Grammars Matuszek1996

• A grammar G is a quadruple G (V, T, S, P)where
  
• V is a finite set of (meta)symbols, or variables.
  
• T is a finite set of terminal symbols.
• S is a distinguished element of V called the
  start symbol.
• P is a finite set of productions (or rules).
• A production has the form where

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String Grammars Regular Grammars

• In a right-linear grammar, all productions have
  one of the two forms
• V -gt TV or V -gt T
• In a left-linear grammar, all productions have
  one of the two forms
• V -gt VT or V -gt T
• A regular grammar is either a right-linear
  grammar or a left-linear grammar.
• Exmp. S -gt null
• S -gt a X
• X -gt S b is not a regular.

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String Grammars Regular Grammars, Finita
Automata

• Regular grammars are equal to regular expressians
  and describe regular languages. Regular
  expressions and grammars are equivalent to NFAs
  (Nondeterministic Finita Automata) by doing two
  things
• For any given regular expression/grammar it is
  possible to build an NFA that accepts the same
  language and vice-versa.
• The following automaton and
• grammar both recognize the
• set of strings consisting of an
• even number of 0's and an even
• number of 1's.
• S -gt null S -gt 0 B S -gt 1 AA -gt 0 C
  A -gt 1 SB -gt 0 S B -gt 1 CC -gt 0 A C
  -gt 1 B

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String Grammars Context-Free Grammars

• A grammar G (V, T, S, P) is a context free
  grammar (cfg) if all productions in P have the
  form
• A -gt x, where
• A V, and
• x (V ? T)
• Since T?(V ? T) and TV?(V ? T), it follows
  that every right-linear grammar is also a
  context-free grammar, similary left-linear and
  thus regular.

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String Grammars Context-Free Grammars,
Derivation Tree

• For example, for the following derivation the
  tree will look like
• S-gtABC-gtaABC-gtaABcC-gtaBcC-gtabBcC-gtabBc-gtabbBc-gtabb
  c
• A shape of the tree does not depend on
  productions application.

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String Grammars Context-Free Grammars

• Chomsky Normal Form
• A -gt BC or A -gt a where A, B, and C are
  variables and a is a terminal. Greibach Normal
  Form
• Greibach Normal Form
• A -gt ax where a is a terminal and x -gt V.
• Greibach Normal Form -gt proving the equivalence
  of CFGs and NPDAs (the grammars are usually
  ugly).
• Any CFG can be put to Greibach, except containing
  null.

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String Grammars Context-Free Grammars,
Nondeterministic Pushdown Automata

• An example of NPDA
• NPDAs recognises the
• same language as
• and Contextual-Free
• grammars.
• The explaining example

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String Grammars the Summarizing Table (The
Chomsky hierarchy)
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Turing Machine

• Turing machine uses a tape, infinite in both
  directions. The tape consists of a series of
  squares, each of which can hold a single symbol.
  The tape head, or read-write head, can read a
  symbol from the tape, write a symbol to the tape,
  and move one square in either direction. Turing
  machines can be deterministic or
  nondeterministic.
• Unlike the other automata, a Turing machine does
  not read "input." Instead, there may be (and
  usually are) symbols on the tape before the
  Turing machine begins the Turing machine might
  read some, all, or none of these symbols. The
  initial tape may, if desired, be thought of as
  "input."
• The transition function of Turing machine

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Turing Machine

• Turing Machine can read a word from input, accept
  it (come to accepting state), reject it (come to
  rejection state) and buzz forever. The last case
  Turing Machine Halt.
• A Turing machine might halt for one input string,
  but go into an infinite loop when given some
  other string. The halting problem asks Is it
  possible to tell, in general, whether a given
  machine will halt for some given input?
• It is possible to imitate a Turing machine by
  another Turing (universal) machine.
• Theorem. Turing machine
• WillHalt(M,w) does not exist.
• The proof is by appeared
• paradox.
• Many things proved by
• reduction to the halt problem.

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Turing Machine

• Turing machines - an attempt to formalize the
  notion of an effective procedure ("algorithm").
  The same time
• Alonzo Church -- Lambda calculus
• Emil Post -- Production systems
• Raymond Smullyan -- Formal systems
• Stephen Kleene -- Recursive function theory
• Noam Chomsky -- Unrestricted grammars
• All of these formalisms were proved equivalent
  to one another.
• Turing's Thesis (weak form) A Turing machine can
  compute anything that can be computed by a
  general-purpose digital computer.
• Turing's Thesis (strong form) A Turing machine
  can compute anything that can be computed.

