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Reasoning about connectivity without paths Alberto Casagrande and Eugenio G. Omodeo Dip. Matematica e Geoscienze DMI Eugenio G. Omodeo Reasoning about Connectivity without Paths 1/24 Reasoning about connectivity without paths 1 Alberto

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Reasoning about connectivity without paths

Alberto Casagrande and Eugenio G. Omodeo

  • Dip. Matematica e Geoscienze — DMI

Eugenio G. Omodeo Reasoning about Connectivity without Paths 1/24

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Reasoning about connectivity without paths1

Alberto Casagrande and Eugenio G. Omodeo

  • Dip. Matematica e Geoscienze — DMI

1Work partially funded by: INdAM/GNCS 2013, FRA-UniTS 2012 PUMA Eugenio G. Omodeo Reasoning about Connectivity without Paths 1/24

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Every connected graph has non-cut vertices

Example

Eugenio G. Omodeo Reasoning about Connectivity without Paths 2/24

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Every connected graph has non-cut vertices

Example

Eugenio G. Omodeo Reasoning about Connectivity without Paths 2/24

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SLIDE 5

Every connected graph has non-cut vertices

Example

Eugenio G. Omodeo Reasoning about Connectivity without Paths 2/24

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SLIDE 6

Every connected graph has non-cut vertices

Example

Eugenio G. Omodeo Reasoning about Connectivity without Paths 2/24

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SLIDE 7

Every connected graph has non-cut vertices

Example ·

Eugenio G. Omodeo Reasoning about Connectivity without Paths 2/24

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Every connected graph has non-cut vertices

Example ·

Eugenio G. Omodeo Reasoning about Connectivity without Paths 2/24

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Every connected graph has non-cut vertices

Example More generally: ‘Every connected (finite) hyper graph has at least one vertex whose removal does not disrupt connectivity’

Eugenio G. Omodeo Reasoning about Connectivity without Paths 2/24

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SLIDE 10

(Ideal) itinerary of this talk i Connectivity and non-cut vertices ii Applications ( one in particular. . . ) iii The proof assistant Ref iv Our proof-verification experiment

http://www2.units.it/eomodeo/NonCutVertices.html http://aetnanova.units.it/scenarios/NonCutVertices/

Eugenio G. Omodeo Reasoning about Connectivity without Paths 3/24

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Connectivity

Connectivity plays a crucial role in many fields.

  • Es. The number of connected components of a graph

is a topological invariant; corresponds to the multiplicity of the eigenvalue 0 in its Laplacian; is related to the number of its claw-free subgraphs [CPR07].

∴ Large scale proof-verification efforts [Wie07, SCO11] must

formally investigate this notion.

Eugenio G. Omodeo Reasoning about Connectivity without Paths 4/24

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How we see a hypergraph

Example a c b d e f g h

Eugenio G. Omodeo Reasoning about Connectivity without Paths 5/24

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How we see a hypergraph

Example a c b d e f g h The edges of G belong to G .

Eugenio G. Omodeo Reasoning about Connectivity without Paths 5/24

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How we see a hypergraph

Example a c b d e f g h The    vertices

  • r

nodes    of G belong to {v : e ∈ G, v ∈ e} .

Eugenio G. Omodeo Reasoning about Connectivity without Paths 5/24

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How we see a hypergraph

Example a c b d e f g h In a graph, the edges have cardinality 2 .

Eugenio G. Omodeo Reasoning about Connectivity without Paths 5/24

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Our formal definition of connectivity

. . .

  • Def. Conn(G)

↔Def {p ⊆ G | nodes(p) ∩ nodes(G\p) = ∅} ⊆ {∅, G} & HGraph(G)

Eugenio G. Omodeo Reasoning about Connectivity without Paths 6/24

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SLIDE 17

Our formal definition of connectivity

.

  • Def. nodes(G)

=Def G .

  • Def. Conn(G)

↔Def {p ⊆ G | nodes(p) ∩ nodes(G\p) = ∅} = {∅, G} & HGraph(G)

Eugenio G. Omodeo Reasoning about Connectivity without Paths 6/24

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SLIDE 18

Our formal definition of connectivity

.

