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copied from http://www.dwheeler.com/secure-class/presentations/Secure-Software-10-Formal-Methods.ppt
SWE 681 / ISA 681 Secure Software Design & Programming Lecture 10: Formal Methods
Dr. David A. Wheeler
2016-04-06
Formal methods (FM)
Introduction
Notations
Tools
Specifications
Verification (general & for proving programs)
Open ProofsSome portions © Institute for Defense Analyses
(the open proofs sections), used by permission.
This material is not intended to endorse particular suppliers or products.
2
Formal methods (FM)
Formal methods (FM) = the use of “mathematically rigorous techniques and tools for the specification, design and verification of software and hardware systems.”
Can be applied to spec, actual software, or model
Mathematically rigorous = “specifications are well-formed statements in a mathematical logic and that the formal verifications [if any] are rigorous deductions in that logic”3
Source: “What is Formal Methods?” by Ricky W. Butler
http://shemesh.larc.nasa.gov/fm/fm-what.html
Why formal methods (FM)?
Imagine it’s critical that software meet some requirement
E.G., “Never give secret information to unauthorized personnel”
So important that it’s a disaster if requirement not met
FM spec language can reduce requirement ambiguity
You cannot get truly high confidence with just:
Dynamic analysis (e.g., testing) – only tiny % of inputs
Vulnerability scanners – only reports some defects
In contrast, FM can prove “always” or “never” (!!!)
Given some assumptions (& you decide what those will be)4
Applying formal methods to security issues
To apply FM in general:
Describe requirements using FM-based spec language
If will verify, choose & apply FM verification approach(es)
Thus, to directly apply FM to security:
Must describe security requirements using FM-based spec language (and verify if applicable)
Verification tool may predefine requirement (“no race condition”) - simplifies use, but limits usage & claim
Verification must usually prove no undefined situation occurs, which by itself can reveal potential issues
Some FM technologies can be “repurposed”5
The idea of using logic to determine truth is an old one!
6
“The only way to rectify our reasonings is to make them as tangible as those of the Mathematicians, so... when there are disputes... we can simply say: Let us calculate... to see who is right.”
– Gottfried Leibniz, The Art of Discovery (1685)
Already achieved in certain areas
Limitations exist, both in theory & practice
A little history will help…Some advances in logic & formalized (math) reasoning
Artistotle’s Organon, esp. “Analytica Priora” (Prior Analytics) – first work on logic (Aristotle lived 384-322 B.C.)
Euclid’s Elements — oldest axiomatic deductive treatment of math (c. 300 BC)
Non-Euclidean geometry (Lobachevsky 1829, Bolyai 1831, Gauss)
Ancient assumptions might not necessarily hold!
George Boole’s “An Investigation of the Laws of Thought…” (1854) — founded boolean algebraic logic (much work done since)
Georg Cantor founded set theory (1874)
Gottlob Frege’s Begriffsschrift… (1879)
Added “quantified” variables (for-all & there-exists)
Added more-general mechanisms to handle functions
Peirce (1881), then Richard Dedekind & Giuseppe Peano (1888), formalized natural numbers7
The foundational crisis of mathematics
Bertrand Russell discovers “Russell’s paradox” in 1901
Mailed to Frege 1902
Shows that naïve set theory leads to paradox, must fix somehow
Let R be the set of all sets that are not members of themselves… is R a member of itself?
“Foundational crisis of mathematics” – what is the proper foundation?
Whitehead & Russell release Principia Mathematica (1910-1913)
Goal: Describe axioms & inference rules from which all mathematical truths could in principle be proven (“logicism” school: Base all math on logic)
Notation shift since (see Linsky’s “The Notation in Principia Mathematica”)
Russell’s paradox solution (a hierarchy of types) unwieldy; today most fundamental math work based on Zermelo–Fraenkel set theory, often with the axiom of choice (ZFC)
Various philosophical views emerged, including:
Logicism (Russell)
Intuitionism/Constructivism (Brouwer & Heyting)
Formalism (Hilbert*)8
Logicomix
Logicomix: An Epic Search for Truth by Apostolos Doxiadis & Christos Papadimitriou
Dramatizes some of the “foundational crisis of mathematics” (the early 20th century search for the proper foundations of mathematics), from the viewpoint of Bertrand Russell
Dramatization - takes licenses with history
Shows people talking when they really converse by letter or papers
It oversimplifies many things
In a few places reorders events
Little coverage of constructivism/ intuitionism
It’s excellent at explaining what people were doing, and why it mattered so much to them
Best non-mathematical introduction to this important aspect of 20th century history
Presents as story, not just dry history
Russell and others desperately wanted absolute truth; their results were valuable, but not what they expected or exactly wanted9
Most of modern mathematics is built on a small set of rules of reasoning & axioms
10
Classical propositional logic
(variables are true or false)
First-order predicate logic
(variables can be objects)
Second/higher-order predicate logic
(variables can be functions)†
Logic
Set Theory
Zermelo
Zermelo-Fraenkel (ZF)
Axiom of Choice (AC) †
(some controversy!)
ZFC
(most math
uses this as
its basis)
Numbers
Theories about numbers can be derived from lower-level axioms, then used as building blocks*
The division between “logic” & “set theory” isn’t as strict as implied by this figure
Other axioms are sometimes added, e.g., Tarski's axiom, continuum hypothesis (CH)
Different logics & axioms are in use, e.g., intuitionism subsets classical logic & rejects AC† Less commonly used
Mathematicians identify
rules of reasoning, axioms (assumptions), and derive proofs using them
Gödel’s incompleteness theorems
David Hilbert’s retirement address at 1930 Königsberg conference concluded with:
“For the mathematician there is no Ignorabimus… in my opinion… there is no unsolvable problem. In contrast to the foolish Ignoramibus, our credo avers: We must know. We shall know!” (“Wir müssen wissen. Wir werden wissen!”)
Gödel’s incompleteness theorems proved this is impossible:
Two theorems proven & published by Kurt Gödel in 1931*
First one was announced by Gödel at the same 1930 conference!
Establishes fundamental limitations of mathematics
First incompleteness theorem:
No consistent system of axioms whose theorems can be listed by an “effective procedure” (e.g., computer program or algorithm) can prove all truths about the relations of the natural numbers (arithmetic)
I.E.: There will always be statements about the natural numbers that are true but unprovable within a consistent system
Second incompleteness theorem:
Such a system cannot demonstrate its own consistency
Note: Gödel proved a different “completeness theorem” in 1929 (as dissertation)
In first-order logic (limited), all logically valid formulas can be formally proved in finite steps
Not the same thing!11
Principia Mathematica and Related Systems I”
(second part never written) as Theorems VI and XI
Halting Problem
Halting problem: Given a description of an arbitrary computer program, decide whether the program finishes running or continues to run forever (for some input)
Alan Turing proved in 1936 that a general algorithm to solve the halting problem for all possible program-input pairs cannot exist
Proof similar to Gödel’s incompleteness theorems
Fundamental limit on computation12
But within these theoretical limits,
a lot can be accomplished.