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Graph Grammars

• The main difference is concatenation.
• Bardohl1999 A visual programming language is a
  programming language with at least some
  constructs which have an either inherently or
  superimposed multi-dimensional representation.
• A string has only two members to concatenate to
  (left and right), but a graph object has many
  choices (dimensions).
• Why can not we represent graph grammars as string
  grammars (despite we can represent a graph as a
  string)?
• A graph change the string representation
  unpedictably with some graph transformations
  (applications of graph grammar productions)

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Graph Grammars

• Generally there are two main graph transformation
  approaches single pushout and double pushout
  Corradini1997.
• If we have finite L?R then L LnR will be
  defined and we can use single pushout

Single pushout
Double pushout
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Graph Grammars - Types

• Context-Free Vertex Replacement graph grammar
  Engelfriet1997

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Graph Grammars - Types

• Context-Free Hypergraph Replacement graph grammar
  Drewes1997 (Nassi-Shneiderman diagram example)

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Graph Grammars - Types

• Unrestricted graph grammars.

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And now - the subject
Logics!!!!
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Structures

• Relational structures (without functions)
• Ds the domain and (Rs) are the relations on
  the domain p(R) is the arity of the relations
• 1) Structure for strings

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Structures Courcelle1997

• 2) Structures for graphs.
• The first structure G1
• this structure can not express a graph with many
  edges between two nodes
• The second structure G2

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Logics

• First Order Logic decidable on finite sets but
  not expressible enough
• Second Order Logic the most expressible, but
  undecidable in full its volume
• Limitations on the SOL is Monadic Second Order
  Logic!

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Monadic Second-Order Logic (MSO or MS)

• The subset of SOL but with quantifiers only on
  unary relations (sets). So it consists of
  formulas on a domain, fixed relations and
  quantified sets, includes FOL quantifiers also.
• The name monadic unary relations mono in
  contrast to relations of higher arity
  polyadic
• The example. The MSO formula
  expresses that a word in A has an odd
  length. The property does not depend on letter
  we do not use relations labx

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MSO

• There are different types of MSO logic
• MSO0 without FOL quantifiers but with relations
  on sets and Sing() singleton function. Has the
  same power (all expressions in MSO can be
  expressed in MSO0). The Sing() function
• The advantage it has simpler syntax.
• All the second order quantifiers can be shifted
  in front of first order quantifiers. For example
  is equivalent to
• Now if such formula has only existentional
  quantifiers in front of first order quantifiers
  it is Existentional Monadic Second-Order formula
  or EMSO formula.
• And at lest weak MSO is MSO on finite sets.

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MSO

• An expressivity of the logic (the question which
  properties we can express with it) depends on a
  logic of course and on structures we apply the
  logic and even on relation properties of the
  structure (transitivity, reflexivity and
  symmetry/asymetry/antysimmetry).
• Which properties on graphs we can express with
  MSO?

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MSO graph expressivity

• MSO with G1 structure can express

etc.
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MSO graph expressivity

• But MSO with G2 can express even more
• But!!! Another important question of a logic is
  decidability. A theory on some structure is
  decidable if it is recursive.

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MSO recognizability and decidability

• There are two main theorems about decidability of
  MSO logic Thomas1996
• Theorem 1 (Buchi, Elgot) A language of finite
  words is recognizable by a finite automation iff
  it is MSOS (another name S1S)- definable, and
  both conversions, from automata to formulas and
  vice versa, are effective.
• MSOS or S1S is a MSO logic on structure with
  one successor (binary relation). The consequence
  of the theorem is equivalence of regular
  languages (they are definable on S1S) and finita
  automata.

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MSO recognizability and decidability

• Theorem 2 (Rabin Tree Theorem) The theory S2S
  (MSO with 2 successor structure) is decidable.
• This theorem is celebrated because many modal
  logics like temporal or mu-calculus were proved
  to be decidable because they can be expressed in
  MSO on structures with 2-successors.
• One of the structures is binary tree with two
  childs 0successor and 1successor.