  • Def. nodes(G)

=Def G

  • Def. HGraph(G)

↔Def

∀e ∈ G | CardAtLeast2(e) &

Finite

  • nodes(G)
  • Def. Conn(G)

↔Def {p ⊆ G | nodes(p) ∩ nodes(G\p) = ∅} = {∅, G} & HGraph(G)

Eugenio G. Omodeo Reasoning about Connectivity without Paths 6/24

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Our formal definition of connectivity

  • Def. CardAtLeast2(E) ↔Def

E ⊆ {arb(E)}

  • Def. nodes(G)

=Def G

  • Def. HGraph(G)

↔Def

∀e ∈ G | CardAtLeast2(e) &

Finite

  • nodes(G)
  • Def. Conn(G)

↔Def {p ⊆ G | nodes(p) ∩ nodes(G\p) = ∅} = {∅, G} & HGraph(G)

Eugenio G. Omodeo Reasoning about Connectivity without Paths 6/24

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Does this hypergraph have a spanning tree ?

a b c d e f

Eugenio G. Omodeo Reasoning about Connectivity without Paths 7/24

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Existence of non-cut vertices

. . .

  • Thm. Conn(G) & G ⊆ {arb(G)} → ∃v ∈ nodes(G) | NonCut(G, v)

Eugenio G. Omodeo Reasoning about Connectivity without Paths 8/24

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Existence of non-cut vertices

. .

  • Def. NonCut(G, V)

↔Def Conn

  • rmv(G, V)
  • & lost(G, V) = ∅
  • Thm. Conn(G) & G ⊆ {arb(G)} → ∃v ∈ nodes(G) | NonCut(G, v)

Eugenio G. Omodeo Reasoning about Connectivity without Paths 8/24

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Existence of non-cut vertices

  • Def. rmv(G, V)

=Def {e\{V} : e ∈ G | CardAtLeast2(e\{V})} .

  • Def. NonCut(G, V)

↔Def Conn

  • rmv(G, V)
  • & lost(G, V) = ∅
  • Thm. Conn(G) & G ⊆ {arb(G)} → ∃v ∈ nodes(G) | NonCut(G, v)

Eugenio G. Omodeo Reasoning about Connectivity without Paths 8/24

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Existence of non-cut vertices

  • Def. rmv(G, V)

=Def {e\{V} : e ∈ G | CardAtLeast2(e\{V})}

  • Def. lost(G, V)

=Def nodes(G)\

  • nodes
  • rmv(G, V)
  • ∪ {V}
  • Def. NonCut(G, V)

↔Def Conn

  • rmv(G, V)
  • & lost(G, V) = ∅
  • Thm. Conn(G) & G ⊆ {arb(G)} → ∃v ∈ nodes(G) | NonCut(G, v)

Eugenio G. Omodeo Reasoning about Connectivity without Paths 8/24

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Ur-application of non-cut vertices: Walking

How can we: walk along an infinite acyclic path? visit all vertices of a finite acyclic path?

Eugenio G. Omodeo Reasoning about Connectivity without Paths 9/24

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Ur-application of non-cut vertices: Walking

Example 1 2 3 4 · · ·

Eugenio G. Omodeo Reasoning about Connectivity without Paths 9/24

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Ur-application of non-cut vertices: Walking

Example 1 2 3 4 · · ·

Eugenio G. Omodeo Reasoning about Connectivity without Paths 9/24

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Ur-application of non-cut vertices: Walking

Example 1 2 3 4 · · ·

Eugenio G. Omodeo Reasoning about Connectivity without Paths 9/24

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Ur-application of non-cut vertices: Walking

Example 2 3 4 · · ·

Eugenio G. Omodeo Reasoning about Connectivity without Paths 9/24

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Ur-application of non-cut vertices: Walking

Example 3 4 · · ·

Eugenio G. Omodeo Reasoning about Connectivity without Paths 9/24

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Ur-application of non-cut vertices: Walking

Example 4 · · ·

Eugenio G. Omodeo Reasoning about Connectivity without Paths 9/24

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Ur-application of non-cut vertices: Walking

Example 4 · · · This amounts to repeatedly picking and removing a non-cut vertex (the only one, in this case) from a graph (infinite in this case)

Eugenio G. Omodeo Reasoning about Connectivity without Paths 9/24

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Ur-application of non-cut vertices: Walking

Example 4 · · · This amounts to repeatedly picking and removing a non-cut vertex (the only one, in this case) from a graph (infinite in this case) Why such a silly example ?

Eugenio G. Omodeo Reasoning about Connectivity without Paths 9/24

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Ur-application of non-cut vertices: Walking

Example 1 2 3 4 · · ·

N N∪{N}

Where does the difference between N and N ∪ {N, N ∪ {N}} lie ?

