Formal Specification and Verification
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1 Formal Specification and Verification Model Checking with Temporal Logic Bernhard Beckert Based on a lecture by Wolfgang Ahrendt and Reiner Hähnle at Chalmers University, Göteborg Formal Specification and Verification: Model Checking B. Beckert 1 / 43
2 Model Checking with Spin model name.pml correctness properties SPIN -a verifier pan.c C compiler executable verifier pan either or random/ interactive / simulation guided failing run model.trail errors: 0 Formal Specification and Verification: Model Checking B. Beckert 2 / 43
3 Stating Correctness Properties model name.pml correctness properties Correctness properties can be stated within, or outside, the model. Formal Specification and Verification: Model Checking B. Beckert 3 / 43
4 Stating Correctness Properties model name.pml correctness properties Correctness properties can be stated within, or outside, the model. stating properties within model, using assertion statements Formal Specification and Verification: Model Checking B. Beckert 3 / 43
5 Stating Correctness Properties model name.pml correctness properties Correctness properties can be stated within, or outside, the model. stating properties within model, using assertion statements meta labels end labels accept labels progress labels Formal Specification and Verification: Model Checking B. Beckert 3 / 43
6 Stating Correctness Properties model name.pml correctness properties Correctness properties can be stated within, or outside, the model. stating properties within model, using assertion statements meta labels end labels accept labels progress labels stating properties outside model, using never claims temporal logic formulas Formal Specification and Verification: Model Checking B. Beckert 3 / 43
7 Stating Correctness Properties model name.pml correctness properties Correctness properties can be stated within, or outside, the model. stating properties within model, using assertion statements meta labels end labels accept labels progress labels stating properties outside model, using never claims temporal logic formulas (today s main topic) Formal Specification and Verification: Model Checking B. Beckert 3 / 43
8 Stating Correctness Properties model name.pml correctness properties Correctness properties can be stated within, or outside, the model. stating properties within model, using assertion statements meta labels end labels accept labels (briefly) progress labels stating properties outside model, using never claims (briefly) temporal logic formulas (today s main topic) Formal Specification and Verification: Model Checking B. Beckert 3 / 43
9 Preliminaries fairness accept labels, accepting runs Formal Specification and Verification: Model Checking B. Beckert 4 / 43
10 Preliminaries I: Fairness Does the following Promela model necessarily terminate? byte n = 0; bool flag = f a l s e ; active proctype P() { do :: flag - > break; :: e l s e -> n = 5 - n; od } active proctype Q() { flag = true } Formal Specification and Verification: Model Checking B. Beckert 6 / 43
11 Preliminaries I: Fairness Does the following Promela model necessarily terminate? byte n = 0; bool flag = f a l s e ; active proctype P() { do :: flag - > break; :: e l s e -> n = 5 - n; od } active proctype Q() { flag = true } Termination guaranteed only if scheduling is (weakly) fair! Formal Specification and Verification: Model Checking B. Beckert 6 / 43
12 Preliminaries I: Fairness Does the following Promela model necessarily terminate? byte n = 0; bool flag = f a l s e ; active proctype P() { do :: flag - > break; :: e l s e -> n = 5 - n; od } active proctype Q() { flag = true } Termination guaranteed only if scheduling is (weakly) fair! Definition (Weak Fairness) A run is called weakly fair iff the following holds: each continuously executable statement is executed eventually. Formal Specification and Verification: Model Checking B. Beckert 6 / 43
13 Preliminaries II: Acceptance Cycles Definition (Accept Location) A location marked with an accept label of the form acceptxxx: is called an accept location. Formal Specification and Verification: Model Checking B. Beckert 7 / 43
14 Preliminaries II: Acceptance Cycles Definition (Accept Location) A location marked with an accept label of the form acceptxxx: is called an accept location. Accept locations can be used to specify cyclic behavior. Formal Specification and Verification: Model Checking B. Beckert 7 / 43
15 Preliminaries II: Acceptance Cycles Definition (Accept Location) A location marked with an accept label of the form acceptxxx: is called an accept location. Accept locations can be used to specify cyclic behavior. Definition (Acceptance Cycle) A run which infinitely often passes through an accept location is called an acceptance cycle. Formal Specification and Verification: Model Checking B. Beckert 7 / 43