This is only a partial summary of formal methods
This is a summary; many important things omitted
Have been decades of research (more than we can cover)
Active research area (so expect changes/new approaches!)
Presentation goal: awareness of key portions of field
Omit details on how they work, but some info necessary to understand their capabilities & limitations
Too many know little, & some courses only explain 1 tool
Assume you know already how to develop software
Especially note open source software & no-cost tools
Identify things you can try & experiment with right now
If you’re interested, whole classes & careers in this field
Need math, especially discrete math & logic13
Making formal methods affordable
Formal methods costly to apply today in many cases
Many backoff approaches exist to limit costs:
Level 0: Formal specification created, then program informally developed from it. “Formal methods lite”
Level 1: Level 0, + prove some select properties or formal refinement from specification towards program
Level 2: Fully prove claims, mechanically checked
Any of the above can apply to a subset of components or properties
E.G., specialized analysis tool to determine one specific (important) property. Narrow properties can be relatively affordable on big systems
Tool support typically needed to scale up
Different tools good for different things (combining may help)
Choose specific toolsuite that meets goal, e.g., use weakest language
Improvements: Increasing CPU power, improved algorithms, & OSS (speeding research, distribution, and multi-approach integration)
We’ll discuss basics of FM notations, then tools/approaches14
Formal methods: Notations
15
Quick review: Set notation
Set: a collection of elements or members
Given sets S = {1, 2, 3} ; T = {1, 2}; R = {1, 5}; L = {a, b}
Is member of: 1 S is true, 7 S is false, 7 S is true
Is subset of: T S is true, R S is false
Union: S R is {1,2,3,5}
Intersection: S ∩ R is {1}
Set difference: S \ R = S – R = {2,3}
Cartesian Product: R x L = {(1,a),(1,b),(5,a),(5,b)}
Common predefined sets (sometimes with blackboard bold):
∅ or {}: Empty set
N: natural numbers {1, 2, 3, …} or {0, 1, 2, …}; ISO 31-11 includes 0
Z: integers (from Zahl, German for integer) {…, -2, -1, 0, 1, 2, …}
Q: rational numbers (from quotient)
R: real numbers
C: complex numbers16
Many whose “N” includes 0 may also append “*” superscript to any of the above letters to mean “except zero” [ISO 31-11; Gullberg, Mathematics: From the Birth of Numbers]
Set builder notation
Sets can be defined using set builder notation. A common format:
{ expression-using-variable
| variable set ∧ condition} |
Can omit “variable set” if understood (e.g., pre-stated “universe”)
Pronounciation:
| is “where” or “such that”
is “in” or “member of” (in Z notation, “:” used instead)
∧ is “and” (some notations use “&” or “,” instead)
E.g., set1 = {x | x A ∧ p(x)}
“set1 is the set of elements in A for which the proposition p(x) is true”
List comprehensions of Python & Haskell are similar
Python: [(x, y) for x in [1,2,3] for y in [3,1,4] if x != y]
Haskell: [(k, x) | k <- ks, x <- xs, p x]17
Many different languages for mathematical logic
Various math logic notations (weakest…strongest):
Predicate logic (just boolean variables)
First-order logic (FOL) (“all X..”, non-booleans, functions,…)
Can add “theories” about integers, real numbers, etc. to FOL
Higher-order logics (HOL) (+ functions can vary/be objects)
Again, can add “theories” about integers, etc. to HOL
Stronger logic notations provide more capability
But tend to be harder to automatically analyze
Often want “weakest” language that meets needs
Can add “temporal logic” (“X will happen eventually”)
Often computation tree logic (CTL) or linear temporal logic (LTL)
These traditional math notations can be used directly or be part of a larger specification language18
Predicate logic
Predicate logic – define expressions with just:
Boolean variables
Operators and, or, not
Parentheses allowed
Predicate logic too limited for many problems
Useful in some cases, e.g., dependency analysis
Can be used to implement more sophisticated systems19
First-order logic (FOL)
FOL widely used mathematical logic
Aka “first-order predicate logic” or “first-order predicate calculus”
Basis of most FM software languages for specs
Including “higher-order” logics that relax FOL restrictions
May add “theories” to describe integers, real numbers, etc.
In traditional FOL, every expression is either a:
Term (an object / “non-boolean”): a variable, a constant, or a function f(term1, term2, …)Typically there’s a “domain of discourse” (aka “universe”), the set of entities over which variables may range. E.G., “integers” or “real numbers”
Formula (a truth value / “boolean”): see next slide, includes predicates (“functions” that return truth value)
Can have variables & constants (must distinguish)
Prolog convention: Variable if begin with uppercase, else constant
Math convention: Variables begin with w, x, y, z
In FOL, functions & predicates can’t be variables20
FOL formula notation
21
Traditional Notation
Alternate Notation
Meaning
¬A
-A, ~A
Not A. In classical (not intuitionist) logic, ¬¬A = A
A ∧ B
A & B
A and B. True iff both true
A ∨ B
A
| B |
A or B. True iff at least one true
A → B
A -> B
A implies B. Aka “if A then B” (1-arm) Classically same as ¬A ∨ B
p(T1, T2, …)
Predicate p true when given those terms
T1 = T2
Term is equal to term. Not in traditional FOL but is a common FOL extension
∀ X …
forall X …
For all X, … is true
∃ X …
exists X …
There exists an X where … is true
A and B are formulas; X is a term variable; T1 & T2 are terms; iff = if and only if
Conventional FOL example
∀ X man(X) → mortal(X)
“For all X, if X is a man, then X is mortal”
I.E., “All men are mortal”
This uses Prolog naming convention (uppercase vars)
man(socrates)
“Socrates is a man”
mortal(socrates)
“Socrates is mortal”
This can be proven from the first two statements with appropriate rules for reasoning
22
More FOL examples
∃ X (bird(X) ∧ ¬fly(X))
“At least one bird can’t fly”
∀ X ∀ Y (mother(X, Y) → younger(Y, X))
“All children are younger than their own mother”
∀ X ∀ Y ((mother(X) ∧ child(Y)) → younger(Y, X))
“All children are younger than all mothers” (FALSE!)
∃ X (brother(X, bob) ∧ likes(alice, X))
“Bob has a brother that Alice likes”
∀ X (brother(X, bob) → likes(alice, X))
“Alice likes all of Bob’s brothers”
∀ X (man(X) → X = socrates)
“All men are Socrates” (FALSE!)23
(Most of these examples are based on Huth & Ryan “Logic in Computer Science” 2006)
Bugs possible in formal methods specifications!
