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Tutorial ??
Formal Methods for
Hardware Verification:
Overview and Application to VHDL
Carlos Delgado Kloos, Peter T. Breue...
Introduction Outline ??
⋆ Formal Hardware Verification Approaches
⋆ Theorem Proving
⋆ Model Checking
⋆ Formal Reasoning wit...
Motivation Citation ??
“If you are faced by
a difficulty or a controversy in science,
an ounce of algebra is worth a ton of ...
Motivation Bryants comparison ??
simulate exhaustively a 256 bit RAM
⇒ 1080 possible combinations of input and state
⋆ use...
Motivation Pentium ??
The top ten reasons to buy a PENTIUM Machine:
10. Your current computer is too accurate.
9. You want...
Motivation Trends ??
⋆ systems are growing larger
⋆ systems are growing more complex
⋆ design teams are growing larger
⋆ t...
Theorem proving Classification ??
Systems that manipulate an object language
⋆ Term rewrite systems
⋆ Transformational syst...
Theorem proving Theorem provers ??
Systems that help to prove theorems
⋆ Proof checkers
a posteriori check (eg. MIZAR)
⋆ P...
Theorem proving Comparison ??
⋆ Degree of interaction
⋆ Object language: underlying logic
⋆ Meta language: command languag...
Theorem proving Nqthm ??
The Boyer-Moore Theorem Prover
⋆ Quantifier-free first-order classical logic with equality
(free va...
Theorem proving The LCF family ??
LCF = Logic for Computable Functions
⋆ Stanford LCF: proof checker with fixed commands (S...
Theorem proving LCF-related provers ??
⋆ Veritas: meta-language: Miranda, object-language: Higher-Order
Intuitionistic Log...
Theorem proving Theory ??
A formal logic consists of
⋆ a notation (a set of well-formed formulas)
⋆ a finite set of axioms
...
Theorem proving HOL expressions ??
The expressions in HOL can be:
⋆ constants 1: num, +: num->num->num
⋆ variables x: num,...
Theorem proving HOL types ??
The types in HOL can be:
⋆ atomic types bool, num
⋆ compound types num*bool, num->num->num
⋆ ...
Theorem proving HOL ??
The Object Language HOL comprises
⋆ Typed λ-Calculus (functions, including higher-order)
⋆ Polymorp...
Theorem proving Axioms ??
There are only 5 axioms in HOL: 4 for the theory of bool
⋆ ⊢ ∀t. (t=T) ∨ (t=F)
⋆ ⊢ ∀t1 t2. (t1⇒t...
Theorem proving Inference rules ??
There are only 8 primitive inference rules:
{t} ⊢ t ⊢ t=t ⊢ (λx.t1)t2 = t1[t2/x]
Γ1 ⊢ t...
Theorem proving Theories ??
All information (types, constants, axioms, theorems, etc.)
is hierarchically structured in the...
Theorem proving Proof styles ??
There are essentially two ways to proceed:
⋆ Forward proof:
(Primitive or derived) inferen...
Theorem proving Example (J. Joyce) ??
i1
i2
x o2
⋆ specify behavioural models for NAND and NOT
⋆ specify intended behaviou...
Theorem proving Example (Specs) ??
Behaviours:
⊢def NAND(i1,i2,o1) ≡ o1 = ¬(i1∧i2)
⊢def NOT(i1,o1) ≡ o1 = ¬ i1
⊢def ANDspe...
Theorem proving Example (Correctness) ??
⋆ Strong correctness
ANDimpl(i1,i2,o1) ≡ ANDspec(i1,i2,o1)
⋆ Weaker correctness
A...
Theorem proving Example (Proof) ??
1) ANDimpl(i1,i2,o1) {initial formula}
2) ∃x. NAND(i1,i2,x) ∧ NOT(x,o1) {def ANDimp}
3)...
Theorem proving Pros ??
⋆ generality
⋆ flexibility
⋆ expresiveness
⋆ exploitation of regularity, hierarchy and abstraction
...
Theorem proving Cons ??
⋆ long learning curve
⋆ large expertise needed
⋆ requires deep knowledge of mathematics and logic
...
Theorem proving Achievements ??
Several microprocessors have been verified
⋆ FM8501 (Nqthm, Warren Hunt, Univ. Texas, 1986)...
Theorem proving The future ??
⋆ higher temporal complexity (pipelines, asynch. systems, real-time)
⋆ higher data complexit...
Theorem proving More info about Nqthm ??
⋆ Computational Logic Inc.
1717 W. 6th St., Suite 290
Austin, TX 78703-4776, USA
...
Theorem proving More info about HOL ??
⋆ Cambridge Univ. Computer Laboratory
Pembroke Street, GB–Cambridge CB2 3QG, Englan...
Theorem proving More info about LAMBDA ??
⋆ LAMBDA: Logic and Mathematics Behind Design Automation
⋆ Abstract Hardware Ltd...
Model checking Model of Computation ??
⋆ Finite state systems are modeled by labelled state-transition graphs
(Kripke stru...
Model checking Temporal Logics ??
⋆ In a linear temporal logic, the operators describe events along a
single computation p...
Model checking CTL: theory ??
Every atomic formula is a CTL formula.
If f and g are CTL formulae, then so are
¬f (not f)
f...
Model checking CTL: some operators ??
M, s AF x M, s EF x M, s EG x
VDHL-Forum for CAD in Europe – Nantes, 24 April 1995 36
Model checking CTL: examples ??
⋆ AG(EF greenNS): always it is possible to get to the greenNS
state (at a traffic light) [li...
Model checking Model Checking Problem ??
⋆ Let M be the Kripke structure representing the behaviour of a
system,
⋆ let f b...
Model checking Model Checking Algorithm ??
M, s0 EGa ∧ AFb?
VDHL-Forum for CAD in Europe – Nantes, 24 April 1995 39
Model checking OBDDs ??
DAG-representation of Boolean functions
VDHL-Forum for CAD in Europe – Nantes, 24 April 1995 40
Model checking OBDDs ??
The importance of the variable ordering
VDHL-Forum for CAD in Europe – Nantes, 24 April 1995 41
Model checking OBDDs ??
⋆ S. Akers: Binary Decision Diagrams, IEEE Trans. Computers C–
27:6, June 78
⋆ R. Bryant: Graph-Ba...
Model checking Symbolic Model Checking ??
⋆ Representing state-transition graphs with OBDDs
⋆ The transition relation can ...
Model checking Pros ??
⋆ decision procedure completely automated: no proofs!
⋆ fast
⋆ counter-examples
⋆ symbolic techniqu...
Model checking Cons ??
⋆ specification is enumeration of desired properties
⋆ completeness problem
⋆ state explosion proble...
Model checking Achievements ??
⋆ Formal verification of the IEEE Futurebus+ cache consistency
protocol
(precise model define...
Model checking Some references ??
⋆ J. Burch, E. Clarke, et al.: Symbolic Model Checking: 1020 States
and Beyond, Conf. Lo...
Conclusion Other approaches ??
⋆ Symbolic trajectory evaluation
similar to conventional simulation
considers symbols rathe...
Conclusion Combining approaches ??