16 Preliminaries II: Acceptance Cycles Definition (Accept Location) A location marked with an accept label of the form acceptxxx: is called an accept location. Accept locations can be used to specify cyclic behavior. Definition (Acceptance Cycle) A run which infinitely often passes through an accept location is called an acceptance cycle. Acceptance cycles are mainly used in never claims (see below), to define forbidden behavior of infinite kind. Formal Specification and Verification: Model Checking B. Beckert 7 / 43
17 Model Checking of Temporal Properties many correctness properties not expressible by assertions Formal Specification and Verification: Model Checking B. Beckert 8 / 43
18 Model Checking of Temporal Properties many correctness properties not expressible by assertions today: model checking of properties formulated in temporal logic Formal Specification and Verification: Model Checking B. Beckert 8 / 43
19 Model Checking of Temporal Properties many correctness properties not expressible by assertions today: model checking of properties formulated in temporal logic Remark: in this course, temporal logic is synonymous to linear temporal logic (LTL) Formal Specification and Verification: Model Checking B. Beckert 8 / 43
20 Beyond Assertions Assertions only talk about the state at their own location in the code. Formal Specification and Verification: Model Checking B. Beckert 10 / 43
21 Beyond Assertions Assertions only talk about the state at their own location in the code. Example: mutual exclusion expressed by adding assertion into each critical section. critical ++; assert ( critical <= 1 ); critical - -; Formal Specification and Verification: Model Checking B. Beckert 10 / 43
22 Beyond Assertions Assertions only talk about the state at their own location in the code. Example: mutual exclusion expressed by adding assertion into each critical section. critical ++; assert ( critical <= 1 ); critical - -; Drawbacks: no separation of concerns (model vs. correctness property) Formal Specification and Verification: Model Checking B. Beckert 10 / 43
23 Beyond Assertions Assertions only talk about the state at their own location in the code. Example: mutual exclusion expressed by adding assertion into each critical section. critical ++; assert ( critical <= 1 ); critical - -; Drawbacks: no separation of concerns (model vs. correctness property) changing assertions is error prone (easily out of synch) Formal Specification and Verification: Model Checking B. Beckert 10 / 43
24 Beyond Assertions Assertions only talk about the state at their own location in the code. Example: mutual exclusion expressed by adding assertion into each critical section. critical ++; assert ( critical <= 1 ); critical - -; Drawbacks: no separation of concerns (model vs. correctness property) changing assertions is error prone (easily out of synch) easy to forget assertions: correctness property might be violated at unexpected locations Formal Specification and Verification: Model Checking B. Beckert 10 / 43
25 Beyond Assertions Assertions only talk about the state at their own location in the code. Example: mutual exclusion expressed by adding assertion into each critical section. critical ++; assert ( critical <= 1 ); critical - -; Drawbacks: no separation of concerns (model vs. correctness property) changing assertions is error prone (easily out of synch) easy to forget assertions: correctness property might be violated at unexpected locations many interesting properties not expressible via assertions Formal Specification and Verification: Model Checking B. Beckert 10 / 43
26 Temporal Correctness Properties properties more conveniently expressed as global properties, rather than assertions: Formal Specification and Verification: Model Checking B. Beckert 12 / 43
27 Temporal Correctness Properties properties more conveniently expressed as global properties, rather than assertions: Mutual Exclusion critical <= 1 holds throughout the run Formal Specification and Verification: Model Checking B. Beckert 12 / 43
28 Temporal Correctness Properties properties more conveniently expressed as global properties, rather than assertions: Mutual Exclusion critical <= 1 holds throughout the run Array Index within Bounds (given array a of length len) 0 <= i <= len-1 holds throughout the run Formal Specification and Verification: Model Checking B. Beckert 12 / 43
29 Temporal Correctness Properties properties more conveniently expressed as global properties, rather than assertions: Mutual Exclusion critical <= 1 holds throughout the run Array Index within Bounds (given array a of length len) 0 <= i <= len-1 holds throughout the run properties impossible to express via assertions: Formal Specification and Verification: Model Checking B. Beckert 12 / 43
30 Temporal Correctness Properties properties more conveniently expressed as global properties, rather than assertions: Mutual Exclusion critical <= 1 holds throughout the run Array Index within Bounds (given array a of length len) 0 <= i <= len-1 holds throughout the run properties impossible to express via assertions: Absence of Deadlock If some processes try to enter their critical section, eventually one of them does so. Formal Specification and Verification: Model Checking B. Beckert 12 / 43