English “and” & “or” often don’t translate exactly
“Or” may mean inclusive (either-or-both) or exclusive (only-one)
Common mistake: Empty sets used with forall
∀ X martian(X) → green(X)
Means “All Martians are green”
∀ X martian(X) → ¬green(X)
Means “All Martians are not green”
If there are no Martians, both expressions are true
Different from non-expert expectations & Aristotelian logic
Though there are good reasons for it
Can easily fix “empty sets” issue once you know about it, e.g.:
(∃ X martian(X)) ∧ (∀ X martian(X) → green(X))
Some notations make this easier & less error-prone
Common mistake: ∃…→… (usually wrong, use ∧ not →)
∃ X (a(X) → b(X)) means ∃ X (¬a(X) ∨ b(X)). Instead try ∃ X (a(X) ∧ b(X))24
Tips on how to create accurate translations to math notations
“Translation Tips” by Peter Suber http://www.earlham.edu/~peters/courses/log/transtip.htm
“First-Order Logic” by C. R. Dyer http://pages.cs.wisc.edu/~dyer/cs540/notes/fopc.html
“Guide to Axiomatizing Domains in First-Order Logic” by Ernest Davis http://cs.nyu.edu/faculty/davise/guide.html
“Logic in Computer Science” by Michael Huth and Mark Ryan (Cambridge University Press)25
FOL extensions
“Traditional” FOL comes from mathematics
Many notations in practice add extensions:
Equality (“=”) is extremely common
Notation for constant “true” & “false”
If-then-else: ite(condition, true-term, false-term)
Traditionally functions can’t accept formulas (booleans)
Various work-arounds exist, but nicer to have it
Allow formulas (booleans) as parameters
F1 ↔ F2, an abbreviation of ((F1 → F2) ∧ (F2 → F1))
Add types/sorts (integers, reals, etc.)
Add “Theories” (e.g., about integers, etc.)
Resulting notations are still first-order
Ways to structure/organize groups of statements26
Beyond FOL
“Second-order” logics & “Higher-order” logics
Enable functions & predicates to themselves be variables
More flexible as a notation…
But harder to automatically verify
Support tools tend to be interactive27
Some approaches for analyzing programs using logic
Hoare logic
Predicate transformer semantics (including weakest precondition)
Separation logic28
Hoare logic (aka Hoare-Floyd logic)
Created by Sir C.A.R. “Tony” Hoare, based on Floyd
Hoare triples: {P} C {Q}
P=precondition, C=Command/code, Q=postcondition
Examples of rules:29
P is
“loop invariant”
Key:
Premises
Conclusion
Predicate transformer semantics (weakest preconditions)
Dijkstra “Guarded commands, nondeterminacy and formal derivation of programs”
Assign every language statement a “predicate transformer” from one predicate to another
Weakest-precondition for statement S maps any postcondition to a (weakest) precondition. Go backwards
Strongest-postcondition, map precondition to postcondition. Go forwards
Actual transformers based on Hoare logic
E.G., wp(if E then S1 else S2 end, R) =
(E → wp(S1, R)) ∧ (¬E → wp(S2, R))30
Separation logic
Extension of Hoare logic developed by John C. Reynolds, Peter O'Hearn, Samin Ishtiaq and Hongseok Yang
Describes “states” consisting of a:
store (“stack-oriented variables”) and a
heap (“dynamically-allocated objects”)
Defines a set of operations about them
“Frame rule” enables local reasoning
A program that executes safely in a small state can also execute in any bigger state and that its execution will not affect the additional part of the state when certain conditions proved
E.G., Coq “Ynot” library implements separation logic
Can apply separation logic concepts using traditional FOL
E.g., Jessie; see Francois Bobot and Jean-Christophe Filliatre. “Separation Predicates: a Taste of Separation Logic in First-Order Logic”. 14th International Conference on Formal Engineering Methods (ICFEM), Nov 201231
Beware of math vs. real world
Mathematical models are not the real world
They are simplified models of the real world
Common issue: Math numbers ≠ computer numbers
In math, infinite number of integers & reals
Computers always finite; cannot exactly represent all numbers, and integers/reals often fixed size
Common issue: Assumptions or goals are wrong
Wrong assumptions can lead to wrong conclusions
If you didn’t ask for it, you might not get it
Testing, inspection, & peer review can help
Models can be vitally useful
Be wary of their limitations32
Formal Methods: Tools
33
Types of formal methods tools
Formal specification tools
Without necessarily verifying
Verification tools
Major verification approaches include:
Theorem-provers: Automated & interactive
Satisfiability (SAT) solvers: Boolean-only or modulo theories
Model-checkers
Abstract interpretation / symbolic execution (for programs)
First discuss in general (e.g., verifying models)
Then, how to verify program meets spec (some dups)
Formalizing & verifying mathematical theorems (not code)
This is just my grouping & is approximate
Active research areas, annual competitions34
Formal specifications of systems (e.g., for level 0)
Any FM-based language can be used for specification
Including PVS, Isabelle/HOL, Coq, ACL2, & many others (we’ll see later)
Many formal specification languages often used for specification without (significant) verification
Language examples: Z, Object-Z (Z+classes), VDM, B, Unified Modeling Language (UML) Object Constraint Language (OCL), Alloy
There’s an ISO standard for Z, but “the version described in Mike Spivey’s book continues to be the most popular” [Jackson]
Each has various tools to help write specs
Goal is often to eliminate unintentional ambiguity
“Alloy” OSS tool is unusual & interesting (Daniel Jackson, MIT)
Relatively easy & unique spec language
Parts loosely based on Z, includes “transitive closure” operator
Includes “checker” to quickly find counter-examples
Not a prover, but easy-to-use & much more rigorous than just reviewing text
Could send to theorem prover (Kelloy->KeY, Prioni->Athena/Otter)35
Z example (birthday book)
36
Source: The Z Notation: A Reference Manual, Second Edition
by J. M. Spivey (University of Oxford)
Schemas
(state space & operation)
Alloy: Sample Screenshot
37
Source: 
Alloy: File system example
File System Model (I)
A file system object in the file system
sig FSObject { parent: lone Dir }
A directory in the file system
sig Dir extends FSObject { contents: set FSObject }
A file in the file system
sig File extends FSObject { }
A directory is the parent of its contents
fact { all d: Dir, o: d.contents
| o.parent = d } |
All FSOs are either files or directories
fact { File + Dir = FSObject }
There exists a root
one sig Root extends Dir { } { no parent }
File system is connected
fact { FSObject in Root.*contents }
The contents path is acyclic
assert acyclic { no d: Dir
| d in d.^contents } |
Now check it for a scope of 5
check acyclic for 5
File system has one root
assert oneRoot { one d: Dir
| no d.parent } |
Now check it for a scope of 5
check oneRoot for 5
Every fs object is in at most one directory
assert oneLocation { all o: FSObject
| lone d: Dir | o in d.contents } |
Now check it for a scope of 5
check oneLocation for 5
38
Source: Alloy tutorial, http://alloy.mit.edu/alloy/tutorials/online/frame-FS-1.html
Alloy notation ~subset of Z, not as pretty or rich
But its special closure operators useful
Can compute & display counter-example(s); “refuter” not “prover”Theorem provers
Theorem prover
Accepts assumptions (“givens”) & goal in some notation
Tries to produce proof of goal, starting from assumptions…
Using only a sequence of allowed inference rules & theorems
Many different possible inference rules, e.g.:Modus ponens: Modus tollens:
Formal proof: Every step fully justified by accepted rule
“Proof checker” can verify proof - easy to build, enabling separate third-party verification. Supports “Proof carrying code”
Theorem proving tools may be either:
Automated
Interactive (“proof assistant”)39
P, P→Q?
