Mathematical Modeling and Logical Framework in Data Computation
Mathematical Modeling
Carsten Schürmann
DemTech
April 8, 2015
IEEE-SA VSSC-1622.6
Objective
Implementation
Administrative Processes
Software
Legislation
Law
Logical Framework
Data
Computation
Logic
Properties
Proofs
1. Discourse
•
Voting Protocol
–
Cryptography
(Privacy, Security)
–
Statistics
(Audits)
–
Logic
(Distribution, Implementation)
•
Social Choice Algorithms
–
Complexity
–
Impossibility
–
Utility
2. Style
•
Algorithmic
–
Algorithm + control flow
–
Justify the adherence with “specification”
•
Declarative
–
Describe the algorithm, no control flow
–
Justify the adherence with “specification”
•
Abstract
–
Describe the “specification”
3. Audience
•
The Public
–
Assumptions of common knowledge
–
Levels of abstraction
•
Experts
–
Completeness
(no choice left to imagination)
–
Soundness (must make sense)
•
Courts
–
Verifiability
–
Dispute resolution
Logical Framework ToolBox
Logical Framework
Data
: Votes, Ballots, Totals
Computation
: Algorithms
Pictoral Presentation
“A picture says more than
100 words”
•
Examples: Category
Theory, Geometry,
Trigonometry
Declarative (Algebraic) Presentation
•
Constructive Logics:
“If there is a solution then
there is a program that
actually computes it!”
Let
P=(x
P
,y
P
)
and
Q =(x
Q
,y
Q
)
Define
R = P + Q
where
R = (x
R
,y
R
)
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
Dataflow Presentation
s = (y
P
– y
Q
)/(x
P
–
x
Q
)
x
R
= s
2
– x
P
– x
Q
y
R
= – y
P
+ s(x
P
– x
Q
)
R = (x
R,
y
R
)
P = Q
-P = Q
R = (2x
R,
2y
R
)
R = ∞
P=(x
P
,y
P
)
Q =(x
Q
,y
Q
)
Formal Presentation
•
Logics
–
Intuitionistic Logic
–
Linear Logic
–
Temporal Logic
•
Dynamic system
•
Model Checker
–
Type theories
•
Active area of research
•
Coq, Agda, Twelf
•
Tool support essential
“A formalization of Elliptic Curves” in Coq
Logic ToolBox
Logic
Properties
: Requirements, Specifications
Proofs
: Arguments, Equivalences
Logics
For all points P + Q it
holds:
P + Q = Q + P
And in the Coq system:
–
First-order logic
–
KeY system
–
Higher-order logics
–
Temporal Logic
–
Model Checker
–
Type theories
–
Active area of research
–
Coq, Agda, Twelf
Case Study
First-Past the Post
Logical Framework: Linear Logic
“The Logic of Food” * -o
Logic: First-order Logic
Objective
Implementation
Administrative Processes
Software
Legislation
Tally all ballots for each hopeful candidate.
Logical Framework
Data
Computation
Declarative Statement
If you
count ballots
find an
uncounted ballot
for C
where C
is still in the race
then 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
N
Objective
Implementation
Administrative Processes
Software
Legislation
Tally all ballots for each hopeful candidate.
Logical Framework
Data
Computation
Logic
Properties
Proofs
Abstract Specification:
Each Vote is Counted Once
Conclusion
•
Narrowing the gap between law and maths
–
Laws easier to interpret, implement
–
Completeness
–
Soundness
–
Analysis becomes easier
•
Long term benefits
This content explores mathematical modeling, legislative frameworks, logical properties, voting protocols, audience assumptions, and formal presentations related to data computation. It delves into algorithmic styles, cryptographic protocols, complexity, and more.
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Presentation Transcript
Mathematical Modeling Carsten Sch rmann DemTech April 8, 2015 IEEE-SA VSSC-1622.6
Objective Legislation Law Logical Framework Data Computation Logic Properties Proofs Implementation Administrative Processes Software
1. Discourse Voting Protocol Cryptography (Privacy, Security) Statistics (Audits) Logic (Distribution, Implementation) Social Choice Algorithms Complexity Impossibility Utility
2. Style Algorithmic Algorithm + control flow Justify the adherence with specification Declarative Describe the algorithm, no control flow Justify the adherence with specification Abstract Describe the specification
3. Audience The Public Assumptions of common knowledge Levels of abstraction Experts Completeness Soundness (must make sense) Courts Verifiability Dispute resolution (no choice left to imagination)
Logical Framework ToolBox Logical Framework Data: Votes, Ballots, Totals Computation: Algorithms
Pictoral Presentation A picture says more than 100 words Examples: Category Theory, Geometry, Trigonometry
Declarative (Algebraic) Presentation Let P=(xP,yP) and Q =(xQ,yQ) Define R = P + Q where R = (xR,yR) Case 1: P Q and P Q xR = s2 xP xQ yR = yP+ s(xP xQ) s = (yP yQ)/(xP xQ) Case 2: -P = Q P + Q = Case 3: P = Q P+ Q = 2P Constructive Logics: If there is a solution then there is a program that actually computes it!
Dataflow Presentation P=(xP,yP) Q =(xQ,yQ) R = (2xR, 2yR) P = Q R = -P = Q s = (yP yQ)/(xP xQ) xR = s2 xP xQ yR = yP+ s(xP xQ) R = (xR, yR)
Formal Presentation Logics Intuitionistic Logic Linear Logic Temporal Logic Dynamic system Model Checker Type theories Active area of research Coq, Agda, Twelf Tool support essential A formalization of Elliptic Curves in Coq
Logic ToolBox Logic Properties: Requirements, Specifications Proofs: Arguments, Equivalences
Logics First-order logic KeY system Higher-order logics Temporal Logic Model Checker Type theories Active area of research Coq, Agda, Twelf For all points P + Q it holds: P + Q = Q + P And in the Coq system:
Case Study First-Past the Post Logical Framework: Linear Logic The Logic of Food * -o Logic: First-order Logic
Objective Legislation Tally all ballots for each hopeful candidate. Logical Framework Data Computation Implementation Administrative Processes Software
Declarative Statement count-ballots (U + 1) H * uncounted-ballot C * hopeful C N -o (hopeful C (N + 1) * count-ballots U H) If you count ballots find an uncounted ballot for C where C is still in the race then increase C s tally and continue counting Number uncounted ballots: U Number hopefuls: H Name of a candidate: C Number of ballots in a tally: N Propositions: count-ballots UH uncounted-ballot C hopeful C N
Objective Legislation Tally all ballots for each hopeful candidate. Logical Framework Data Computation Logic Properties Proofs Implementation Administrative Processes Software
Abstract Specification: Each Vote is Counted Once
Conclusion Narrowing the gap between law and maths Laws easier to interpret, implement Completeness Soundness Analysis becomes easier Long term benefits