31 Temporal Correctness Properties properties more conveniently expressed as global properties, rather than assertions: Mutual Exclusion critical <= 1 holds throughout the run Array Index within Bounds (given array a of length len) 0 <= i <= len-1 holds throughout the run properties impossible to express via assertions: Absence of Deadlock If some processes try to enter their critical section, eventually one of them does so. Absence of Starvation If one process tries to enter its critical section, eventually that process does so. Formal Specification and Verification: Model Checking B. Beckert 12 / 43
32 Temporal Correctness Properties properties more conveniently expressed as global properties, rather than assertions: Mutual Exclusion critical <= 1 holds throughout the run Array Index within Bounds (given array a of length len) 0 <= i <= len-1 holds throughout the run properties impossible to express via assertions: Absence of Deadlock If some processes try to enter their critical section, eventually one of them does so. Absence of Starvation If one process tries to enter its critical section, eventually that process does so. all these are temporal properties Formal Specification and Verification: Model Checking B. Beckert 12 / 43
33 Temporal Correctness Properties properties more conveniently expressed as global properties, rather than assertions: Mutual Exclusion critical <= 1 holds throughout the run Array Index within Bounds (given array a of length len) 0 <= i <= len-1 holds throughout the run properties impossible to express via assertions: Absence of Deadlock If some processes try to enter their critical section, eventually one of them does so. Absence of Starvation If one process tries to enter its critical section, eventually that process does so. all these are temporal properties use temporal logic Formal Specification and Verification: Model Checking B. Beckert 12 / 43
34 Boolean Temporal Logic talking about numerical variables (like in critical <= 1 or 0 <= i <= len-1) requires variation of propositional temporal logic which we call Boolean temporal logic: Boolean expressions (over Promela variables), rather than propositions, form basic building blocks of the logic Formal Specification and Verification: Model Checking B. Beckert 13 / 43
35 Boolean Temporal Logic over Promela Set For BTL of Boolean Temporal Formulas (simplified) all Promela variables and constants of type bool/bit are For BTL Formal Specification and Verification: Model Checking B. Beckert 15 / 43
36 Boolean Temporal Logic over Promela Set For BTL of Boolean Temporal Formulas (simplified) all Promela variables and constants of type bool/bit are For BTL if e1 and e2 are numerical Promela expressions, then all of e1==e2, e1!=e2, e1<e2, e1<=e2, e1>e2, e1>=e2 are For BTL Formal Specification and Verification: Model Checking B. Beckert 15 / 43
37 Boolean Temporal Logic over Promela Set For BTL of Boolean Temporal Formulas (simplified) all Promela variables and constants of type bool/bit are For BTL if e1 and e2 are numerical Promela expressions, then all of e1==e2, e1!=e2, e1<e2, e1<=e2, e1>e2, e1>=e2 are For BTL if P is a process and l is a label in P, then P@l is For BTL (P@l reads P is at l ) Formal Specification and Verification: Model Checking B. Beckert 15 / 43
38 Boolean Temporal Logic over Promela Set For BTL of Boolean Temporal Formulas (simplified) all Promela variables and constants of type bool/bit are For BTL if e1 and e2 are numerical Promela expressions, then all of e1==e2, e1!=e2, e1<e2, e1<=e2, e1>e2, e1>=e2 are For BTL if P is a process and l is a label in P, then P@l is For BTL (P@l reads P is at l ) if φ and ψ are formulas For BTL, then all of are For BTL! φ, φ && ψ, φ ψ, φ > ψ, φ < > ψ [ ]φ, <>φ, φ U ψ Formal Specification and Verification: Model Checking B. Beckert 15 / 43
39 Semantics of Boolean Temporal Logic A run σ through a Promela model M is a chain of states L 0, I 0 L 1, I 1 L 2, I 2 L 3, I 3 L 4, I 4 L j maps each running process to its current location counter. From L j to L j+1, only one of the location counters has advanced (exception: channel rendezvous). I j maps each variable in M to its current value. Formal Specification and Verification: Model Checking B. Beckert 16 / 43
40 Semantics of Boolean Temporal Logic A run σ through a Promela model M is a chain of states L 0, I 0 L 1, I 1 L 2, I 2 L 3, I 3 L 4, I 4 L j maps each running process to its current location counter. From L j to L j+1, only one of the location counters has advanced (exception: channel rendezvous). I j maps each variable in M to its current value. Arithmetic and relational expressions are interpreted in states as expected; e.g. L j, I j = x<y iff I j (x) < I j (y) Formal Specification and Verification: Model Checking B. Beckert 16 / 43