Q
P→Q?, ¬Q
¬P
Premises
Conclusion
You’ve already done this manually in high school Geometry…
Automated theorem provers
Automatically prove goal, given assumptions
Often like chess programs; try many options & guided by algorithm/heuristics
Many let humans provide guiding “hints”
Many good theorem provers for traditional FOL
Harder to do as add theories or move to a HOL
Annual CADE ATP System Competition (CASC)
Uses subsets of TPTP Problem Library
Example automated theorem provers for traditional FOL with equality:
Prover9 (William McCune, University of New Mexico) (GPLv2).
E theorem prover (aka “Eprover”) (Stephan Schulz, TU Munich) (GPLv2)
Other examples of automated theorem provers:
Princess: FOL, linear + integer arithmetic (no general multiplication) (GPLv3)
gappa: Tool to analyze numerical calculation bounds (CeCILL or GPL)
Uses interval arithmetic and forward error analysis to bound math expressions
Generates a theorem and its proof for each verified enclosure
Satallax: HOL (Church's simple type theory+ extensionality + choice) (MIT)40
Prover9
Created by William McCune, University of New Mexico
Descended from “Otter” prover
Supports FOL with equality, plus list operations
Approach:
Starts with assumptions & negated goal
Transitively generates all facts it can
If it finds a contradiction, reports that chain as a proof
Also supports:
mace4 (tries to find counter-example)
ivy - proof-checker (checks output—counters tool error)41
Prover9 example – prove square root of 2 is irrational
formulas(assumptions). % Note: Universe = integers > 0, “forall” assumed
1*x = x. % identity
x*1 = x.
x*(y*z) = (x*y)*z. % associativity
x*y = y*x. % commutativity
(x*y = x*z) -> y = z. % cancellation (0 is not allowed, so x!=0).
% divides(x,y) true iff x divides y. E.G., divides(2,6) is true because 2*3=6.
divides(x,y) <-> (exists z x*z = y).
divides(2,x*y) -> (divides(2,x)
| divides(2,y)). % If 2 divides x*y, it divides x or y. |
2 != 1. % Original author almost forgot this. Wheeler thinks needing it is a bug.
% Now, assert that we can have a rational fraction for sqrt(2), reduced
% to lowest terms (this will fail, and that's the point of the proof):
a*a = (2*(b*b)). % a/b = sqrt(2), so a^2 = 2 * b^2.
(x != 1) -> -(divides(x,a) & divides(x,b)). % a/b is in lowest terms
end_of_list. % Will show proof by contradiction.
42
Source: “The Seventeen Provers of the World” compiled by Freek Wiedijk. Otter/Ivy section. Original proof created by Larry Wos, Michael Beeson, and William McCune?. Heavily modified/simplified by David A. Wheeler. Note: Prover9 descends from “Otter”
Interactive theorem provers (proof assistants)
Humans enter symbol-manipulation commands to derive goals from assumptions
Tend to support richer notations, e.g., rich theories and/or HOL and/or…
But require expert human guidance proof in non-trivial cases
Sample tools:
PVS (by S. Owre, N. Shankar, & J. M. Rushby at SRI)
LCF-style family (HOL 4, HOL Light, Isabelle/HOL)
Coq (will discuss later)43
PVS Example #1
mortality: THEORY
BEGIN
man: TYPE+ % The "+" means there's at least one man.
mortal(m: man): bool % Returns True if m is mortal.
% Socrates is a man.
Socrates: man
% All men are mortal.
all_men_mortal: AXIOM FORALL (m: man): mortal(m)
% Socrates is mortal.
socrates_mortal: CLAIM mortal(Socrates)
END mortality
44
In normal PVS use it'd be better
to omit the axiom and say:
mortal(m: man): bool = true
but this tries to stay close
to the traditional example.
Prove using: (rewrite "all_men_mortal") or (grind :rewrites ("all_men_mortal"))
PVS Example #2
Phone_4 : THEORY
BEGIN
N: TYPE % names
P: TYPE % phone numbers
B: TYPE = [N -> setof[P]] % phone books
VB: TYPE = {b:B | (FORALL (x,y:N): x /= y => disjoint?(b(x), b(y)))}nm, x: VAR N
pn: VAR P
bk: VAR VB
FindPhone(bk,nm): setof[P] = bk(nm)
UnusedPhoneNum(bk,pn): bool = (FORALL nm: NOT member(pn, FindPhone(bk,nm)))
AddPhone(bk,nm,pn): VB =
IF UnusedPhoneNumb(bk,pn) THEN bk WITH [(nm) := add(pn, bk(nm))]
ELSE bk
ENDIF
…
45
Source: PhoneBook? Example, John Rushby
http://www.csl.sri.com/papers/wift-tutorial/
Logic for Computable Functions (LCF)-Style
Family of interactive theorem provers
Based on concepts of LCF (Robin Milner et al 1972), which used general-purpose programming language ML to allow users to write theorem-proving tactics
Library implements an abstract data type of proven theorems - new objects of this type can only be created using the functions which correspond to inference rules
If these functions are correctly implemented, all theorems proven in the system must be valid
Large system can be built on top of a small trusted kernel with “tactics” that automate many tasks
Includes HOL 4, HOL Light, Isabelle/HOL
Isabelle/HOL used to prove seL4 operating system kernel46
Isabelle/HOL proved a microkernel
“In 2009, the L4.verified project at NICTA produced the first formal proof of functional correctness of a general-purpose operating system kernel: the seL4 (secure embedded L4) microkernel.
The proof is constructed and checked in Isabelle/HOL and comprises over 200,000 lines of proof script to verify 8,700 lines of C and 600 lines of assembler.
The verification covers code, design, and implementation, and the main theorem states that the C code correctly implements the formal specification of the kernel.
The proof uncovered 160 bugs in the C code of the seL4 kernel, and about 150 issues in each of design and specification.”Source: Gerwin Klein et al, "seL4: Formal verification of an OS kernel". 22nd ACM Symposium on Operating System Principles. October 2009
47
Isabelle/HOL sample
lemma prime-not-square:
p prime (k: 0 < k m * m ≠ p * (k * k))
apply (induct m rule: nat-less-induct)
apply clarify
apply (frule prime-dvd-other-side; assumption)
apply (erule dvdE)
apply (simp add: nat-mult-eq-cancel-disj prime-nonzero)
apply (blast dest: rearrange reduction)
done
48
Source: “The Seventeen Provers of the World”
compiled by Freek Wiedijk (Isabelle/HOL script version)
(great for comparing some prover notations!)
