Carsten Schürmann DemTech April 8 2015 IEEESA VSSC16226 Objective Implementation Administrative Processes Software Legislation Law Logical Framework Data Computation Logic Properties ID: 483345
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Slide1
Mathematical Modeling
Carsten Schürmann
DemTech
April 8, 2015
IEEE-SA VSSC-1622.6Slide2
Objective
Implementation
Administrative Processes
Software
Legislation
Law
Logical Framework
Data
Computation
Logic
Properties
ProofsSlide3
1. Discourse
Voting Protocol
Cryptography
(Privacy, Security)
Statistics (Audits)Logic (Distribution, Implementation)Social Choice Algorithms Complexity
ImpossibilityUtilitySlide4
2.
Style
Algorithmic
A
lgorithm + control flowJustify the adherence with “specification”DeclarativeDescribe the algorithm, no control flowJustify the adherence with “specification”
AbstractDescribe the “specification”Slide5
3.
Audience
The Public
Assumptions of common knowledge
Levels of abstractionExpertsCompleteness (no choice left to imagination)Soundness (must make sense)Courts
VerifiabilityDispute resolutionSlide6
Logical Framework ToolBox
Logical Framework
Data
: Votes, Ballots, Totals
Computation
: AlgorithmsSlide7
Pictoral Presentation
“A picture says more than 100 words”
Examples: Category Theory, Geometry, TrigonometrySlide8
Declarative (Algebraic) Presentation
Constructive Logics:
“If there is a solution then there is a program that actually computes it!”
Let
P=(
x
P,yP
) and Q =(xQ,y
Q)Define R = P + Q
where R = (xR,yR)
Case 1:
P
≠
Q
and
–P
≠
Q
x
R
= s
2
–
x
P
–
x
Q
y
R
= –
y
P
+ s(
x
P
–
x
Q
)
s = (
y
P
–
y
Q
)/(
x
P
–
x
Q
)
Case 2:
-
P = Q
P + Q = ∞
Case 3:
P
=
Q
P+ Q = 2P Slide9
Dataflow Presentation
s = (
y
P
– y
Q)/(xP – x
Q)xR
= s2 – x
P – xQ y
R
= –
y
P
+ s(
x
P
–
x
Q
)
R = (
x
R
,
yR)
P = Q
-P = Q
R = (2xR, 2yR)
R
= ∞
P=(
x
P
,y
P)
Q =(
x
Q
,y
Q
)Slide10
Formal Presentation
Logics
Intuitionistic Logic
Linear Logic
Temporal Logic
Dynamic system Model CheckerType theories
Active area of researchCoq, Agda, TwelfTool support essential“A formalization of Elliptic Curves” in CoqSlide11
Logic ToolBox
Logic
Properties
: Requirements, Specifications
Proofs
: Arguments, EquivalencesSlide12
Logics
For all points P + Q it holds:
P + Q = Q + P
And in the Coq system:
First-order logic
KeY
system
Higher-order logicsTemporal LogicModel CheckerType theoriesActive area of researchCoq, Agda
, TwelfSlide13
Case Study
First-Past the Post
Logical Framework: Linear Logic
“The Logic of Food” * -o
Logic: First-order LogicSlide14
Objective
Implementation
Administrative Processes
Software
Legislation
Tally all ballots for each hopeful candidate.
Logical Framework
Data
ComputationSlide15
Declarative Statement
If you
count ballots
find an
uncounted ballot
for Cwhere C
is still in the racethen increase C’s tally and continue counting
count-ballots (U + 1) H
*uncounted-ballot C
*hopeful C N -o (hopeful
C
(
N + 1
)
*
count-ballots
U
H
)
Number uncounted ballots:
U
Number hopefuls:
H
Name of a candidate: C
Number of ballots in a tally:
N
Propositions:
count
-ballots
U
H uncounted-ballot
C hopeful C
NSlide16
Objective
Implementation
Administrative Processes
Software
Legislation
Tally all ballots for each hopeful candidate.
Logical Framework
Data
Computation
LogicPropertiesProofsSlide17
Abstract Specification:
Each Vote is Counted OnceSlide18
Conclusion
Narrowing the gap between law and
maths
Laws easier to interpret, implement
CompletenessSoundnessAnalysis becomes easierLong term benefits