⋆ HOL with a model checker
⋆ HOL with COSMOS [Bryant, Seger]
⋆ HOL with VOSS [Joyce, Se...
Conclusion Conclusions ??
⋆ the field of formal methods is old
⋆ first breakthroughs obtained recently
⋆ still primarily aca...
Conclusion Citation ??
“It is now a well-established phenomenon
that what is highly abstract
for a generation of mathemati...
Semantics Classical VHDL simulations ??
A VHDL signal is associated with a driver.
0 1 2 3 4 . . . future time →
Signal
VH...
Semantics Example 1 ??
Initial driver.
0 1 2 3 4 ...
Signal
future time
X <= transport 1.0 after 3 ns
Driver is altered.
0...
Semantics Example 2 ??
Initial driver.
0 1 2 3 4 ...
Signal
future time
wait until X=1
Final driver:
0 1 2 3 4 ...
Signal
...
Semantics Semantics of VHDL ??
VHDL statements relate:
1. an old driver set to a new driver set;
2. an old current timepoi...
Semantics Example 3 ??
X=0
X=0
X=0
X=0
X=1
T=1
T=2
T=3
X=0X=0 T=0
X=0X=0 T=-1
X <= transport 1.0 after 2 ns
X=1
Time
Initi...
Semantics Example 4 ??
X=0X=0
X=1
T=1
T=2
T=3
X=0X=0 T=0
X=0X=0 T=-1
X=1
Time
Initial WL Final WL
wait until X=1
X=1
X=1
⋆...
Semantics Example 5 ??
X=0
X=1
X=0
X=0
X=1
Final WL
wait until X=1
X=0 T=1
T=2
T=3
X=0 T=0
X=0 T=-1
X=1
X=1
X=0 T=1
T=2
T=...
Semantics Process semantics ??
A process relates:
⋆ An initial worldline to a final worldline.
Note that:
⋆ The initial tim...
Semantics A simple oscillator circuit ??
?
?
X <=not X after 2 ns
wait on X
begin
X <= transport not X after 2 ns
wait on ...
Semantics Example 6 ??
X=0
X=1
X=0
X=0
X=1
Final WL
X=0 T=1
T=2
T=3
X=0 T=0
X=0 T=-1
X=1
X=1
X=0 T=1
T=2
T=3
X=0 T=0
X=0 T...
Semantics Example 7 ??
X=0
X=1
X=0
X=0
X=1
X=0 T=1
T=2
T=3
X=0 T=0
X=0 T=-1
X=1
X=1
X=0 T=1
T=2
T=3
X=0 T=0
X=0 T=-1
Time
...
Semantics Time and processes ??
A process relates the initial world line with T=0 and the ultimate
world line with T=∞, bu...
Semantics Process semantics ??
. . .
. . . .
. . . . . . ..
.
.
.
.
. . ..
.
.
.
.
. . ..
.
.
.
.
. . ..
.
.
.
. . ..
.
.
...
Semantics Processes and parallelism ??
⋆ Processes in parallel have the semantics of the intersection of
relations.
⋆ They...
Semantics A simple follower circuit ??
?
?
Y <= X after 1 ns
wait on X
wait on X;
Y <= transport X after 1 ns;
process beg...
Semantics Oscillator and follower in parallel ??
?
?
?
?
wait on X
Y <= X after 1 ns
wait on X
process
begin
X <= not X af...
Semantics Example 8 ??
X=0
X=1
X=0
X=1
Final WL
X=0 T=1
T=2
T=3
X=0 T=0
T=-1
X=1
X=1
X=0 T=1
T=2
T=3
X=0 T=0
T=-1
Time
Ini...
Semantics A useful theorem ??
⋆ Take the ultimate world line and feed it back in again to a process
as its initial world l...
Logic Plan ??
⋆ Insert predicative assertions and timing information in the gaps
between VHDL statements.
⋆ Predicates con...
Logic Meaning ??
The Hoare triple
Sρ : {P, T1} a {Q, T2}
means
if a begins to execute at time T1 and condition P holds the...
Logic Example 9 ??
Look at the oscillator process using the logic of termination.
{X= ⊙X,T}
X <= transport (not X) after 2...
Logic Meaning ??
The Hoare triple
S′ρ : {P, T1} a {Q, T2}
means
if a begins to execute at time T1 and condition P holds th...
Logic Example 10 ??
Now look using the logic of suspension. Suspension can only happen
in a blocked wait statement; everyt...
Logic Process logic ??
A process never terminates; it can only be suspended.
Suspension occurs within the process body, af...
Logic Example 11 ??
Each oscillator cycle takes 2ns. At this time, termination establishes
and thereafter re-establishes t...
Logic The use of a useful theorem ??
{⊙−2X= ⊙−1X=X= ⊙X ∧ even(T)} ∨ {⊙−2X= ⊙−1X=X= ⊙X ∧ odd(T)}
is an invariant of the osc...
Logic Logical Rules ??
We reason by deriving one Hoare triple of the programming logic from
earlier derived Hoare triples....
Logic The S-logic for termination - seq ??
[∀T ∈ [T1, T2]]
Sρ : {P, T1} a {QAT, T} Sρ : {QBT, T} b {R, T2}
Sρ : {P, T1} a ...
Logic Example 12 ??
Take two consecutive signal assignments, each delayed by 1ns.
After the second, it will be the case th...
Logic Example 13 ??
Waiting for 1ns makes the things that are promised to happen next,
happen.
Sx : {⊙x = 1, 3} null {⊙x =...
Logic The S-logic for termination - wait for ??
Sρ : {⊙P, T1} null {⊙Q, T2 − 1}
Sρ : {P, T1} wait for 1 {Q, T2}
Sρ : {P, T...
Logic The S-logic for termination - wait on ??
A wait on can be viewed as a loop:
wait on x = do wait for 1ns while x = ⊙−...
Logic The S-logic for termination - if ??
This logic requires us to be able to get to a desired postcondition
along either...
Logic The S-logic for termination - while ??
While loops can be read as sequences of if branches.
while c do b = if c then...
Logic The S-logic for termination - null ??
The simplest things are the hardest!
A null command won’t execute over a non-z...
Logic The S′-logic for suspension - seq ??
[∀T ∈ [T1, T2]]
Sρ : {P, T1} a {QT, T} S′ρ : {QT, T} b {R, T2}
S′ρ : {P, T1} a ...
Logic Other parts of the S′-logic for suspension ??
The sequence logic of suspension is the only part that refers back to
...
Logic The S′-logic for suspension - wait for ??
S′ρ : {P, T} wait for 1 {Q, T}
[⊙T(P → Q)] (9)
Example:
S′x : {x = 1, 3} w...
Logic Summary ??
The logic used here is weak in the sense that if we prove
S′ : {P, T1}foo{Q, T2}
then we have not proved ...
Logic Conclusion ??
Logic is useful for reasoning about the properties of VHDL descrip-
tions.
VDHL-Forum for CAD in Europ...
Algebra Another approach ??
Another approach to VHDL is to construct a process algebra.
An algebra is a set of equations a...
Algebra Synthesis ??