Coq
Proof assistant for a higher-order logic (HOL), developed in France
Can define computational function in special language
Coq’s spec language is called “Gallina”
Coq can generate OCaml code from it
Rich type system
Uses “calculus of inductive constructions” (a HOL)
Native support for inductive datatypes + “calculus of constructions” (a higher-order typed lambda calculus by Thierry Coquand from ~1986)
Based on intuitionist logic (Brouwer et al) – must be able to construct. In intuitionist logic, “A ∨ ¬A” (law of excluded middle) is not an axiom. Can import “Classical” library
5 is an instance of the type “Z” (math integers)
Specification is a type for a program
If prove program of that type, then program meets spec
Defines a large set of “tactics”
Successes: Java Card EAL7 certified, CompCert C verified compiler
Gerard P Huet won 2013 ACM Software System Award for Coq
http://awards.acm.org/award_winners/huet_1246701.cfm49
Coq example
Lemma foo : ∀n, ble_nat 0 n = true. (* ∀n : naturals, 0 ≤ n *) Proof. intros. destruct n. (* Leaves two subgoals, which are discharged identically... *) Case "n=0". simpl. reflexivity. Case "n=Sn'". simpl. reflexivity. Qed.
50
Source: Benjamin C. Pierce et al., “Software Foundations”
http://www.cis.upenn.edu/~bcpierce/sf/Imp.html
Boolean satisfiability (SAT) solvers
Boolean SAT solvers are automated tools that:
Given predicate logic expressions (boolean variables, and, or, not)…
Find variable assignments to make true OR report unsatisfiable
Proven to be an NP-complete problem (first known example)
Cook, S.A., 1971, “The complexity of theorem proving procedures”, Proceedings Third Annual ACM Symp. on the Theory of Computing, pp. 151-158
But multiple algorithm breakthroughs (e.g., Chaff in 2001) now make SAT solvers remarkably fast for most real problems
Many good ones free & available as OSS
Other tools can be built on these or their approaches
Annual SAT competition; many use DIMACS CNF input format
Sample tools: Chaff*, MiniSAT, Sat4j, PicoSAT, … (*=not OSS)51
The rise of practical SAT solvers is a key technological breakthrough;
many tools have been rewritten to use SAT solvers.
DIMACS CNF format for Boolean SAT Solvers
Every propositional formula can be converted to conjunctive normal form (CNF):
An expression = 1+ clauses connected by “AND”
A clause = 1+ non-repeated terms connected with “OR”
A term = A boolean variable, possibly negated
DIMACS CNF represents CNF
Line-oriented, initial c=comment
First non-comment line is “problem”: p FORMAT #VARIABLES #CLAUSES
Lines represent clauses with whitespace-separated terms (number=boolean term)
For example, given this CNF expression:(x1 | -x5 | x4) &
(-x1 | x5 | x3 | x4) &
(-x3 | x4).
Its DIMACS CNF form could be:
c Here is a comment.
p cnf 5 3
1 -5 4 0
-1 5 3 4 0
-3 -4 0
52
DIMACS=Center for
Discrete Mathematics &
Theoretical Computer Science,
an NSF S&T center
More info here:
ftp://dimacs.rutgers.edu/pub/ challenge/satisfiability/doc/
Satisfiability Modulo Theories (SMT) solvers
SMT solvers are also automated tools
Given expressions in richer notation beyond predicate logic
Typically FOL + “theories” (variables may be integers, reals, etc.)
E.g., (x+y ≥ 0) ∧ (y > 0) is satisfiable with integers x=1,y=2
Reports satisfiable (“sat”) (maybe with satisfying variables) or “unsat” or “unknown” (e.g., ran out of time/memory)
To determine if “X is always true”, supply “not X”… returns unsat
Some can also provide proof (if can’t, how verify results?)
Internally similar to SAT solvers, may be built on one
Tools often theory-specific & restrict input language
Less flexible & more efficient vs. general theorem provers
Annual competition SMT-COMP, SMT-LIB input format
Sample tools: CVC4 (successor to CVC3), alt-ergo, STP, OpenSMT, Z3, Yices* (*=not OSS)53
SMT-LIB version 2 example (1 of 2)
> (set-logic QF_LIA) ; Basic arithmetic on integers
> (declare-fun x () Int) ; Functions x & y return Int
> (declare-fun y () Int)
> (assert (= (+ x (* 2 y)) 20)) ; Assert x+2*y = 20
> (assert (= (- x y) 2)) ; Assert x-y = 2
> (check-sat) ; Is this satisfiable?
sat ; Yes, it is.
> (get-value (x y)) ; What’s an example?
((x 8)(y 6)) ; Here’s one of many examples.
Source: David R. Cok, “The SMT-LIB v2 Language and Tools: A Tutorial”.
The “success” replies are omitted. The initial “>” is a prompt.
54
SMT-LIB version 2 example (2 of 2)
> (set-logic QF_UF)
> (declare-fun p () Bool)
> (declare-fun q () Bool)
> (declare-fun r () Bool)
> (assert (=> p q)) ; p -> q
> (assert (=> q r)) ; q -> r. This means p-> r.
> (assert (not (=> p r))) ; !(p -> r)
> (check-sat)
unsat ; Not possible given the previous assertions
Source: David R. Cok, “The SMT-LIB v2 Language and Tools: A Tutorial”
The “success” replies are omitted. The initial “>” is a prompt.
55
Model-checkers (aka property checkers)
Given a system model, exhaustively check if meets a given requirement
Requirement is often narrow property, often temporal requirement
System is represented as a finite-state machine (FSM)
Exhaustively explore state (conceptually)
Clever approaches orders-of-magnitude faster vs. brute force. E.g.:
Symbolically represent FSM, e.g., using binary decision diagrams (BDDs)
Abstraction (simplify system for this specific property)
Bounded model checking - unroll FSM for fixed number of steps (build on SAT)
Only shows true/false for that many steps!!