Eugenio G. Omodeo Reasoning about Connectivity without Paths 9/24

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A more revealing application

Example We can get a spanning tree for a connected nonnull graph by:

Eugenio G. Omodeo Reasoning about Connectivity without Paths 10/24

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A more revealing application

Example We can get a spanning tree for a connected nonnull graph by:

1 Picking & removing a non-cut vertex from a connected graph Eugenio G. Omodeo Reasoning about Connectivity without Paths 10/24

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A more revealing application

Example We can get a spanning tree for a connected nonnull graph by:

1 Picking & removing a non-cut vertex from a connected graph 2

recursively getting a spanning tree for the resulting graph

Eugenio G. Omodeo Reasoning about Connectivity without Paths 10/24

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A more revealing application

Example We can get a spanning tree for a connected nonnull graph by:

1 Picking & removing a non-cut vertex from a connected graph 2

recursively getting a spanning tree for the resulting graph

3 restoring the removed vertex, along with one of the edges

incident to it

Eugenio G. Omodeo Reasoning about Connectivity without Paths 10/24

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SLIDE 39

A more revealing application

Example We can get a spanning tree for a connected nonnull graph by:

1 Picking & removing a non-cut vertex from a connected graph 2

recursively getting a spanning tree for the resulting graph

3 restoring the removed vertex, along with one of the edges

incident to it

0 In the base case , the spanning tree consists of the (sole) edge Eugenio G. Omodeo Reasoning about Connectivity without Paths 10/24

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SLIDE 40

A more revealing application

Example We can get a spanning tree for a connected nonnull graph by:

1 Picking & removing a non-cut vertex from a connected graph 2

recursively getting a spanning tree for the resulting graph

3 restoring the removed vertex, along with one of the edges

incident to it

0 In the base case , the spanning tree consists of the (sole) edge

This example is paradigmatic: Inductive proofs on connected graphs usually follow this pattern

Eugenio G. Omodeo Reasoning about Connectivity without Paths 10/24

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Direct motivating application

An achievement, but also a pending proof obligation:

Eugenio G. Omodeo Reasoning about Connectivity without Paths 11/24

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Direct motivating application

An achievement, but also a pending proof obligation:

Eugenio G. Omodeo Reasoning about Connectivity without Paths 11/24

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SLIDE 43

Two earlier proof-pearl scenarios

Two fully formal reconstructions of results on connected claw-free graphs have been achieved by means of Ref.

  • E. G. Omodeo and A. I. Tomescu.

Set graphs. V. On representing graphs as membership digraphs. To appear on J. Log. Comput.

  • Cf. http://www2.units.it/eomodeo/GraphsViaMembership.html
  • E. G. Omodeo and A. I. Tomescu.

Set graphs. III. Proof Pearl: Claw-free graphs mirrored into transitive hereditarily finite sets.

  • J. Autom. Reason., 52(1), pp.1–29, 2014.
  • Cf. http://www2.units.it/eomodeo/ClawFreeness.html
  • E. G. Omodeo and A. I. Tomescu.

Appendix: Claw-free graphs as sets. In: M. Davis, E. Schonberg (eds.) From Linear Operators to Computational Biology: Essays in Memory of Jacob T. Schwartz, pp. 131–167, Springer, 2012.

Eugenio G. Omodeo Reasoning about Connectivity without Paths 12/24

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SLIDE 44

Two earlier proof-pearl scenarios

Two fully formal reconstructions of results on connected claw-free graphs have been achieved by means of Ref.

  • E. G. Omodeo and A. I. Tomescu.

Set graphs. V. On representing graphs as membership digraphs. To appear on J. Log. Comput.

  • Cf. http://www2.units.it/eomodeo/GraphsViaMembership.html
  • E. G. Omodeo and A. I. Tomescu.

Set graphs. III. Proof Pearl: Claw-free graphs mirrored into transitive hereditarily finite sets.

  • J. Autom. Reason., 52(1), pp.1–29, 2014.
  • Cf. http://www2.units.it/eomodeo/ClawFreeness.html
  • E. G. Omodeo and A. I. Tomescu.

Appendix: Claw-free graphs as sets. In: M. Davis, E. Schonberg (eds.) From Linear Operators to Computational Biology: Essays in Memory of Jacob T. Schwartz, pp. 131–167, Springer, 2012.

Eugenio G. Omodeo Reasoning about Connectivity without Paths 12/24

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SLIDE 45

Two earlier proof-pearl scenarios

Two fully formal reconstructions of results on connected claw-free graphs have been achieved by means of Ref.