The algebra has a “pure” component that only refers to events and
processes, and a “code” component t...
Algebra State transition diagram for follower ??
'
T
c

©
PQ
RS
Y!0 X?0
Y!0 X?0
Y!1 X?1
Y!0 X?1 Y!1 X?0
Entry = Y!0 X?0 PQ...
Algebra Code/diagram equivalence ??
-
??
?
?
C!0 C!0
C!1
C!1 C!0
[C!
0
0];a
=
[C!
0
1];a
=
[C!
0
0];a
=
a = C = not C afte...
Algebra Summary ??
Algebras make reasoning about VHDL programs via equalities possi-
ble.
The transformation can go in bot...
Conclusion Conclusion ??
Formal methods allow VHDL to be handled in ways that correspond
to classical activities, but base...
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Tutorial: Formal Methods for Hardware Verification - Overview and Application to VDHL

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Tutorial: Formal Methods for Hardware Verification - Overview and Application to VDHL

  1. 1. Tutorial ?? Formal Methods for Hardware Verification: Overview and Application to VHDL Carlos Delgado Kloos, Peter T. Breuer Universidad Polit´ecnica de Madrid <{cdk,ptb}@dit.upm.es> VDHL-Forum for CAD in Europe – Nantes, 24 April 1995 1
  2. 2. Introduction Outline ?? ⋆ Formal Hardware Verification Approaches ⋆ Theorem Proving ⋆ Model Checking ⋆ Formal Reasoning with VHDL ⋆ Semantics ⋆ Logic ⋆ Algebra VDHL-Forum for CAD in Europe – Nantes, 24 April 1995 2
  3. 3. Motivation Citation ?? “If you are faced by a difficulty or a controversy in science, an ounce of algebra is worth a ton of verbal argument.” J.B.S. Haldane VDHL-Forum for CAD in Europe – Nantes, 24 April 1995 3
  4. 4. Motivation Bryants comparison ?? simulate exhaustively a 256 bit RAM ⇒ 1080 possible combinations of input and state ⋆ use all matter in galaxy to build computers (1017 kg) ⋆ let each computer have the size of an electron (10−30 kg) ⋆ let each computer simulate 1012 cases per second ⋆ start simulation at the time of Big Bang (1010 years ago) by now, we would have simulated 0, 05% of all cases VDHL-Forum for CAD in Europe – Nantes, 24 April 1995 4
  5. 5. Motivation Pentium ?? The top ten reasons to buy a PENTIUM Machine: 10. Your current computer is too accurate. 9. You want to get into the Guiness book as “Owner of Most Expensive Paperweight”. 8. Math errors add zest to life. 7. You need an alibi for the I.R.S. 6. You want to see what all the fuss is about. 5. You’ve always wondered what it would be like to be a plaintiff. 4. The “Intel Inside” logo matches your decor perfectly. 3. You no longer have to worry about CPU overheating. 2. You got a great deal from JPL. 1. It’ll probably work. VDHL-Forum for CAD in Europe – Nantes, 24 April 1995 5
  6. 6. Motivation Trends ?? ⋆ systems are growing larger ⋆ systems are growing more complex ⋆ design teams are growing larger ⋆ time to market is getting more critical VDHL-Forum for CAD in Europe – Nantes, 24 April 1995 6
  7. 7. Theorem proving Classification ?? Systems that manipulate an object language ⋆ Term rewrite systems ⋆ Transformational systems ⋆ Theorem provers There is a convergence of these kinds of systems VDHL-Forum for CAD in Europe – Nantes, 24 April 1995 7
  8. 8. Theorem proving Theorem provers ?? Systems that help to prove theorems ⋆ Proof checkers a posteriori check (eg. MIZAR) ⋆ Proof assistants user guided proof, strategies can be defined, forward and back- ward proof (eg. LCF, HOL, Isabelle, Veritas+) ⋆ Automatic theorem provers (eg. Nqthm) VDHL-Forum for CAD in Europe – Nantes, 24 April 1995 8
  9. 9. Theorem proving Comparison ?? ⋆ Degree of interaction ⋆ Object language: underlying logic ⋆ Meta language: command language ⋆ Kinds of proofs ⋆ Proof management VDHL-Forum for CAD in Europe – Nantes, 24 April 1995 9
  10. 10. Theorem proving Nqthm ?? The Boyer-Moore Theorem Prover ⋆ Quantifier-free first-order classical logic with equality (free variables are implicitly universally quantified) ⋆ ⋆ ⋆ VDHL-Forum for CAD in Europe – Nantes, 24 April 1995 10
  11. 11. Theorem proving The LCF family ?? LCF = Logic for Computable Functions ⋆ Stanford LCF: proof checker with fixed commands (Scott 71–72) ⋆ Edinburgh LCF: meta-language: ML, object-language: PPλ (Mil- ner 75–79) ⋆ Cambridge LCF: meta-lenguage: Standard ML, object-language: PPλ (improved) (Paulson 84) ⋆ G¨oteborg LCF: supports Martin-L¨of’s type theory (Petersson 82) ⋆ Cambridge HOL: meta-lenguage: ML, object-language: Higher- Order Logic (Gordon 80–) VDHL-Forum for CAD in Europe – Nantes, 24 April 1995 11
  12. 12. Theorem proving LCF-related provers ?? ⋆ Veritas: meta-language: Miranda, object-language: Higher-Order Intuitionistic Logic (Hanna, Daeche 85–) ⋆ Isabelle: meta-language: ML, object-language: parametrizable (Paulson 86–) ⋆ Lambda: meta-language: Poly-ML, object-language: Higher-Order Polymorphic Predicate Calculus of Partial Terms, interfaced to CAD- system VDHL-Forum for CAD in Europe – Nantes, 24 April 1995 12
  13. 13. Theorem proving Theory ?? A formal logic consists of ⋆ a notation (a set of well-formed formulas) ⋆ a finite set of axioms ⋆ a finite set of inference rules A formal proof is a sequence of well-formed formulas f1, f2, ..., fn, such that for all i ⋆ fi is an axiom, or ⋆ fi can be derived from {f1, f2, ..., fn} using an inference rule VDHL-Forum for CAD in Europe – Nantes, 24 April 1995 13
  14. 14. Theorem proving HOL expressions ?? The expressions in HOL can be: ⋆ constants 1: num, +: num->num->num ⋆ variables x: num, x: num->bool ⋆ abstractions λx.(λy.x+y) ⋆ applications (λx.(λy.x+y)1)2 VDHL-Forum for CAD in Europe – Nantes, 24 April 1995 14
  15. 15. Theorem proving HOL types ?? The types in HOL can be: ⋆ atomic types bool, num ⋆ compound types num*bool, num->num->num ⋆ polymorphic types ’a->’b, (’a->bool)->bool VDHL-Forum for CAD in Europe – Nantes, 24 April 1995 15