Counterexample guided abstraction refinement (CEGAR)
Pros: Fully automated & easy-to-use (compared to theorem-provers)
Cons: Can quickly become infeasible & often limited to narrow properties
Eventually state explosion can overwhelm clever optimizations
Model-checkers (excluding code analysis – discuss separately):
SPIN – verify distributed systems, Promela language (license probably not OSS)
Gerard Holzmann created of SPIN, won 2001 ACM Software System Award
DiVinE: Distributed execution, can accept C/C++ too (BSD 3-clause)
Others include NuSMV 2 (LGPL 2.1)56
A low-level data structure: Binary decision diagrams (BDDs)
BDD = data structure, can represent a Boolean function in compressed form
Can perform operations directly on BDDs
Easily determine equivalence & combine boolean functions
Variable order matters, heuristics help determine order
Some tools use BDDs to compute in reasonable time
Don Knuth: “one of the only really fundamental data structures that came out in the last twenty-five years”57
Source: Images from Wikipedia. Key paper: Randal E. Bryant. "Graph-Based Algorithms for
Boolean Function Manipulation". IEEE Transactions on Computers, C-35(8):677–691, 1986 (widely cited)
Promela: Mars Pathfinder model
Promela = Notation of the SPIN model-checker
Next slide shows Promela model of Mars Pathfinder scheduling algorithm (from SPIN source code)
Explains recurring reset problem during mission on Mars
Situation:
High priority process that consumes data produced by a low priority process
Data consumption and production happens under the protection of a mutex lock
Mutex lock conflicts with the scheduling priorities, can deadlock the system if high() starts up while low() has the lock set
This is called “priority inversion”
Model has 12 reachable states in the full (non-reduced) state space - two of which are deadlock states58
Promela: Mars Pathfinder example
mtype = { free, busy, idle, waiting, running };
show mtype h_state = idle;
show mtype l_state = idle;
show mtype mutex = free;
active proctype high() { /* can run any time */
end: do
:: h_state = waiting;
atomic { mutex == free -> mutex = busy };h_state = running;
/* critical section - consume data */
atomic { h_state = idle; mutex = free }od
}
active proctype low() provided (h_state == idle) { /* Note scheduling rule */
end: do
:: l_state = waiting;
atomic { mutex == free -> mutex = busy};l_state = running;
/* critical section - produce data */
atomic { l_state = idle; mutex = free }od
}
59
Source: SPIN source Version 6.2.5 (4 May 2013), file Test/pathfinder.pml
Abstract interpretation & symbolic execution
Abstract interpretation = sound approximation programs. I.E., a partial execution without performing all calculations
In some definitions, must be based on monotonic functions over ordered sets
E.G., track “is variable +, 0, or –”… that’s enough to determine signs from multiplication (without overflow)
Symbolic execution = analyze program by tracking symbolic rather than actual values60
So you want to prove that a program meets a specification…
61
So you want to prove that a program meets a specification…
Often must write program with that goal in mind
Challenges: Pointers & threads
Variable assignment can be handled, but complicates
Loops can be handled, but complicates (loop invariants)
Functional languages common (at least underneath), including Lisp, ML, OCaml, Haskell
Various tools exist that can be used to do this
Already seen tools that can prove programs, including PVS, HOL4, Isabelle/HOL, HOL Light, Coq
Following are some tools (and notations) specifically for it
E.g., ACL2, Toccata/ProVal (Why3), Frama-C + (Jessie or WP), JML (supported by many tools), SPARK, model-checking
As with everything else here, omits much62
ACL2
Automated theorem-prover (BSD 3-clause)
But often must guide via lemmas (“waypoints”)
Models programs in Lisp-based language
Can be used to model other implementations & hardware
Specializes in automatic proofs of recursive programs
“Industrial strength” version of Boyer–Moore theorem prover NQTHM
Given large set of proven patterns, repeatedly substitute
Creators Robert S. Boyer, Matt Kaufmann, & J. Strother Moore won 2005 ACM Software System Award (began in Edinburgh, Scotland, later at University of Texas, Austin)
Many successes, esp. hardware / microprocessor design (division, multiplication, etc.), proof of ivy63
ACL2 examples (1 of 2)
(defun factorial (n) ; Trivial factorial implementation
(if (zp n)
1
(* n (factorial (- n 1)))))
(thm (> (factorial n) 0)) ; Prove that factorial always produces >0
; Prove that append is associative, that is, that
; (append (append xs ys) zs) equals (append xs (append ys zs))
(thm (equal (append (append xs ys) zs)
(append xs (append ys zs))))
Source: http://tryacl2.org/
64
ACL2 examples (2 of 2)
(defun rev (xs) ; Return list in reverse order
(if (endp xs) nil
(append (rev (rest xs)) (list (first xs)))))
(defthm rev-rev ; Prove reverse(reverse(x))=x
(implies (true-listp xs)
(equal (rev (rev xs)) xs)))
Source: http://tryacl2.org/
65
Toccata (née ProVal?) approach
66
Krakatoa with
Jessie
WP
JML-annotated
Java
ACSL-annotated C
Why3
Eprover
CVC4
Automated
provers
Interactive
provers
Coq
PVS
…
…
Why3 can compute goals needed to
prove the code implements spec.
Encodes to & manages tools to prove goals.
Alt-Ergo
Isabelle/HOL
Z3
SPARK 2014
tools
gappa
Annotated
SPARK Ada
Specialty prover
for floating point
WhyML? program + spec
…
veriT
WP has short-circuits
for alt-ergo & Coq
Jessie
Frama-C
…
Plug-ins
Why3 Proof Session Example
67
Source: “Why3: Shepherd Your Herd of Provers” by Bobot et al.
Java Modeling Language (JML) example
public class BankingExample?{
/*@ spec_public */ private Integer balance;
//@ invariant balance != null
//@ invariant 0 <= balance && balance <= MAX_BALANCE
//@ ensures this.balance = 0
public BankingExample { balance = 0; } //@ requires amount != null
//@ requires 0 < amount && amount + balance < MAX_BALANCE
//@ modifies balance
//@ ensures this.balance = \old(this.balance) + amount
public void credit(Integer amount) {...}}
68
Source: “Automated Program Verification” 2011,
https://courses.cs.washington.edu/courses/cse331/11wi/lectures/lect20-program-verification.pdf
Many tools support JML, both for dynamic (run-time) checking & formal proofs, including
Krakatoa (with Jessie/Why3), OpenJML?, KeY?, Sireum/Kiasan, TACO, ESC/Java2..
Okay to refer to private data
in publicly-viewable spec
JML uses Hoare-style preconditions, post-conditions, and invariants. JML supports the design by contract paradigm, & builds on ideas from Eiffel, Larch, & Refinement Calculus.
ACSL (C) example: Swap
File swap.c:
/*@ requires \valid(a) && \valid(b);
@ ensures A: *a == \old(*b) ;
@ ensures B: *b == \old(*a) ;
@ assigns *a,*b ;
@*/
void swap(int *a,int *b)
{
int tmp = *a ;
return ;
}
69
Source: http://frama-c.com/wp.html
Precondition
Postconditions
ACSL = “ANSI/ISO C Specification Language”,
created for & used by Frama-C for C.
It’s inspired by the Java Modeling Language (JML)
used by many tools for Java.