41 Semantics of Boolean Temporal Logic A run σ through a Promela model M is a chain of states L 0, I 0 L 1, I 1 L 2, I 2 L 3, I 3 L 4, I 4 L j maps each running process to its current location counter. From L j to L j+1, only one of the location counters has advanced (exception: channel rendezvous). I j maps each variable in M to its current value. Arithmetic and relational expressions are interpreted in states as expected; e.g. L j, I j = x<y iff I j (x) < I j (y) L j, I j = P@l iff L j (P) is the location labeled with l Formal Specification and Verification: Model Checking B. Beckert 16 / 43
42 Semantics of Boolean Temporal Logic A run σ through a Promela model M is a chain of states L 0, I 0 L 1, I 1 L 2, I 2 L 3, I 3 L 4, I 4 L j maps each running process to its current location counter. From L j to L j+1, only one of the location counters has advanced (exception: channel rendezvous). I j maps each variable in M to its current value. Arithmetic and relational expressions are interpreted in states as expected; e.g. L j, I j = x<y iff I j (x) < I j (y) L j, I j = P@l iff L j (P) is the location labeled with l Evaluating other formulas For BTL in runs σ: see lecture 2. Formal Specification and Verification: Model Checking B. Beckert 16 / 43
43 Boolean Temporal Logic Support in Spin Spin supports Boolean temporal logic Formal Specification and Verification: Model Checking B. Beckert 18 / 43
44 Boolean Temporal Logic Support in Spin Spin supports Boolean temporal logic but Formal Specification and Verification: Model Checking B. Beckert 18 / 43
45 Boolean Temporal Logic Support in Spin Spin supports Boolean temporal logic but arithmetic operators (+,-,*,/,...), relational operators (==,!=,<,<=,...), label operators cannot appear directly in TL formulas given to Spin Formal Specification and Verification: Model Checking B. Beckert 18 / 43
46 Boolean Temporal Logic Support in Spin Spin supports Boolean temporal logic but arithmetic operators (+,-,*,/,...), relational operators (==,!=,<,<=,...), label operators cannot appear directly in TL formulas given to Spin instead Boolean expressions must be abbreviated using #define Formal Specification and Verification: Model Checking B. Beckert 18 / 43
47 Safety Properties formulas of the form [ ]φ are called safety properties Formal Specification and Verification: Model Checking B. Beckert 20 / 43
48 Safety Properties formulas of the form [ ]φ are called safety properties state that something good, φ, is guaranteed throughout each run Formal Specification and Verification: Model Checking B. Beckert 20 / 43
49 Safety Properties formulas of the form [ ]φ are called safety properties state that something good, φ, is guaranteed throughout each run special case: [ ] ψ states that something bad, ψ, never happens Formal Specification and Verification: Model Checking B. Beckert 20 / 43
50 Safety Properties formulas of the form [ ]φ are called safety properties state that something good, φ, is guaranteed throughout each run special case: [ ] ψ states that something bad, ψ, never happens example: [](critical <= 1) Formal Specification and Verification: Model Checking B. Beckert 20 / 43
51 Safety Properties formulas of the form [ ]φ are called safety properties state that something good, φ, is guaranteed throughout each run special case: [ ] ψ states that something bad, ψ, never happens example: [](critical <= 1) it is guaranteed throughout each run that at most one process visits its critical section Formal Specification and Verification: Model Checking B. Beckert 20 / 43
52 Safety Properties formulas of the form [ ]φ are called safety properties state that something good, φ, is guaranteed throughout each run special case: [ ] ψ states that something bad, ψ, never happens example: [](critical <= 1) it is guaranteed throughout each run that at most one process visits its critical section or equivalently: more than one process visiting its critical section will never happen Formal Specification and Verification: Model Checking B. Beckert 20 / 43
53 Applying Temporal Logic to Critical Section Problem We want to verify [](critical<=1) as correctness property of: active proctype P() { do :: /* non - critical activity */ atomic {! incriticalq ; incriticalp = true } critical ++; /* critical activity */ critical - -; incriticalp = f a l s e od } /* similarly for process Q */ Formal Specification and Verification: Model Checking B. Beckert 22 / 43
54 Model Checking a Safety Property with jspin 1. add #define mutex (critical <= 1) to Promela file 2. open Promela file 3. enter []mutex in LTL text field 4. select Translate to create a never claim, corresponding to the negation of the formula 5. ensure Safety is selected 6. select Verify 7. (if necessary) select Stop to terminate too long verification Formal Specification and Verification: Model Checking B. Beckert 24 / 43