  16. 16. Theorem proving HOL ?? The Object Language HOL comprises ⋆ Typed λ-Calculus (functions, including higher-order) ⋆ Polymorphic objects (parametric polymorphism) ⋆ Higher-order Logic (quantifiers over values, predicates, etc.) It can be manipulated from the Metalanguage ML (which is quite similar). VDHL-Forum for CAD in Europe – Nantes, 24 April 1995 16
  17. 17. Theorem proving Axioms ?? There are only 5 axioms in HOL: 4 for the theory of bool ⋆ ⊢ ∀t. (t=T) ∨ (t=F) ⋆ ⊢ ∀t1 t2. (t1⇒t2)⇒(t2⇒t1)⇒(t1=t2) ⋆ ⊢ ∀t. (λx. t x)=t ⋆ ⊢ ∀P x. P x ⇒P(ǫ P) and one for the theory of ind ⋆ ⊢ ∃f. ONE ONE f ∧ ¬ ONTO f VDHL-Forum for CAD in Europe – Nantes, 24 April 1995 17
  18. 18. Theorem proving Inference rules ?? There are only 8 primitive inference rules: {t} ⊢ t ⊢ t=t ⊢ (λx.t1)t2 = t1[t2/x] Γ1 ⊢ t1 ⇒ t2 Γ2 ⊢ t1 Γ1 ∪ Γ2 ⊢ t2 Γ ⊢ t1=t2 Γ ⊢ (λx.t1)=(λx.t2) Γ ⊢ t2 Γ − {t1} ⊢ t1 ⇒ t2 Γ1 ⊢ t1=t2 Γ2 ⊢ t[t1] Γ1 ∪ Γ2 ⊢ t[t2] Γ ⊢ t Γ ⊢ t[s1, ... sn/’a, ... ’n] VDHL-Forum for CAD in Europe – Nantes, 24 April 1995 18
  19. 19. Theorem proving Theories ?? All information (types, constants, axioms, theorems, etc.) is hierarchically structured in theories: ⋆ pairs ⋆ natural numbers ⋆ lists ⋆ primitive recursion ⋆ arithmetic ⋆ trees ⋆ etc. VDHL-Forum for CAD in Europe – Nantes, 24 April 1995 19
  20. 20. Theorem proving Proof styles ?? There are essentially two ways to proceed: ⋆ Forward proof: (Primitive or derived) inference rules are applied to (axioms or) theorems until the desried theorem is proved ⋆ Backward proof: A goal (a sequent to be proved into a theorem) is successively decomposed into subgoals, until there are already proved theo- rems VDHL-Forum for CAD in Europe – Nantes, 24 April 1995 20
  21. 21. Theorem proving Example (J. Joyce) ?? i1 i2 x o2 ⋆ specify behavioural models for NAND and NOT ⋆ specify intended behaviour of AND ⋆ specify implementation of AND in terms of NAND and NOT ⋆ prove that implementation satisfies intended behaviour for AND VDHL-Forum for CAD in Europe – Nantes, 24 April 1995 21
  22. 22. Theorem proving Example (Specs) ?? Behaviours: ⊢def NAND(i1,i2,o1) ≡ o1 = ¬(i1∧i2) ⊢def NOT(i1,o1) ≡ o1 = ¬ i1 ⊢def ANDspec(i1,i2,o1) ≡ o1 = i1∧i2 Structure: ⊢def ANDimpl(i1,i2,o1) ≡ ∃x. NAND(i1,i2,x) ∧ NOT(x,o1) VDHL-Forum for CAD in Europe – Nantes, 24 April 1995 22
  23. 23. Theorem proving Example (Correctness) ?? ⋆ Strong correctness ANDimpl(i1,i2,o1) ≡ ANDspec(i1,i2,o1) ⋆ Weaker correctness ANDimpl(i1,i2,o1) ⇒ ANDspec(i1,i2,o1) VDHL-Forum for CAD in Europe – Nantes, 24 April 1995 23
  24. 24. Theorem proving Example (Proof) ?? 1) ANDimpl(i1,i2,o1) {initial formula} 2) ∃x. NAND(i1,i2,x) ∧ NOT(x,o1) {def ANDimp} 3) NAND(i1,i2,x) ∧ NOT(x,o1) {strip off ∃x} 4) NAND(i1,i2,x) {left conjunt of 3)} 5) x=¬(i1∧i2) {def NAND} 6) NOT(x,o1) {right conjunt of 3)} 7) o1=¬x {def NOT} 8) o1=¬(¬(i1∧i2)) {subst. 5) in 7)} 9) o1=(i1∧i2) {simplify ¬¬t=t} 10) AND(i1,i2,o1) {def AND} 11) ANDimpl(i1,i2,o1) ⇒ AND(i1,i2,o1) {discharge assumption 1)} VDHL-Forum for CAD in Europe – Nantes, 24 April 1995 24
  25. 25. Theorem proving Pros ?? ⋆ generality ⋆ flexibility ⋆ expresiveness ⋆ exploitation of regularity, hierarchy and abstraction ⋆ proof security ⋆ user extensibility VDHL-Forum for CAD in Europe – Nantes, 24 April 1995 25
  26. 26. Theorem proving Cons ?? ⋆ long learning curve ⋆ large expertise needed ⋆ requires deep knowledge of mathematics and logic ⋆ tedious proofs VDHL-Forum for CAD in Europe – Nantes, 24 April 1995 26
  27. 27. Theorem proving Achievements ?? Several microprocessors have been verified ⋆ FM8501 (Nqthm, Warren Hunt, Univ. Texas, 1986) ⋆ Viper (HOL, Avra Cohn, Univ. Cambridge, 1988) ⋆ Tamarack-3 (HOL, Jeff Joyce, Univ. Cambridge, 1989) ⋆ AVM–1 (HOL, Phil Windley, Univ. California, Davis, 1990) VDHL-Forum for CAD in Europe – Nantes, 24 April 1995 27
  28. 28. Theorem proving The future ?? ⋆ higher temporal complexity (pipelines, asynch. systems, real-time) ⋆ higher data complexity (IEEE floating point std, ...) ⋆ higher-level specifications (hardware/software verification, ...) ⋆ verification of classes of designs (microproc. families, ...) ⋆ verification of an ATM network (Fairisle) ⋆ TkHolWorkbench (a GUI for HOL) ⋆ BDDs in HOL ⋆ HOL 2000 initiative ⋆ several logic embeddings (CCS, TLA, Unity, Noden, ...) VDHL-Forum for CAD in Europe – Nantes, 24 April 1995 28
  29. 29. Theorem proving More info about Nqthm ?? ⋆ Computational Logic Inc. 1717 W. 6th St., Suite 290 Austin, TX 78703-4776, USA ⋆ <Software-Request@cli.com> http://www.cli.com/ ⋆ R.S. Boyer, J.S. Moore: A Computational Logic Handbook, Academic Press 1988 VDHL-Forum for CAD in Europe – Nantes, 24 April 1995 29
  30. 30. Theorem proving More info about HOL ?? ⋆ Cambridge Univ. Computer Laboratory Pembroke Street, GB–Cambridge CB2 3QG, England (UK) ⋆ Sara Kalvala <sk@cl.cam.ac.uk> http://www.comlab.ox.ac.uk/archive/formal-methods/hol.html (info) http://lal.cs.byu.edu/lal/getting-hol.html (tool) ⋆ M. Gordon, T. Melham (eds.): Intr. to HOL: A Theorem Proving Environment for Higher Order Logic, Cambridge Univ. Press 93 ⋆ 8th International Workshop on Higher Order Logic Theorem Prov- ing and its Applications, Utah September 11–14, 1995 VDHL-Forum for CAD in Europe – Nantes, 24 April 1995 30
  31. 31. Theorem proving More info about LAMBDA ?? ⋆ LAMBDA: Logic and Mathematics Behind Design Automation ⋆ Abstract Hardware Ltd. The Howell Building, Brunel University Science Park GB–Uxbridge UB8 3PH, England (UK) ⋆ <lambda@ahl.co.uk> VDHL-Forum for CAD in Europe – Nantes, 24 April 1995 31
