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Introduction to CS 270
Math Foundations of CS
Verification of Computer Systems
Mark Boady, Jeremy Johnson &
Kurt Schmidt
Drexel University
Course Description
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Emphasizes analytic problem-solving and
introduction of mathematical material necessary
for later courses in algorithms, systems
architecture, programming languages, software
engineering, concurrent programming and
artificial intelligence. Includes topics such as
logic, theorem-proving, language operations,
context-free grammars and languages,
recurrence relations, and analysis of algorithms.
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Course Description (revised)
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Introduces formal logic and its connections to
Computer Science. Students learn to translate
statements about the behavior of computer
programs into logical claims and to prove such
assertions both by hand and using automated
tools. Considers approaches to proving
termination, correctness, and safety for
programs. Discusses propositional and
predicate logic, logical inference, recursion and
recursively defined sets, mathematical
induction, and structural induction.
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Course Goals
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For students to learn how to formally specify
and reason about properties of computer
systems. To appreciate what it means to prove
something and the value of formalism. To
become aware of tools for formal specification
and automatic deduction. To use logical
thinking to become better programmers and
systems designers.
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Course Objectives
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To use recursion and divide and conquer to solve problems
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To provide recursive definitions of patterns and data structures
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To formally specify the input/output requirements of programs
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To use induction and other proof techniques to prove properties of
algorithms, data structures, programs, and computer systems
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To use logic to describe the state of systems and to use logical
deduction (by hand and using tools) to prove properties of systems
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To understand the power and limitations of formal logic.
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Course Topics
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Propositional and Predicate Logic
Formal Proof using Natural Deduction
Applications of Logic to Computer Science
Functional Programming
Recursion, Recursive Definitions and Induction
Divide and Conquer Algorithms and Recurrence Relations
Program Specification and Verification
Automated Reasoning
Termination Analysis
Test Case and Counter Example Generation
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Course Lectures
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Week 1 [Propositional calculus, deduction and SAT solvers]
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Week 2 [Propositional calculus, deduction and SAT solvers]
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Course Introduction (formal reasoning, proof, and computer verification)
Proposotional Calculus (boolean functions and boolean expressions, syntax
and semantics and equivalence)
Satisfiability and tautology checking (truthlab)
Natural deduction (prooflab)
Week 3 [Propositional calculus, deduction and SAT solvers]
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Proof strategies
Normal forms and SAT solvers (minisat)
Soundness and completeness
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Course Lectures
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Week 4 [Propositional calculus, deduction and SAT solvers]
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DPLL algorithm for satisfiability
Overview of predicate calculus (comparison to propositional calculus)
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Course Lectures
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Week 5 [Induction, recursion and recursively defined sets]
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Week 6 [Induction, recursion and recursively defined sets]
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ACL2 expressions (syntax and semantics)
ACL2 recursion
ACL2 list processing
Recursion and induction (recursive summation, tower of Hanoi)
Properties of lists
Boolean expressions and evaluation
Week 7 [Induction, recursion and recursively defined sets]
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Equivalent boolean expressions and simplification
Proof of completeness and soundness
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Course Lectures
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Week 8 [Automated reasoning and computer verification]
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Week 9 [Automated reasoning and computer verification]
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Equational reasoning
Equational reasoning
Inductive reasoning
Week 10 [Automated reasoning and computer verification]
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Formal specifications (input and output contracts)
Logic in ACL2
Proving in ACL2
Inductive reasoning
Week 11 [Automated reasoning and computer verification]
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Inductive reasoning
Well ordering and proof of termination
Termination proofs in ACL2
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Textbook (first half)
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Logic & Proof (CMU OLI)
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Course key: drexel-landp
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Textbook (second half)
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Logic & Computation (Northeastern)
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Software (ProofLab)
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Software (MiniSAT)
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Software (ACL2)
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ACL2 >QUERY
(thm (implies (and (true-listp x) (true-listp y) (true-listp z))
(equal (app (app x y) z) (app x (app y z)))))
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<< Starting proof tree logging >>
^^^ Checkpoint Goal ^^^
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*1 (the initial Goal, a key checkpoint) is pushed for proof by induction.
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…
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Subgoal *1/3
Subgoal *1/3'
Subgoal *1/2
Subgoal *1/1
Subgoal *1/1'
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*1 is COMPLETED!
Thus key checkpoint Goal is COMPLETED!
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Q.E.D.
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Prerequisites and Grading
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Programming skills (CS 172)
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Course Requirements and Grading
Class Participation (10%)
 Weekly homework assignments (40%)
 Two Midterms (30%) and Final (20%) exam
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• Midterms tentatively weeks 5 and 8
• Final exam during finals week
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Getting Help
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Office Hours
Jeremy Johnson [MWF 1-2], Mark Boady [T
2-5, W 2-3],
 Samantha Bewley [M 5-7], Richa Bhartia [T
12-2, R 11-1], Brian Lee [R 6-8]
 www.cs.drexel.edu/clc
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Piazza
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piazza.com/drexel/fall2015/cs270/home
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Getting Help (piazza)
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Software Bugs
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In 1980, NORAD reported that the US was under missile attack.
The problem was caused by a faulty circuit, a possibility the
reporting software hadn’t taken into account.
The Therac-25 medical radiation therapy device was involved in
several cases where massive overdoses of radiation were
administered to patients in 1985-87, a side effect of the buggy
software powering the device.
In 1996, a European Ariane 5 rocket was set to deliver a payload of
satellites into Earth orbit, but problems with the software caused
the launch rocket to veer off its path a mere 37 seconds after
launch.