ACSL (C) example: Binary search
/*@ requires n >= 0 && \valid (t+(0..n -1));
@ assigns \nothing ;
@ ensures -1 <= \result <= n -1;
@ behavior success:
@ ensures \result >= 0 ==> t[ \result ] == v;
@ behavior failure:
@ assumes t_is_sorted: \forall integer k1 , int k2;
@ 0 <= k1 <= k2 <= n -1 ==> t[k1] <= t[k2];
@ ensures \result == -1 ==>
@ \forall integer k; 0 <= k < n ==> t[k] != v;
@*/
int bsearch ( double t[], int n, double v) {
int l = 0, u = n -1;
/*@ loop invariant 0 <= l && u <= n -1;
@ for failure: loop invariant
@ \forall integer k; 0 <= k < n && t[k] == v ==> l <= k <= u;
@*/
while (l <= u ) {int m = l + (u-l )/2; // better than (l+u)/2
if (t[m] < v) l = m + 1;
else if (t[m] > v) u = m - 1;
else return m;
}
return -1;
}
70
Source: Baudin et al, “ACSL: ANSI/ISO C Specification Language Version 1.6” Example 2.23
Jessie vs. WP
Same purpose & general approach
Accept C+ACSL annotations, create proofs, apply weakest precondition (WP) approach, use Why/Why3
Different memory modeling of C in math
Jessie uses separation predicates inspired by separation logic
WP focuses on memory model parameterization
Different implementation approach
Jessie translates C directly into Why3’s language
WP designed to cooperate with other Frama-C plug-ins such as the value analysis plug-in
WP presumes there are no run-time errors (RTEs); use a separate plug-in (e.g., rte generation) to check for RTEs71
Some key WP options
Memory model (how map C memory values to math)
Hoare model: C variables mapped directly to logical variables. Very efficient, easy & concise proofs. Can’t use heap or pointer read/write
Typed model*: Heap values stored in separated, one for each atomic type. Pointers are indexes into these arrays
Bytes model: Heap is array of bytes, pointers are memory addresses. Very precise, but proof obligations hard to discharge automatically“Use the simpler models whenever… possible, and [use] more involved models on the remaining more complex parts”
Arithmetics models (how to model C arithmetic in math)
Natural Model*: Integer operations use mathematical integers
Machine Integer Model: Integer operations performed modulo; proof obligations hard to discharge
Real Model*: Floating-point operations have no rounding (unsound)
Float Model: Floating-point operations are mathematical with rounding, consistent with IEEE. Most automated provers can’t handle; use gappa72
Sample GUI: Frama-C and WP
73
Source: “WP 0.8” on Frama-C website
SPARK (Ada)
Designed for high-reliability applications
Originally targeted at highly constrained run-time environments
Pre-2014 version was subset/superset of Ada. Pre-2014 version:
No dynamic memory allocation
Until recently, no threads (has added Ravenscar)
Adds specification language in comments
Developed by Bernard Carré and Trevor Jennings; much done by Rod Chapman, principal engineer of Altran
Exploits Ada’s subtyping to automate many proofs
Ada’s strict typing & additional information on ranges makes proof automation easier than for some alternative implementation languages; often can enumerate all cases
Speed of C, but its strong typing counters errors & simplifies proofs
Successes reported, “Tokeneer” public, US lunar project “CubeSat”
SPARK 2014 version changes & relaxes many restrictions
Uses Ada 2012 aspect notation (instead of specially formatted comments)
Integrates with testing - contracts be proved, or they can be compiled and executed as run-time assertations
Bigger subset: Adds bounded types with dynamic bounds, recursion, etc.
SPARK 2014 has switched to Why3/Toccata approach – multi-prover supportSource: http://www.spark-2014.org
74
SPARK 2014 Example Spec (1 of 2)
pragma SPARK_Mode;
package Search is
type Index is range 1 .. 10;
type Element is new Integer;
type Arr is array (Index) of Element;
type Search_Result (Found : Boolean := False) is record
case Found is
when True =>
At_Index : Index;
when False =>
null;
end case;
end record;
function Value_Found_In_Range(A : Arr;
Val : Element; Low, Up : Index) return Boolean
is (for some J in Low .. Up => A(J) = Val);
75
SPARK 2014 Example Spec (2 of 2)
function Linear_Search(A : Arr; Val : Element) return Search_Result
with
Pre => Val >= 0,
Post => (if Linear_Search'Result.Found then
A (Linear_Search'Result.At_Index) = Val),
Contract_Cases =>
(A(1) = Val =>
Linear_Search'Result.At_Index = 1,
A(1) /= Val and then Value_Found_In_Range (A, Val, 2, 10) =>
Linear_Search'Result.Found,
(for all J in Arr'Range => A(J) /= Val) =>
not Linear_Search'Result.Found);
end Search;
76
SPARK 2014 Example Body
pragma SPARK_Mode;
package body Search is
function Linear_Search(A : Arr; Val : Element) return Search_Result is
Pos : Index'Base := A'First;
Res : Search_Result;
begin
while Pos <= A'Last loop
if A(Pos) = Val then
Res := (Found => True, At_Index => Pos);
return Res;
end if;
pragma Loop_Invariant(Pos in A'Range and then not Value_Found_In_Range(A, Val, A'First, Pos));
pragma Loop_Variant (Increases => Pos);
Pos := Pos + 1;
end loop;
Res := (Found => False);
return Res;
end Linear_Search;
end Search;
77
Source for SPARK 2014 example: SPARK 2014 Tutorial,
http://docs.adacore.com/spark2014-docs/html/ug/tutorial.html
Some other toolsuites
Dafny (Microsoft Research)
Programming language with built-in specification constructs
Dafny verifier is run as part of the compiler to verify functional correctness
Dafny compiler produces code for .NET platform
Appears to be OSS: Dafny & underlying Boogie are Microsoft Public License; Z3 is MIT license
KeY (http://www.key-project.org/)
Proofs of Java programs annotated with JML
OSS (GPL)
OpenJML (http://jmlspecs.sourceforge.net/)
Proofs of Java programs annotated with JML
Translates into SMT-LIB format, passes to backend SMT solvers
OSS (GPLv2)
KIV system
Specs can refine down to Java
Proprietary, free for noncommercial use78
Model-checking FM tools for programs (code analysis)
Model-checking can be applied to code, too:
In practice, often limited to looking for specific properties/defects (e.g., TOCTOU, temp files)
Approximations – can be sound (always finds problem under certain assumptions) but with false positives
Java PathFinder (NASA)
State software model checker for Java™ bytecode
Weird NASA license
MOPS – Analyze for very specific security vulnerabilities
For C, has analyzed Linux kernel & even Linux distro!
[Benjamin Schwarz et al, “Model Checking An Entire Linux Distribution for Security Violations” – 6 vulnerability patterns]
DiVinE (BSD 3-clause; accepts C/C++, builds on LLVM)
Other tools: BLAST, CPAchecker (Apache 2.0), Microsoft SLAM*, …79
MOPS – Checking a Linux distribution
“Model Checking An Entire Linux Distribution for Security Violations”
by Benjamin Schwarz, Hao Chen, David Wagner, Geoff Morrison*, Jacob West*, Jeremy Lin, Wei Tu (*=Fortify, rest are UC Berkeley); ACSAC 2005
Did software model checking for security properties on big scale
Entire Linux distro, 839 packages, 60 million LOC
Discovered 108 exploitable bugs
“MOPS errs on the conservative side: MOPS will catch all the bugs for a property (… it is sound, subject to certain requirements), but it [reports] spurious warnings.”