  32. 32. Model checking Model of Computation ?? ⋆ Finite state systems are modeled by labelled state-transition graphs (Kripke structures) ⋆ Given an initial state, the structure can be unwound to an in- finite tree (computation tree), whose paths represent possible behaviours ⋆ A temporal logic is used to express properties of behaviours ⋆ Verification is carried out by exhaustive search of the state space ⋆ To speed up verification, efficient representation techniques are used based on binary decision diagrams. VDHL-Forum for CAD in Europe – Nantes, 24 April 1995 32
  33. 33. Model checking Temporal Logics ?? ⋆ In a linear temporal logic, the operators describe events along a single computation path ⋆ In a branching temporal logic, the operators describe events along several computation paths ⋆ path quantifiers: A (for every path), E: there exists a path ⋆ linear time operators: Xf (f holds next time) Ff (f holds sometime in the future) Gf (f holds globally in the future) fUg (f holds until g holds) VDHL-Forum for CAD in Europe – Nantes, 24 April 1995 34
  34. 34. Model checking CTL: theory ?? Every atomic formula is a CTL formula. If f and g are CTL formulae, then so are ¬f (not f) f ∧ g (f and g) AXf (for all paths, f holds in the next state) EXf (for some path, in which f holds in the next state) AFf (for all paths, f holds eventually) EFf (for some path, in which f holds eventually) AGf (for all paths, f holds in every state) EGf (for some path, in which f holds in every state) A(fUg) (for all paths, f holds until g holds) E(fUg) (for some path, f holds until g holds) VDHL-Forum for CAD in Europe – Nantes, 24 April 1995 35
  35. 35. Model checking CTL: some operators ?? M, s AF x M, s EF x M, s EG x VDHL-Forum for CAD in Europe – Nantes, 24 April 1995 36
  36. 36. Model checking CTL: examples ?? ⋆ AG(EF greenNS): always it is possible to get to the greenNS state (at a traffic light) [liveness property] ⋆ AG(¬(greenNS∧greenEW )): never both greenNS and greenEW hold (both lights are green)[safety property] ⋆ AG(req ⇒ AF ack): if a request occurs, it will be eventually acknowledged ⋆ EF(started∧¬ready): it is possible to get to a state where started holds, but ready does not hold. VDHL-Forum for CAD in Europe – Nantes, 24 April 1995 37
  37. 37. Model checking Model Checking Problem ?? ⋆ Let M be the Kripke structure representing the behaviour of a system, ⋆ let f be a temporal logic formula representing a property to check, ⋆ the objective is to find all states s of M that satisfy the formula f: M, s f ⋆ in fact, there exist very efficient algorithms for the logic CTL (Clarke, Emerson and Sistla, ACM TOPLAS 8:2, 1986) ⋆ complexity linear in size of M and f VDHL-Forum for CAD in Europe – Nantes, 24 April 1995 38
  38. 38. Model checking Model Checking Algorithm ?? M, s0 EGa ∧ AFb? VDHL-Forum for CAD in Europe – Nantes, 24 April 1995 39
  39. 39. Model checking OBDDs ?? DAG-representation of Boolean functions VDHL-Forum for CAD in Europe – Nantes, 24 April 1995 40
  40. 40. Model checking OBDDs ?? The importance of the variable ordering VDHL-Forum for CAD in Europe – Nantes, 24 April 1995 41
  41. 41. Model checking OBDDs ?? ⋆ S. Akers: Binary Decision Diagrams, IEEE Trans. Computers C– 27:6, June 78 ⋆ R. Bryant: Graph-Based Algorithms for Boolean Function Ma- nipulation, IEEE Trans. Computers C–35:8, August 86 VDHL-Forum for CAD in Europe – Nantes, 24 April 1995 42
  42. 42. Model checking Symbolic Model Checking ?? ⋆ Representing state-transition graphs with OBDDs ⋆ The transition relation can be seen as a boolean formula ⋆ T(v1, ..., vn, v′ 1, ..., v′ n), where (v1, ..., vn) represents the current state and (v′ 1, ..., v′ n) the next state ⋆ T is represented by a OBDD. VDHL-Forum for CAD in Europe – Nantes, 24 April 1995 43
  43. 43. Model checking Pros ?? ⋆ decision procedure completely automated: no proofs! ⋆ fast ⋆ counter-examples ⋆ symbolic techniques allow to handle a big number of states VDHL-Forum for CAD in Europe – Nantes, 24 April 1995 44
  44. 44. Model checking Cons ?? ⋆ specification is enumeration of desired properties ⋆ completeness problem ⋆ state explosion problem ⋆ large data paths can introduce many states ⋆ no taking advantage of parametrization ⋆ temporal formulas can be difficult to understand VDHL-Forum for CAD in Europe – Nantes, 24 April 1995 45
  45. 45. Model checking Achievements ?? ⋆ Formal verification of the IEEE Futurebus+ cache consistency protocol (precise model defined, bugs found) Clarke et al. 93 ⋆ ⋆ ⋆ VDHL-Forum for CAD in Europe – Nantes, 24 April 1995 46
  46. 46. Model checking Some references ?? ⋆ J. Burch, E. Clarke, et al.: Symbolic Model Checking: 1020 States and Beyond, Conf. Logic in Computer Science 1990. ⋆ O. Coudert, J.C. Madre, C. Berthet: Verifying Teporal Properties of Sequential Machines without Building their State Diagram, DIMACS Worksh. Computer-Aided Verification, June 1990 ⋆ Th. Filkorn: A Method for Symbolic Verification of Synchronous Circuits, CHDL’91, April 1991 VDHL-Forum for CAD in Europe – Nantes, 24 April 1995 47
  47. 47. Conclusion Other approaches ?? ⋆ Symbolic trajectory evaluation similar to conventional simulation considers symbols rather than actual values VOSS, COSMOS ⋆ Automata-based Systems COSPAN ⋆ Tautology checkers Checking of combinational circuits TACHE VDHL-Forum for CAD in Europe – Nantes, 24 April 1995 48
  48. 48. Conclusion Combining approaches ?? ⋆ HOL with a model checker ⋆ HOL with COSMOS [Bryant, Seger] ⋆ HOL with VOSS [Joyce, Seger] ⋆ embedding VHDL in HOL [van Tassel, Kropf] ⋆ a VDHL simulator in Acl2 [Boyer, Hunt] ⋆ interfacing HOL to GENESIL (silicon compiler) [Rushby] ⋆ Prevail calling Nqthm or Tache [Borrione, Pierre] VDHL-Forum for CAD in Europe – Nantes, 24 April 1995 49
  49. 49. Conclusion Conclusions ?? ⋆ the field of formal methods is old ⋆ first breakthroughs obtained recently ⋆ still primarily academic work ⋆ increasing interest of industry (Siemens, Bull, some CAD vendors) ⋆ still a long way to go VDHL-Forum for CAD in Europe – Nantes, 24 April 1995 50