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Software Bugs
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In 1994 in Scotland, a Chinook helicopter crashed and killed all 29
passengers. While initially the pilot was blamed for the crash, that
decision was later overturned since there was evidence that a
systems error had been the actual cause.
One of the subcontractors NASA used when building its Mars
climate orbiter had used English units instead of the intended
metric system, which caused the orbiter’s thrusters to work
incorrectly. Due to this bug, the orbiter crashed almost immediately
when it arrived at Mars in 1999. The cost of the project was $327
million, not to mention the lost time (it took almost a year for the
orbiter to reach Mars).
In 2002 NIST estimated that programming errors cost the US
economy $60B annually
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Hardware Bug
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Intel FDIV Bug
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Intel P5 Pentium floating point unit
$500M
Error as high as the fourth significant digit of a
decimal number, but the possibilities of this
happening are 1 in 360 billion.
Approximately 8000 bugs introduced in during
design of Pentium 4.
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Verification and Validation
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Verification and Validation is the process
of checking that a SW/HW system meets
specifications and fulfills its intended
purpose
Features
Tie-Line Flows
PowerG rid
Real Time Analysis
Power System
Vulnerabilities
Real Time Analysis
Power System
Reliability
Excellent Training
Facility for Concerned
Parties
Hardware Power System
Model
Decision Making
Station
Engineering
Analysis Station
Economic Analysis
Station
Ultra-Fast Bus
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Empirical Testing
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Traditionally, errors in hardware and software
have been detected empirically by testing
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Number of possibilities too large so only a small
subset can be tested
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E.G. Testing arithmetic operations on all 264 double
precision floating point numbers is infeasible
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Formal Methods
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In the context of hardware and software
systems, formal verification is the act of
proving or disproving the correctness of
intended algorithms underlying a system
with respect to a certain formal
specification or property, using formal
methods of mathematics
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Success Stories
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Verified the cache coherence protocol in the
IEEE Futurebus+ Standard
Analysis of Microsoft Windows device drivers
using SLAM
Non-overflow proof for Airbus A380 flight control
software
Verification of Pentium 4 floating-point unit with
a mixture of STE and theorem proving
NICTA’s embedded L4 microkernel
Compcert compiler
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Approaches
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Model Checking
Temporal logic, BDD, Z notation, …
Static Analysis
Type Checking
Logical Inference
 Automated theorem proving
Proof Checking
Program Derivation
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Model Checking
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model checking refers to the following problem: Given a model of
a system, test automatically whether this model meets a given
specification. Typically, the systems one has in mind are hardware
or software systems, and the specification contains safety
requirements such as the absence of deadlocks and similar critical
states that can cause the system to crash. Model checking is a
technique for automatically verifying correctness properties of
finite-state systems.
An important class of model checking methods have been
developed for checking models of hardware and software designs
where the specification is given by a temporal logic formula.
Pioneering work in the model checking of temporal logic formulae
was done by E. M. Clarke and E. A. Emerson[1][2][3] and by J. P.
Queille and J. Sifakis.
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Automated
Theorem Proving
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Formal proof by hand is difficult
Have proof checked or generated
automatically by a computer
Higher Order Logic, or HOL, is a widelyused tool for creating formal specifications
of systems, and for proving properties
about them. It has been used in both
industry and academia to support formal
reasoning in many areas, including
hardware and software verification. It can
be used to support any project which can
be defined in higher order logic, an
expressive logic originally developed as a
foundation for mathematics.
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Proof Carrying Code
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Proof-carrying code (PCC) is a software mechanism that allows a
host system to verify properties about an application via a formal
proof that accompanies the application's executable code. The host
system can quickly verify the validity of the proof, and it can
compare the conclusions of the proof to its own security policy to
determine whether the application is safe to execute. This can be
particularly useful in ensuring memory safety, i.e. preventing buffer
overflows and other vulnerabilities common in some programming
languages.
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Proof-carrying code was originally described in 1996 by George
Necula and Peter Lee.
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Static Analysis
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Static program analysis (also static code analysis or SCA) is the
analysis of computer software that is performed without actually
executing programs built from that software. The term is usually
applied to the analysis performed by an automated tool.
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A growing commercial use of static analysis is in the verification of
properties of software used in safety-critical computer systems and
locating potentially vulnerable code[3]. For example the following
industries have identified the use of static code analysis as a
means of improving the quality of increasingly sophisticated and
complex software: Medical software, Nuclear software.
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Program Generation
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program derivation is the derivation of a program from its
specification, by mathematical means.
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To derive a program means to write a formal specification, which is
usually non-executable, and then apply mathematically correct
rules in order to obtain an executable program satisfying that
specification. The program thus obtained is then correct by
construction. Program and correctness proof are constructed
together.
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Hoare logic, stepwise refinement, Bird-Meertens Formalism,
parallel program design, FLAME, SPIRAL
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References
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http://en.wikipedia.org/wiki/Verification_and_validation_(software)
http://en.wikipedia.org/wiki/Formal_verification
http://en.wikipedia.org/wiki/Model_checking
http://en.wikipedia.org/wiki/Temporal_logic
http://en.wikipedia.org/wiki/Automated_theorem_proving
http://en.wikipedia.org/wiki/Proof-carrying_code
http://en.wikipedia.org/wiki/Static_program_analysis
http://en.wikipedia.org/wiki/List_of_tools_for_static_code_analysis
http://en.wikipedia.org/wiki/Program_derivation
E. Allen Emerson, The Beginning of Model Checking: A Personal
Perspective
John Harrison, Formal verification of floating-point arithmetic at Intel, June
2006.
John Harrison, Formal Verification in Industry (I), 1999.
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