Looked for: TOCTTOU (filesystem race), Standard File Descriptors, Temporary Files, strncpy (misuse), Chroot Jails, Format String80
Abstract interpretation & symbolic execution
Example: Kestrell CodeHawk C Analyzer*
Given targeted (critical) vulnerability types…
Tries to mathematically prove the absence of those vulnerabilities in all relevant code, using abstract interpretation
If cannot prove it, warning issued with remaining proof obligations
GrammaTech CodeSonar*
Has built-in set of (critical) vulnerability types
Uses symbolic execution engine to explore program paths, reasoning about program variables & relations
Dataflow analysis prunes infeasible program paths
Procedure summaries are refined and compacted as-go81
Formalizing & verifying mathematical theorems (not code)
Mathematicians can make mistakes
Euclidean geometry: Omission of Pasch's axiom went unnoticed for 2000 years
Mathematicians normally “sketch” a proof without details
We can formalize math itself so it’s mechanically proved at every step
Freek Wiedijk has compiled info on several math formalization tools
“Formalizing 100 theorems” lists tools used to formalize math, & some theorems they’ve formalized - http://www.cs.ru.nl/~freek/100/
“The Seventeen Provers of the World” demos 17 tools in more detail
Many different tools
Some tools for formalizing math can also prove programs/specs, e.g., HOL Light, Isabelle, Coq, ACL2
Some specialized for just math proofs, e.g., Mizar & Metamath
Formalized math can sometimes be a basis for proving specs/programs82
Mizar & Metamath (tools for formalizing math)
Mizar (proprietary program, OSS theorems)
Notation designed to be similar to traditional math notation
Publish a math journal focused on formalization
Metamath (OSS programs, public domain theorems) – (I find this one intriguing!)
Only 1 built-in logic rule (substitution), ability to define axioms & proofs
Tiny kernel (can verify proofs with ~350 lines of Python)
Very general logic system, can then specify axioms & proofs to build up from there
Designed to be easier for non-mathematician (esp. computer-literate) to follow steps
Founded on a Alfred Tarski formalization exactly equivalent to traditional textbook formalization, but without “proper substitution” and “free variable” (eases mechanization)
Includes a modern “Principia Mathematica”-like set of proofs that’s easier to read than P.M.
States a very few axioms (sets & logic), e.g., modus ponens
Grows from there, e.g., proves basics about numbers & their properties
Proves 2+2=4 in 10 steps, transitively uses 2,452 subtheorems & 25,933 steps http://us.metamath.org/mpegif/2p2e4.html
You can learn a lot about math from its documentation!
Esp. “Metamath” book and “Metamath Proof Explorer” http://us.metamath.org
Both Mizar & Metamath’s set.mm use ZFC + Tarski’s axiom (for category theory)
Both provide little automation for proof creation
Don’t handle goal change well
Makes them much less applicable to software, which does change
But again, some of their math results may be useful in computing83
General caveats when considering formal methods
Many approaches require significant math knowledge
Education needed (comparable to engineers)
Some exceptions (e.g., sometimes model-checking & abstract interpretation, where spec can be pre-canned)
Typically must apply during development
To reduce requirements ambiguity – need to apply when creating requirements
All tools have limits – write so can easily apply to them
Only proves what you ask (“it doesn’t answer questions you don’t ask”†)
Only as good as their assumptions – are they justifiable?84
† Credit: Paul E. Black, “A Brief Introduction to Formal Methods”
Current status of formal methods
Formal methods already useful & used in some circumstances
Tool problems (often hard-to-use/outdated UI, lack of integration/standards, …)
Handling scale – full “level 2” rigor historically small programs
Can get larger by only applying at requirement level, analyzing models (e.g., design), just specialized properties, dropping soundness – useful!
Improved algorithms, more CPU power, combining algorithms, OSS, & more research have potential for scaling up in future
Hybrid approaches seem promising, e.g., the Toccata (ProVal) suite
You’ve sampled some the FM notations/tools available, e.g.:
Z, Alloy; Prover9, E theorem prover; Coq, HOL4, HOL Light, Isabelle/HOL, PVS; CVC4, alt-ergo; SPIN; ACL2, SPARK, Toccata (ProVal)
Valuable for high-assurance security & in some cases today
With potential for far larger applications
Tools/techniques can be repurposed, e.g., to increase assurance85
Open Proofs
86
What’s slowing FM maturation?
Much research & some use, but FM tools are often:
Hard to install, hard to learn to use
Hard to use, time-consuming, & don’t scale
Poorly integrate with other tools/existing environments
Need to mature FM if they’re to be broadly used
Hard problem, relatively few research $ ... but decades?
FM maturation hindered by “culture of secrecy”
Details of FM use often unpublished, classified
Details of FM tools (& the tools!) often unshared/lost
Result (broadly stated):
Researchers/toolmakers receive inadequate feedback
From developers & other researchers/toolmakers
Researchers/toolmakers waste time/$ rebuilding tools
Educators difficulty explaining (esp. without examples)
Developers don’t understand, uncertain value
Evaluators/end-users don’t know what to look for87
Over-generalized
Researchers/toolmakers need more than papers: LIMMAT to NANOSAT
Researchers/toolmakers suffer from lack of information
LIMMAT/NANOSAT developers: “From the publications alone, without access to the source code, various details were still unclear... what we did not realize, and which hardly could be deduced from the literature, was [an optimization] employed in GRASP and CHAFF [was critically important]... Only [when CHAFF's source code became available did] our unfortunate design decision became clear... The lesson learned is, that important details are often omitted in publications and can only be extracted from source code. It can be argued, that making source code of SAT solvers available is as important to the advancement of the field as publications”
88
Need: Working ecosystem
Researchers/Toolmakers/Educators
Learn details from others (papers often inadequate) – share code!!
Build on/experiment with existing tools (vs. rebuilding)
Developers of implementations to be proved
Learn from other developers
Build on/experiment with proven systems/components
Share issues with toolmakers (so tools can improve)
Evaluators/End-users
Evaluate evidence (determine adequacy, give feedback)
Evaluate other systems based on this experience89
Researchers/
Toolmakers/
Educators
Developers
Evaluators/
End-users
“Open proof” idea
“Open proof” (new term):
Source code, proofs, and required tools: OSS
Anyone can examine/critique, improve upon, collaborate with others for improvements
Not just software, but what’s proved & tools
Example for training, or as useful component
Extends OSS idea for high assurance
Enables legal collaboration
Similar to mathematics field
Method for speeding up tech transition
Encourage/require government-funded results be open proofs
By default – evaluate exceptions
Application of “open access” applied broadly
See: http://www.phdcomics.com/comics/archive.php?comicid=1533
Goal: Make supplier identity irrelevant
Don’t need everything to be an open proof
Examples & building blocks (inc. standards’ API)90
Some open proofs
“Tokeneer” (SPARK Ada)
seL4 microkernel (Isabelle/HOL)
ACL2 library (ACL2)More info:
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