  50. 50. Conclusion Citation ?? “It is now a well-established phenomenon that what is highly abstract for a generation of mathematicians is just commonplace for the next one.” J. Dieudonn´e VDHL-Forum for CAD in Europe – Nantes, 24 April 1995 51
  51. 51. Semantics Classical VHDL simulations ?? A VHDL signal is associated with a driver. 0 1 2 3 4 . . . future time → Signal VHDL signal assignments write to the driver. VHDL wait statements read the driver and suspend execution of the process until a time determined by the evolving condition of the driver. VDHL-Forum for CAD in Europe – Nantes, 24 April 1995 52
  52. 52. Semantics Example 1 ?? Initial driver. 0 1 2 3 4 ... Signal future time X <= transport 1.0 after 3 ns Driver is altered. 0 1 2 3 4 ... Signal future time VDHL-Forum for CAD in Europe – Nantes, 24 April 1995 53
  53. 53. Semantics Example 2 ?? Initial driver. 0 1 2 3 4 ... Signal future time wait until X=1 Final driver: 0 1 2 3 4 ... Signal future time Driver is essentially unaltered, but time has moved on. VDHL-Forum for CAD in Europe – Nantes, 24 April 1995 54
  54. 54. Semantics Semantics of VHDL ?? VHDL statements relate: 1. an old driver set to a new driver set; 2. an old current timepoint to a new current timepoint; 3. a previous history to an extended history. H × DS × T ↔ H × DS × T The combination of history plus driver set is called a ‘world line’. WL × T ↔ WL × T VDHL-Forum for CAD in Europe – Nantes, 24 April 1995 55
  55. 55. Semantics Example 3 ?? X=0 X=0 X=0 X=0 X=1 T=1 T=2 T=3 X=0X=0 T=0 X=0X=0 T=-1 X <= transport 1.0 after 2 ns X=1 Time Initial WL Final WL ⋆ A transport assignment re- lates two worldlines and two current timepoints. ⋆ The timepoint has to be the same either side, because the statement takes no physical time to execute. VDHL-Forum for CAD in Europe – Nantes, 24 April 1995 56
  56. 56. Semantics Example 4 ?? X=0X=0 X=1 T=1 T=2 T=3 X=0X=0 T=0 X=0X=0 T=-1 X=1 Time Initial WL Final WL wait until X=1 X=1 X=1 ⋆ A wait statement relates two worldlines with the same sig- nal values – for the signals of the controlling process. ⋆ Other signals may differ in any way possible. ⋆ But time moves on. VDHL-Forum for CAD in Europe – Nantes, 24 April 1995 57
  57. 57. Semantics Example 5 ?? X=0 X=1 X=0 X=0 X=1 Final WL wait until X=1 X=0 T=1 T=2 T=3 X=0 T=0 X=0 T=-1 X=1 X=1 X=0 T=1 T=2 T=3 X=0 T=0 X=0 T=-1 Time Initial WL Intermediate WL X=0 X=0 X<=1 after 2 ns; ⋆ Two statements in se- quence compose via re- lational composition. VDHL-Forum for CAD in Europe – Nantes, 24 April 1995 58
  58. 58. Semantics Process semantics ?? A process relates: ⋆ An initial worldline to a final worldline. Note that: ⋆ The initial timepoint is zero. The final timepoint is ∞ ⋆ The body of the process repeats ad infinitum. VDHL-Forum for CAD in Europe – Nantes, 24 April 1995 59
  59. 59. Semantics A simple oscillator circuit ?? ? ? X <=not X after 2 ns wait on X begin X <= transport not X after 2 ns wait on X end VDHL-Forum for CAD in Europe – Nantes, 24 April 1995 60
  60. 60. Semantics Example 6 ?? X=0 X=1 X=0 X=0 X=1 Final WL X=0 T=1 T=2 T=3 X=0 T=0 X=0 T=-1 X=1 X=1 X=0 T=1 T=2 T=3 X=0 T=0 X=0 T=-1 Time Initial WL Intermediate WL X=0 X=0 X<=not X after 2 ns; wait on X VDHL-Forum for CAD in Europe – Nantes, 24 April 1995 61
  61. 61. Semantics Example 7 ?? X=0 X=1 X=0 X=0 X=1 X=0 T=1 T=2 T=3 X=0 T=0 X=0 T=-1 X=1 X=1 X=0 T=1 T=2 T=3 X=0 T=0 X=0 T=-1 Time Initial WL X=0 X=0 X=0 Final WL X=0 Initial WL X=1 X<=¬ X after 2 ns; X <=¬ X after 2 ns; X=1 X=1 X=1 X=0 X=0 X=0 X=0 X=1 X=1 process begin X <= transport ¬ X after 2 ns ; wait on X ; end wait on X wait on X T=1 T=2 T=3 T=4 T=5 T=5 T=4 T=3 T=2 T=1 VDHL-Forum for CAD in Europe – Nantes, 24 April 1995 62
  62. 62. Semantics Time and processes ?? A process relates the initial world line with T=0 and the ultimate world line with T=∞, but we are interested in what happens before then. A logical treatment will require two kinds of logic: ⋆ execution until termination; ⋆ execution until suspension. Suspension corresponds to looking at intermediate worldlines. Pressing ‘Ctrl-Z’. VDHL-Forum for CAD in Europe – Nantes, 24 April 1995 63
  63. 63. Semantics Process semantics ?? . . . . . . . . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . . . . . .. . . . . . . .. . . . . . . . . . .. . . . . WL0 WL1 WL2 T=3 T=2 T=1 T=0 T=1 T=2 T=3 X=1 X=1 X=0 X=0 X=0 X=1 X=1 X=0 X=1 X=1 X=0 X=0 X=1 X=1 A process relates developing worldlines to each other VDHL-Forum for CAD in Europe – Nantes, 24 April 1995 64
  64. 64. Semantics Processes and parallelism ?? ⋆ Processes in parallel have the semantics of the intersection of relations. ⋆ They have to agree on how world lines change and how long the change takes. ⋆ Knowledge of the driver set, history, and time are all shared ‘in- stantaneously’ between processes. VDHL-Forum for CAD in Europe – Nantes, 24 April 1995 65
  65. 65. Semantics A simple follower circuit ?? ? ? Y <= X after 1 ns wait on X wait on X; Y <= transport X after 1 ns; process begin end VDHL-Forum for CAD in Europe – Nantes, 24 April 1995 66
  66. 66. Semantics Oscillator and follower in parallel ?? ? ? ? ? wait on X Y <= X after 1 ns wait on X process begin X <= not X after 2 ns; end process begin Y <= X after 1 ns; wait on X; end wait on X; X <= not X after 2 ns VDHL-Forum for CAD in Europe – Nantes, 24 April 1995 67
  67. 67. Semantics Example 8 ?? X=0 X=1 X=0 X=1 Final WL X=0 T=1 T=2 T=3 X=0 T=0 T=-1 X=1 X=1 X=0 T=1 T=2 T=3 X=0 T=0 T=-1 Time Initial WL Intermediate WL wait on XY<= X after 1 ns; X<= ¬X after 2 lns; wait on X X=1 X=1 X=1 X=1 X=1 Y=1 Y=1 Y=1 Y=1 Y=0 Y=0 Y=0 Y=1 Y=1 Y=1 Y=1 Y=1 Y=0 Y=0 Y=0 VDHL-Forum for CAD in Europe – Nantes, 24 April 1995 68
