FASE 2011 - Formalizing and Operationalizing Industrial Standards

Formalizing and Operationalizing Industrial Standards

Dominik Dietrich Lutz Schr¨der
o Ewaryst Schulz

DFKI Bremen, Germany
ewaryst.schulz@dfki.de

International Conference on
Fundamental Approaches to Software Engineering
Saarbr¨cken, Germany
u
March 30 2011

Formalizing Industrial Standards German Research Center
D. Dietrich, L. Schr¨der, E. Schulz
o for Artificial Intelligence

Overview

The problem
Assure proper functioning of safety critical systems
Required functional properties of mechanical sub-components must
be satisfied

Our solution
Use engineering calculation methods
Formal language for engineering calculations
Architecture allowing efficient execution and formal verification


Outline

1 Motivation

2 The Industrial Standard EN 1591

3 The engineering calculation language EnCL

4 Formal verification of calculations

5 Integration into Hets framework

6 Conclusion


Reliable Mechanical Engineering

Principal motivation
Assure proper functioning of system, e.g., pipeline of chemical plant
Verification of functional properties of sub-components, e.g.,
flange connection withstands some given pressure

Some possible approaches
Formulating properties from first principles (mechanics, geometry)
→ level of abstraction not adequate
Instead: Relying on established practice in engineering
→ industrial standards, engineering calculations, e.g.,
standard for flange connections EN 1591


The Industrial Standard EN 1591

A standard for gasketed circular flange connections
Consists of applicability, nomenclature and calculation method
Assures impermeability and mechanical strength of the system


Calculation Method Parameter

The input parameters to the calculation method
Flange data, e.g., dimensions and material constants

Data for operating states such as pressure and temperature

Calculation Method Control Flow

Explicit back-jumps require
conditional loops
Evaluation order determined by
dependency in definitions
Piecewise function definitions
require conditionals


Calculation Method analyzed

The standard involves calculations using
√
Real arithmetic with real functions, e.g., cos, n

Special functions such as maximization

Requirements for formalizing the calculation method
Specify dependencies in arbitrary order
(subject to well-formedness requirements)
Imperative control flow
Schematic expressions

Observation: little control but a lot of dependencies
→ division into program part and dependency store

The Engineering Calculation
Language EnCL

The main building blocks
Terms over a {bool, real}-sorted Signature Σ with predefined part
Σpre = {��, cos, ...} and user-defined part Σuser = {c1 , c2 , ...}
Special term constructions
Predefined binders: e.g., solve(t = s, x)
convergence predicate in loop conditions: convergence(0.001, c)

Dependency stores are sets of dependencies c(x1 , ..., xn ) = t
Programs constructs
Assignment: c(x1 , ..., xn ) := t
Sequence: p1 ; ...; pn
Loop: repeat p until b
Conditional: case b : p

EnCL and Computer Algebra
Systems

Running EnCL program p with dependency store ��

CAS provides many predefined Interpreter CAS
functions
start session
CAS supports dependency stores
CAS-interface send(��)

evaluate terms
assign terms to constants ⎧ assign(c, t)

Send dependencies to the CAS
⎪ t′
run
⎨
.
.
.
Run program p program⎪ eval(u)
u′
⎩
The interpreter maintains dependency
store (in parallel to CAS)


A Small Example

Calculating a root of cos in EnCL using Newton’s Method

The EnCL specification Dependency Graph
x = 10 %(A)%
y = cos(x) %(B)% A
z = sin(x) %(C)% x
------------------------ B C
repeat y z
x := x + y/z %(D)%
until convergence(0.001, x) D

Behaves like
A;B;C;repeat x’:=x;D;B;C; until reldistLe(x, x’, 0.001)

EnCL Semantics

Σ-algebras with standard interpretation for predefined part Σpre

[[t]]�� ∈ R is the interpretation of t in the Σ-algebra ��

[[t]]�� is the term t ′ after full substitution of t w.r.t. ��

If �� is a model of �� then [[[[t]]�� ]]�� = [[t]]��

[[c(x1 , ..., xn ) := t]]�� = �� [c(x1 , ..., xn ) = [[t]]�� ]


Formal Verification

Correctness of calculations crucial for safety critical applications

The CAS cannot be fully trusted
However, results of the CAS can be formally verified
Mark selected subterms as verification points
Produce verification conditions

Use Hets to prove verification conditions
EnCL term semantics defined in HO-CASL,
i.e., axiomatization of Σpre

Checking solutions is easier than finding them


Verification Conditions: Example

EnCL program Insert verification point at solve
. CAS computes solve(t = s, x) in
. Dependency Store = ��
. context �� and returns r
y := solve(t=s, x) Verification condition
.
.
⋀︀
. �� ⇒ solve(t = s, x) = r
Semantics of solve in HO-CASL
Translate to HO-CASL for proving


Integration of EnCL into Hets

The Hets Framework
HO-CASL
Higher Order Logic

EnCL
Specification Interpreter
Language

CAS Interface


Conclusion

Formal verification of functional properties of mechanical systems
Formal executable language for engineering calculations EnCL
Integration into Institution Framework
Implementation based on Hets Framework
Generic CAS interface in Hets instantiated for Mathematica,
Maple and Reduce
Support for uncertain numerical values
EnCL-formalization of calculation method from EN 1591
Future Work
Statement and proof of properties of calculation method
Partial instantiations of the standard ensuing simplification
Structuring of multiple calculation methods

Thank you for your attention.


Uncertainty

Some situations require dealing with uncertain numeric values

Input parameters up to an error value, e.g., 1.53 ± 0.01
CAS returns only approximation
Require tracking of uncertainty throughout the CAS session
→ Mathematica’s Numerical-Precision Tracking (NPT)
Adapt verification condition generation
→ replace numbers by intervals


FASE 2011 - Formalizing and Operationalizing Industrial Standards

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