55 Never Claims you may ignore them, but if you are interested: a never claim tries to show the user wrong it defines, in terms of Promela, all violations of a wanted correctness property it is semantically equivalent to the negation of the wanted correctness property jspin adds the negation for you using Spin directly, you have to add the negation yourself Formal Specification and Verification: Model Checking B. Beckert 25 / 43
56 Model Checking a Safety Property with Spin directly Command Line Execution make sure #define mutex (critical <= 1) is in safety1.pml > spin -a -f "!([] mutex )" safety1. pml > gcc -DSAFETY - o pan pan. c >./ pan Formal Specification and Verification: Model Checking B. Beckert 27 / 43
57 Liveness Properties formulas of the form <>φ are called liveness properties Formal Specification and Verification: Model Checking B. Beckert 29 / 43
58 Liveness Properties formulas of the form <>φ are called liveness properties state that something good, φ, eventually happens in each run Formal Specification and Verification: Model Checking B. Beckert 29 / 43
59 Liveness Properties formulas of the form <>φ are called liveness properties state that something good, φ, eventually happens in each run example: <>csp (with csp a variable only true in the critical section of P) Formal Specification and Verification: Model Checking B. Beckert 29 / 43
60 Liveness Properties formulas of the form <>φ are called liveness properties state that something good, φ, eventually happens in each run example: <>csp (with csp a variable only true in the critical section of P) in each run, process P visits its critical section eventually Formal Specification and Verification: Model Checking B. Beckert 29 / 43
61 Applying Temporal Logic to Starvation Problem We want to verify <>csp as correctness property of: active proctype P() { do :: /* non - critical activity */ atomic {! incriticalq ; incriticalp = true } csp = true ; /* critical activity */ csp = f a l s e ; incriticalp = f a l s e od } /* similarly for process Q */ /* here using csq */ Formal Specification and Verification: Model Checking B. Beckert 31 / 43
62 Model Checking a Liveness Property with jspin 1. open Promela file 2. enter <>csp in LTL text field 3. select Translate to create a never claim, corresponding to the negation of the formula 4. ensure that Acceptance is selected (Spin will search for accepting cycles through the never claim) 5. for the moment uncheck Weak Fairness (see discussion below) 6. select Verify Formal Specification and Verification: Model Checking B. Beckert 33 / 43
63 Verification Fails Verification fails. Why? Formal Specification and Verification: Model Checking B. Beckert 34 / 43
64 Verification Fails Verification fails. Why? The liveness property on one process had no chance. Not even weak fairness was switched on! Formal Specification and Verification: Model Checking B. Beckert 34 / 43
65 Model Checking Liveness with Weak Fairness! Always switch Weak Fairness on when checking for liveness! 1. open Promela file 2. enter <>csp in LTL text field 3. select Translate to create a never claim, corresponding to the negation of the formula 4. ensure that Acceptance is selected (Spin will search for accepting cycles through the never claim) 5. ensure Weak Fairness is checked 6. select Verify Formal Specification and Verification: Model Checking B. Beckert 36 / 43
66 Model Checking Lifeness with Spin directly Command Line Execution > spin - a - f "!csp " liveness1. pml > gcc -o pan pan.c >./ pan -a -f Formal Specification and Verification: Model Checking B. Beckert 38 / 43
67 Verification Fails Verification fails again. Why? Formal Specification and Verification: Model Checking B. Beckert 40 / 43
68 Verification Fails Verification fails again. Why? Weak fairness is still too weak. Formal Specification and Verification: Model Checking B. Beckert 40 / 43
69 Verification Fails Verification fails again. Why? Weak fairness is still too weak. Note that!incriticalq is not continuously executable! Formal Specification and Verification: Model Checking B. Beckert 40 / 43
70 Temporal MC Without Ghost Variables We want to verify mutual exclusion without using ghost variables #define && bool incriticalp = f a l s e, incriticalq = f a l s e ; active proctype P() { do :: atomic {! incriticalq ; incriticalp = true } cs: /* critical activity */ incriticalp = f a l s e od } /* similarly for process Q */ /* with same label cs: */ Formal Specification and Verification: Model Checking B. Beckert 42 / 43
71 Temporal MC Without Ghost Variables We want to verify mutual exclusion without using ghost variables #define && bool incriticalp = f a l s e, incriticalq = f a l s e ; active proctype P() { do :: atomic {! incriticalq ; incriticalp = true } cs: /* critical activity */ incriticalp = f a l s e od } /* similarly for process Q */ /* with same label cs: */ Verify []mutex with jspin. Formal Specification and Verification: Model Checking B. Beckert 42 / 43
72 Literature for this Lecture Ben-Ari Chapter 5 Formal Specification and Verification: Model Checking B. Beckert 43 / 43
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