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Transductions Courcelle1997

• A binary relation where A and B
  are sets (typically of words or trees) can be
  considered as multivalued partial mapping
  associating with certain elements of A one or
  more elements of B. This is transduction A-gtB
  (rational transduction in language theory).
• There is no convenient notion of graph
  transduction. But with interpretation we can
  define transformations on structures.
• The formulas defining a structure T inside
  another one S (or k copies of S) will be MS
  formula.

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Transductions Courcelle1997

• R and l - two finit ranked sets of relation
  symbols, W set of variables (parameters). A
  (R,l) definition scheme is a tulpe of formulas

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Transductions Courcelle1997

• The example
• Transduction that maps a word u in a,b to the
  word u3.
• The 3-coping definition scheme without
  parameters is
• 
  

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F-magma Courcelle1997

• F-magma or f-algebra generalization of
  context-free grammars for graphs.
• Let S be a set of sorts, S-signature is a set F
  with two mappings
• A F-magma is
  where Ms is a nonempty set called the domain
  of sort s of M and for each f belonging to F the
  object fM is a total mapping
• - operations on M
  

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F-magma

• A term ts of some sort s is a combination of
  functions f.
• A polynomial system over F is a sequence of
  equations S ltu1p1,,unpngt where u1un are
  S-sorted variables or unknowns of S. Each term p
  is polynomial i.e. term of the form null or
  t1st2sstm. An operation s is like a union.

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F-magma an example

• The example is the context-free grammar
• The system of equations for the grammar will be

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F-magma, MSO and transductions

• If a transformation system can be described with
  a sequence of equations of a f-magma and every
  terminal can be described in MSO logic and there
  is a transduction from the terminal to the final
  sort can be formulated as MSO transduction then
  the transformation system is recognizable.
• Vertex Replacement and Hyperedge Replacement
  transformation systems can be described in S2S
  the way proposed above. So they are recognizable
  and we can make model checking on them (the profe
  is in Courcelle1997)

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Unrestricted Grammars and Turing Machines

• The initial task for Tropos
• We have an unrestricted grammar G and some
  requirements to the language of the grammar R. Is
  this a decidable task in common case?
• Lets say we have a Turing machine for G (a
  logical description or anything else) and we have
  a Turing machine CheckGR(G,R) which should check
  if the language of G if it is appropriate to the
  requirements R as a string
• According to the Turing machine halt problem this
  task is undecidable for random G and R.

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Conclusions

• It seems like there is logic description only for
  VR and HR grammars. They are the easiest type of
  graph grammars
• But the task of checking a grammar G of arbitrary
  kind is undecidable in common
• I should look for some restrictions on the graph
  grammar and the metamodel requirements to make
  the task real.

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References

• Bresciani2003 Paolo Bresciani, Paolo Georgini,
  Fausto Giunchiglia, John Mylopolous, Anna Perini,
  Trpos An Agent-Oriented Software Development
  Methodology, Jan 16 2003
• Bardohl1999 R. Bardohl and others, Application
  of Graph Transformation to Visual Languages.
  Handbook of Graph Grammars and Computing by Graph
  Transformation, Volume 2. 1999.
• Matuszek1996 David Matuszek, Syllabus for CSC
  4170-50 Theory of Computation, 1996,
  http//www.netaxs.com/people/nerp/automata/syllabu
  s.html
• Corradini1997 A. Corradini and others,
  Algebraic Approach to Graph Transformation.
  Handbook of Graph Grammars and Computing by Graph
  Transformation, Volume 1. 1997.
• Engelfriet1997 J. Engelfriet and others, Node
  Replacement Graph Grammars. Handbook of Graph
  Grammars and Computing by Graph Transformation,
  Volume 1. 1997.

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References

• Drewes1997 F. Drewes and others, Hyperedge
  Replacement Graph Grammars. Handbook of Graph
  Grammars and Computing by Graph Transformation,
  Volume 1. 1997.
• Courcelle1997 B. Courcelle, The Expression of
  Graph Properties and Graph Transformations in
  Monadic Second-Order Logic. Handbook of Graph
  Grammars and Computing by Graph Transformation,
  Volume 1. 1997.
• Thomas1996 Wolfgang Thomas, Languages,
  Automata, and Logic. 1996.