  • E. G. Omodeo and A. I. Tomescu.

Set graphs. V. On representing graphs as membership digraphs. To appear on J. Log. Comput.

  • Cf. http://www2.units.it/eomodeo/GraphsViaMembership.html
  • E. G. Omodeo and A. I. Tomescu.

Set graphs. III. Proof Pearl: Claw-free graphs mirrored into transitive hereditarily finite sets.

  • J. Autom. Reason., 52(1), pp.1–29, 2014.
  • Cf. http://www2.units.it/eomodeo/ClawFreeness.html
  • E. G. Omodeo and A. I. Tomescu.

Appendix: Claw-free graphs as sets. In: M. Davis, E. Schonberg (eds.) From Linear Operators to Computational Biology: Essays in Memory of Jacob T. Schwartz, pp. 131–167, Springer, 2012.

Eugenio G. Omodeo Reasoning about Connectivity without Paths 12/24

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A nice class of graphs: claw-free & connected

Definition A graph (V , E) is said to be claw-free if none of its subgraphs induced by 4 vertices has the shape of a ‘Y’

Eugenio G. Omodeo Reasoning about Connectivity without Paths 13/24

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A nice class of graphs: claw-free & connected

Definition A graph (V , E) is said to be claw-free if none of its subgraphs induced by 4 vertices has the shape of a ‘Y’

  • Figure: Forbidden claw K1,3

Eugenio G. Omodeo Reasoning about Connectivity without Paths 13/24

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A nice class of graphs: claw-free & connected

Definition A graph (V , E) is said to be claw-free if none of its subgraphs induced by 4 vertices has the shape of a ‘Y’

Figure: Worse than a claw

Eugenio G. Omodeo Reasoning about Connectivity without Paths 13/24

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A nice class of graphs: claw-free & connected

Definition A graph (V , E) is said to be claw-free if none of its subgraphs induced by 4 vertices has the shape of a ‘Y’

  • Figure: A claw-free graph

Eugenio G. Omodeo Reasoning about Connectivity without Paths 13/24

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Classical results about claw-free graphs

“Every connected claw-free graph admits a perfect matching and has a Hamiltonian cycle in its square”.

(1970s / 1980s)

Also: Each connected claw-free graph has a vertex-pancyclic square

Eugenio G. Omodeo Reasoning about Connectivity without Paths 14/24

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SLIDE 51

Classical results about claw-free graphs

“Every connected claw-free graph admits a perfect matching and has a Hamiltonian cycle in its square”.

(1970s / 1980s)

Also: Each connected claw-free graph has a vertex-pancyclic square

Eugenio G. Omodeo Reasoning about Connectivity without Paths 14/24

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SLIDE 52

Novel proofs based on a repr’n theorem

Martin Milanič and A. I. Tomescu found new, simpler proofs of the results just mentioned via a theorem about the representation of edges as (directed!) membership arcs.

  • M. Milanič and A. I. Tomescu.

Set graphs. I. Hereditarily finite sets and extensional acyclic orientations. Discrete Applied Mathematics, 161(4-5):677–690, 2013.

  • A. I. Tomescu.

A simpler proof for vertex-pancyclicity of squares of connected claw-free graphs. Discrete Mathematics, 312(15):2388–2391, 2012.

Eugenio G. Omodeo Reasoning about Connectivity without Paths 15/24

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SLIDE 53

Novel proofs based on a repr’n theorem

Martin Milanič and A. I. Tomescu found new, simpler proofs of the results just mentioned via a theorem about the representation of edges as (directed!) membership arcs.

  • M. Milanič and A. I. Tomescu.

Set graphs. I. Hereditarily finite sets and extensional acyclic orientations. Discrete Applied Mathematics, 161(4-5):677–690, 2013.

  • A. I. Tomescu.

A simpler proof for vertex-pancyclicity of squares of connected claw-free graphs. Discrete Mathematics, 312(15):2388–2391, 2012.

Eugenio G. Omodeo Reasoning about Connectivity without Paths 15/24

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SLIDE 54

Novel proofs based on a repr’n theorem

Martin Milanič and A. I. Tomescu found new, simpler proofs of the results just mentioned via a theorem about the representation of edges as (directed!) membership arcs.

  • M. Milanič and A. I. Tomescu.

Set graphs. I. Hereditarily finite sets and extensional acyclic orientations. Discrete Applied Mathematics, 161(4-5):677–690, 2013.