  68. 68. Semantics A useful theorem ?? ⋆ Take the ultimate world line and feed it back in again to a process as its initial world line, then the same world line comes out again. ⋆ So, look for invariant world lines. ⋆ Especially helpful when calculating for parallel processes. ⋆ A world line developed by process 1 can be used as a background against which process 2 is evaluated. VDHL-Forum for CAD in Europe – Nantes, 24 April 1995 69
  69. 69. Logic Plan ?? ⋆ Insert predicative assertions and timing information in the gaps between VHDL statements. ⋆ Predicates contain temporal modalities: “it will rain tomorrow”. ⋆ ⊙(x = 1) means “x = 1 will hold in the next instant”. This is the same as ⊙x = 1. ⋆ x = ⊙x means “x will change in the next instant”. ⋆ Timed pre- and post- assertions {p, t1} s {q, t2} across statements s are connected via a formal programming logic of triples. VDHL-Forum for CAD in Europe – Nantes, 24 April 1995 70
  70. 70. Logic Meaning ?? The Hoare triple Sρ : {P, T1} a {Q, T2} means if a begins to execute at time T1 and condition P holds then, then, if it finishes at time T2, Q will hold then. This is the logic of termination, denoted by S. VDHL-Forum for CAD in Europe – Nantes, 24 April 1995 71
  71. 71. Logic Example 9 ?? Look at the oscillator process using the logic of termination. {X= ⊙X,T} X <= transport (not X) after 2 ns ; {X= ⊙X= ⊙2X= ⊙3X,T} wait on X ; {⊙−2X= ⊙−1X=X= ⊙X,T+2} The final condition (X has been stable and now will be stable with a different value) has been forced by the initial condition (X is mo- mentarily stable). Note that the initial condition is re-established as the second part of the final condition. VDHL-Forum for CAD in Europe – Nantes, 24 April 1995 72
  72. 72. Logic Meaning ?? The Hoare triple S′ρ : {P, T1} a {Q, T2} means if a begins to execute at time T1 and condition P holds then, then, if it is suspended at time T2, Q will hold then. This is the logic of suspension, denoted by S′. VDHL-Forum for CAD in Europe – Nantes, 24 April 1995 73
  73. 73. Logic Example 10 ?? Now look using the logic of suspension. Suspension can only happen in a blocked wait statement; everything else takes zero time. Start with the condition established under the logic of termination., {⊙−2X= ⊙−1X=X= ⊙X,T} X <= transport (not X) after 2 ns ; {⊙−2X= ⊙−1X=X= ⊙X= ⊙2X= ⊙3X,T} wait on X ; {⊙−2X= ⊙−1X=X= ⊙X,T} ∨ {⊙−2X= ⊙−1X=X= ⊙X,T+1} The final condition is that X either has just changed or is just about to change. VDHL-Forum for CAD in Europe – Nantes, 24 April 1995 74
  74. 74. Logic Process logic ?? A process never terminates; it can only be suspended. Suspension occurs within the process body, after some non-negative number of executions of the body to termination. VDHL-Forum for CAD in Europe – Nantes, 24 April 1995 75
  75. 75. Logic Example 11 ?? Each oscillator cycle takes 2ns. At this time, termination establishes and thereafter re-establishes the condition: {⊙−2X= ⊙−1X=X= ⊙X,T} and then suspension sets up {⊙−2X= ⊙−1X=X= ⊙X,T} ∨ {⊙−2X= ⊙−1X=X= ⊙X,T+1} VDHL-Forum for CAD in Europe – Nantes, 24 April 1995 76
  76. 76. Logic The use of a useful theorem ?? {⊙−2X= ⊙−1X=X= ⊙X ∧ even(T)} ∨ {⊙−2X= ⊙−1X=X= ⊙X ∧ odd(T)} is an invariant of the oscillator process body under the termination and suspension semantics. VDHL-Forum for CAD in Europe – Nantes, 24 April 1995 77
  77. 77. Logic Logical Rules ?? We reason by deriving one Hoare triple of the programming logic from earlier derived Hoare triples. The rules of reasoning take the form of top bottom [condition] in which the bottom is allowed to be derived from the top when condition holds. Several hypotheses may appear: top1 top2 top3 . . . bottom [condition] or none bottom [condition] in which case the rule represents an axiom. VDHL-Forum for CAD in Europe – Nantes, 24 April 1995 78
  78. 78. Logic The S-logic for termination - seq ?? [∀T ∈ [T1, T2]] Sρ : {P, T1} a {QAT, T} Sρ : {QBT, T} b {R, T2} Sρ : {P, T1} a ; b {R, T2} [QA → QB] (1) If a sequence a;b runs to termination between times T1 and T2, then it does so by running a from T1 to termination at some intermediate time T, then running b to termination at T2. VDHL-Forum for CAD in Europe – Nantes, 24 April 1995 79
  79. 79. Logic Example 12 ?? Take two consecutive signal assignments, each delayed by 1ns. After the second, it will be the case that x is planned to be equal to 2 next. Sx : {true, 3} x <= 1 after 1ns {⊙x = 1, 3} Sx : {true, 3} x <= 2 after 1ns {⊙x = 2, 3} Sx : {true, 3} x <= 1 after 1ns ; x <= 2 after 1ns {⊙x = 2, 3} The precondition for the second assignment is true, so it does not matter what condition the first statement sets up. VDHL-Forum for CAD in Europe – Nantes, 24 April 1995 80
  80. 80. Logic Example 13 ?? Waiting for 1ns makes the things that are promised to happen next, happen. Sx : {⊙x = 1, 3} null {⊙x = 1, 3} Sx : {⊙x = 1, 3} wait for 1 {x = 1, 4} Waiting for 2ns is waiting for 1ns twice. Sx : {⊙x = 1, 3} wait for 1 ; wait for 1 {⊙−1x = 1, 5} Sx : {⊙x = 1, 3} wait for 2 {⊙−1x = 1, 5} VDHL-Forum for CAD in Europe – Nantes, 24 April 1995 81
  81. 81. Logic The S-logic for termination - wait for ?? Sρ : {⊙P, T1} null {⊙Q, T2 − 1} Sρ : {P, T1} wait for 1 {Q, T2} Sρ : {P, T1} wait for 1 ; wait for n {Q, T2} Sρ : {P, T1} wait for n + 1 {Q, T2} (2) A wait for 1ns will terminate (in 1ns). The (local) conditions that hold then are those that are promised to hold now. Longer waits are sequences of shorter ones. VDHL-Forum for CAD in Europe – Nantes, 24 April 1995 82
  82. 82. Logic The S-logic for termination - wait on ?? A wait on can be viewed as a loop: wait on x = do wait for 1ns while x = ⊙−1x Sρ : {P∧x=⊙x, T1} wait for 1; wait on x {Q, T2} Sρ : {P∧x=⊙x, T1} wait for 1 {Q, T2} Sρ : {P, T1} wait on x {Q, T2} (3) To run a wait to termination between times T1 and T2 either the waited on variable has to be about to change, in which case we do a wait for 1ns and terminate at T2=T1+1, or it isn’t, in which case we have to wait for 1ns and then wait longer. VDHL-Forum for CAD in Europe – Nantes, 24 April 1995 83
  83. 83. Logic The S-logic for termination - if ?? This logic requires us to be able to get to a desired postcondition along either path down an if. In each branch we can assume the appropriate extra precondition. Sρ : {P ∧ c, T1} b1 {Q, T2} Sρ : {P ∧ ¬c, T1} b0 {Q, T2} Sρ : {P, T1}if c then b1 else b0{Q, T2} (4) Example: Sxy : {true ∧ x = 0, 4} y <= 1 after 1ns {y = 0, 4} Sxy : {true ∧ x = 0, 4} y <= 2 after 1ns {y = 0, 4} Sxy : {true, 4}if x = 0 then y <= 1 after 1ns else y <= 2 after 1ns{y = 0, 4} VDHL-Forum for CAD in Europe – Nantes, 24 April 1995 84
  84. 84. Logic The S-logic for termination - while ?? While loops can be read as sequences of if branches. while c do b = if c then b; while c do b else null Sρ : {P ∧ c, T1} b; while x do b {Q, T2} Sρ : {P ∧ ¬c, T1} null {Q, T2} Sρ : {P, T1}while c do b{Q, T2} (5) Example: Sx : {true ∧ x = 0, 4} wait on x; while x = 0 do wait on x {x = 0, 5} Sx : {true ∧ x = 0, 4} null {x = 0, 5} Sx : {true, 4}while x = 0 do wait on x{x = 0, 5} VDHL-Forum for CAD in Europe – Nantes, 24 April 1995 85
  85. 85. Logic The S-logic for termination - null ?? The simplest things are the hardest! A null command won’t execute over a non-zero time, so anything we care to say about this (impossible) situation is valid. Sρ : {P, T1} null {Q, T2} [T1 = T2] (6) Over a zero time interval, a null command does nothing, so getting from precondition P to postcondition Q requires that P entails Q at that time. Sρ : {P, T} null {Q, T} [⊙T(P → Q)] (7) VDHL-Forum for CAD in Europe – Nantes, 24 April 1995 86
  86. 86. Logic The S′-logic for suspension - seq ?? [∀T ∈ [T1, T2]] Sρ : {P, T1} a {QT, T} S′ρ : {QT, T} b {R, T2} S′ρ : {P, T1} a {R, T2} S′ρ : {P, T1} a ; b {R, T2} (8) If a sequence a;b runs to suspension between times T1 and T2, then it does so by either 1. running a from T1 to suspension at T2, or 2. running a to completion at some intermediate time T, then running b to suspension at T2. VDHL-Forum for CAD in Europe – Nantes, 24 April 1995 87
  87. 87. Logic Other parts of the S′-logic for suspension ?? The sequence logic of suspension is the only part that refers back to the termination logic. In general, suspension logic is simpler than termination logic. ⋆ Many constructs cannot suspend at all, so have no rules for rea- soning about suspensions! NULL and signal assignment are examples of constructs that can- not suspend. WHILE loops can only suspend in the body. ⋆ WAIT statements cannot exit under suspension logic. They have to be suspended strictly before they exit. VDHL-Forum for CAD in Europe – Nantes, 24 April 1995 88
  88. 88. Logic The S′-logic for suspension - wait for ?? S′ρ : {P, T} wait for 1 {Q, T} [⊙T(P → Q)] (9) Example: S′x : {x = 1, 3} wait for 1 {x = 0, 3} [⊙3(x = 1 → x = 0)] VDHL-Forum for CAD in Europe – Nantes, 24 April 1995 89
  89. 89. Logic Summary ?? The logic used here is weak in the sense that if we prove S′ : {P, T1}foo{Q, T2} then we have not proved that statement foo will suspend at time T2. We have proved that if it is suspended at time T2, then condition Q will hold then. (Ditto for termination). But processes can be suspended at any time. So the logic is always applicable. VDHL-Forum for CAD in Europe – Nantes, 24 April 1995 90
  90. 90. Logic Conclusion ?? Logic is useful for reasoning about the properties of VHDL descrip- tions. VDHL-Forum for CAD in Europe – Nantes, 24 April 1995 91
  91. 91. Algebra Another approach ?? Another approach to VHDL is to construct a process algebra. An algebra is a set of equations asserting behavioural equivalences between different code fragments. For example: x <= 2 after 3ns; x <= 1 after 2ns = x <= 1 after 2ns The algebra can be used to prove or disprove equivalences between different formulations. VDHL-Forum for CAD in Europe – Nantes, 24 April 1995 92
  92. 92. Algebra Synthesis ?? The algebra has a “pure” component that only refers to events and processes, and a “code” component that contains only VHDL. Generally, algebra expressions are mixed (impure). A pure process algebra description can be represented as a state transition diagram. The diagram can be transformed slowly via the algebraic laws into VHDL code. What comes out is code that implements the state transition diagram. VDHL-Forum for CAD in Europe – Nantes, 24 April 1995 93
  93. 93. Algebra State transition diagram for follower ?? ' T c © PQ RS Y!0 X?0 Y!0 X?0 Y!1 X?1 Y!0 X?1 Y!1 X?0 Entry = Y!0 X?0 PQ PQ = Y!0 X?0 PQ | Y!0 X?1 RS RS = Y!1 X?1 RS | Y!1 X?0 PQ VDHL-Forum for CAD in Europe – Nantes, 24 April 1995 94
  94. 94. Algebra Code/diagram equivalence ?? - ?? ? ? C!0 C!0 C!1 C!1 C!0 [C! 0 0];a = [C! 0 1];a = [C! 0 0];a = a = C = not C after 1ns; wait on C; a VDHL-Forum for CAD in Europe – Nantes, 24 April 1995 95
  95. 95. Algebra Summary ?? Algebras make reasoning about VHDL programs via equalities possi- ble. The transformation can go in both directions. We are beginning to find that algebra/transition diagram specifica- tions are useful starting points for the synthesis of VHDL code. The VHDL code can be formally derived by a calculus of refinement from the initial diagram. VDHL-Forum for CAD in Europe – Nantes, 24 April 1995 96
  96. 96. Conclusion Conclusion ?? Formal methods allow VHDL to be handled in ways that correspond to classical activities, but based on secure foundations. VDHL-Forum for CAD in Europe – Nantes, 24 April 1995 97

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