  • A. I. Tomescu.

A simpler proof for vertex-pancyclicity of squares of connected claw-free graphs. Discrete Mathematics, 312(15):2388–2391, 2012.

Eugenio G. Omodeo Reasoning about Connectivity without Paths 15/24

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ÆtnaNova aka Ref eree:

  • Cf. [SCO11]

( On-line worksheet )

Eugenio G. Omodeo Reasoning about Connectivity without Paths 16/24

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SLIDE 56

Interaction with our proof-verifier ( Input )

Eugenio G. Omodeo Reasoning about Connectivity without Paths 17/24

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Interaction with our proof-verifier( Output )

Eugenio G. Omodeo Reasoning about Connectivity without Paths 18/24

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SLIDE 58

Interaction with our proof-verifier( Output )

Eugenio G. Omodeo Reasoning about Connectivity without Paths 18/24

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3 basic constituents of a scenario ( examples )

Definition: ( shorthand )

  • - After the celebrated paper Sur les ensembles fini ( Tarski, 1924 )
  • Def. Finite(F)

↔Def

∀g ∈ P(P(F))\{∅}, ∃m | g ∩ P(m) = {m}

Eugenio G. Omodeo Reasoning about Connectivity without Paths 19/24

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SLIDE 60

3 basic constituents of a scenario ( examples )

Definition: (∈-recursion here! )

  • - “The cardinality of S exceeds M”
  • Def. Exc(S, M)

↔Def S=∅ &

  • p ∈ M | ¬Exc
  • S\{arb(S)}, p
  • = ∅

Eugenio G. Omodeo Reasoning about Connectivity without Paths 19/24

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3 basic constituents of a scenario ( examples )

Theorem and Proof: ( Monotonicity of finitude ) Thm fin0. Y ⊇ X & Finite(Y) → Finite(X). Proof:

Suppose_not(y0, x0) = ⇒

y0 ⊇ x0 & Finite(y0) & ¬Finite(x0)

y0, x0֒

→Tpow1 = ⇒

Py0 ⊇ Px0

Use_def(Finite) = ⇒

Stat1 : ¬∀g ∈ P(Px0)\{∅}, ∃m | g ∩ Pm = {m} & ∀g′ ∈ P(Py0)\{∅}, ∃m | g′ ∩ Pm = {m}

Py0, Px0֒

→Tpow1 = ⇒

P(Py0) ⊇ P(Px0)

g0, g0֒

→Stat1(Stat1⋆) = ⇒

¬∃m | g0 ∩ Pm = {m} &

∃m | g0 ∩ Pm = {m}

Discharge = ⇒

Qed

Eugenio G. Omodeo Reasoning about Connectivity without Paths 19/24

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4th, major constituent of a scenario (example)

A construct for proof reuse Theory finite_image (s0 , g(X)) Finite(s0) End finite_image Enter_theory finite_image . . . . . . . . . . . . Enter_theory Set_theory Within a scenario, the discourse can momentarily digress into a ‘Theory’ that enforces certain local assumptions. At the end of the digression, the upper theory will be re-entered.

Eugenio G. Omodeo Reasoning about Connectivity without Paths 20/24

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SLIDE 63

4th, major constituent of a scenario (example)

A construct for proof reuse Theory finite_image (s0 , g(X)) Finite(s0) End finite_image Enter_theory finite_image . . . . . . . . . . . . Enter_theory Set_theory Within a scenario, the discourse can momentarily digress into a ‘Theory’ that enforces certain local assumptions. At the end of the digression, the upper theory will be re-entered.

Eugenio G. Omodeo Reasoning about Connectivity without Paths 20/24

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SLIDE 64

4th, major constituent of a scenario (example)

A construct for proof reuse Theory finite_image (s0 , g(X)) Finite(s0) = ⇒

  • fΘ
  • Finite
  • { g(x) : x ∈ s0 }
  • fΘ ⊆ s0

✫ ∀ t ⊆ fΘ | g(t) = g(s0) ↔ t = fΘ End finite_image As an outcome of the digression, the Theory will be able to instantiate new theorems

Eugenio G. Omodeo Reasoning about Connectivity without Paths 20/24

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SLIDE 65

4th, major constituent of a scenario (example)

A construct for proof reuse Theory finite_image (s0 , g(X)) Finite(s0) = ⇒

  • fΘ
  • Finite
  • { g(x) : x ∈ s0 }
  • fΘ ⊆ s0

✫ ∀ t ⊆ fΘ | g(t) = g(s0) ↔ t = fΘ End finite_image As an outcome of the digression, the Theory will be able to instantiate new theorems: possibly involving new symbols, whose definition it encapsulates.

Eugenio G. Omodeo Reasoning about Connectivity without Paths 20/24

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SLIDE 66

Our experiment, in digits

The script-file containing our verified formal derivation of the existence of non-cut vertices in hypergraphs: comprises 13 definitions; proves 46 theorems (only two whose length exceeds 50 lines),

  • rganized in 3 Theorys.

Its processing takes ca. 4 seconds; the overall number of proof lines is 905. http://www2.units.it/eomodeo/NonCutVertices.html http://aetnanova.units.it/scenarios/NonCutVertices/

Eugenio G. Omodeo Reasoning about Connectivity without Paths 21/24

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SLIDE 67

Conclusions and future work

Proof-verification can highly benefit from representation theorems

  • f the kind illustrated by the Milanič–Tomescu result about

connected, claw-free graphs. On the human side, such results disclose new insights by shedding light on a discipline from unusual angles

  • n the technological side, they enable the transfer of proof

methods from one realm of mathematics to another.

  • This contribution closes a cycle of activities related to claw-free
  • graphs. . .

. . . and paves the way to an extensive exploration on how to formalize hypergraphs .

Eugenio G. Omodeo Reasoning about Connectivity without Paths 22/24

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SLIDE 68

Conclusions and future work

Proof-verification can highly benefit from representation theorems

  • f the kind illustrated by the Milanič–Tomescu result about

connected, claw-free graphs. On the human side, such results disclose new insights by shedding light on a discipline from unusual angles

  • n the technological side, they enable the transfer of proof

methods from one realm of mathematics to another.

  • This contribution closes a cycle of activities related to claw-free
  • graphs. . .

. . . and paves the way to an extensive exploration on how to formalize hypergraphs .

Eugenio G. Omodeo Reasoning about Connectivity without Paths 22/24

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SLIDE 69

Conclusions and future work

Proof-verification can highly benefit from representation theorems

  • f the kind illustrated by the Milanič–Tomescu result about

connected, claw-free graphs. On the human side, such results disclose new insights by shedding light on a discipline from unusual angles

  • n the technological side, they enable the transfer of proof

methods from one realm of mathematics to another.

  • This contribution closes a cycle of activities related to claw-free
  • graphs. . .

. . . and paves the way to an extensive exploration on how to formalize hypergraphs .

Eugenio G. Omodeo Reasoning about Connectivity without Paths 22/24

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SLIDE 70

Conclusions and future work

Proof-verification can highly benefit from representation theorems

  • f the kind illustrated by the Milanič–Tomescu result about

connected, claw-free graphs. On the human side, such results disclose new insights by shedding light on a discipline from unusual angles

  • n the technological side, they enable the transfer of proof

methods from one realm of mathematics to another.

  • This contribution closes a cycle of activities related to claw-free
  • graphs. . .

. . . and paves the way to an extensive exploration on how to formalize hypergraphs .

Eugenio G. Omodeo Reasoning about Connectivity without Paths 22/24

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SLIDE 71

Conclusions and future work

Proof-verification can highly benefit from representation theorems

  • f the kind illustrated by the Milanič–Tomescu result about

connected, claw-free graphs. On the human side, such results disclose new insights by shedding light on a discipline from unusual angles

  • n the technological side, they enable the transfer of proof

methods from one realm of mathematics to another.

  • This contribution closes a cycle of activities related to claw-free
  • graphs. . .

. . . and paves the way to an extensive exploration on how to formalize hypergraphs .

Eugenio G. Omodeo Reasoning about Connectivity without Paths 22/24

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SLIDE 72

Thank you for your attention!

Eugenio G. Omodeo Reasoning about Connectivity without Paths 23/24

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SLIDE 73

Jacob T. Schwartz, Domenico Cantone, and Eugenio G.

  • Omodeo. Computational Logic and Set Theory –

Applying formalized Logic to Analysis.

Springer, 2011. Foreword by Martin Davis. Freek Wiedijk. The QED Manifesto revisited. Studies in Logic, Grammar and Rhetoric, 10(23):121–133, 2007. Gab-Byung Chae, Edgar M. Palmer, and Robert W. Robinson. Counting labeled general cubic graphs. Discrete Mathematics, 307(23):2979–2992, 2007.

Eugenio G. Omodeo Reasoning about Connectivity without Paths 24/24