Semantically Equivalent Formulas

1. Semantically Equivalent Formulas • Let Φ and ψbe formulas of propositional logic. We say that Φ and ψare semantically equivalent iff Φ ╞ ψ ψ ╞ Φ hold. In that case we write Φ≡ψ. Further, we call Φ valid if ╞ Φ holds.

2. Examples of equivalent formulas • p → q ≡ ¬p  q • p → q ≡ ¬q → ¬p • p  q → p ≡ r  ¬r • p  q → r ≡ p → (q →r)

3. Lemma • Given propositional logic formulas Φ1, Φ2, …, Φn, ψ, we have Φ1, Φ2, …, Φn ╞ ψ iff ╞ Φ1 →(Φ2 → (Φ3 → … → (Φn → ψ)))

4. Literal • A literal is either an atom p or the negation of an atom ¬p.

5. Conjunctive Normal Form (CNF) • A formula Φ is in conjunctive normal form (CNF) if it is of the form ψ1ψ2 …. ψn for some n ≥ 1, such that ψi is a literal, or a disjunction of literal, for all 1 ≤ i ≤ n.

6. Examples for CNF formulas • (¬q  p  r)  (¬p  r)  q • (p  r)  (¬p  r)  (p  ¬r)

7. Lemma • A disjunction of literals L1 L2 ….  Lm is valid (i.e., ╞ L1 L2 ….  Lm) iff there are 1 ≤ i, j ≤ m such that Li is ¬Lj.

8. Satisfiable formulas • Given a formula Φ in a propositional logic, we say that Φ is satisfiable if there exists an assignment of truth values to its propositional atoms such that Φ is true.

9. Proposition • Let Φ be a formula of propositional logic. Then Φ is satisfiable iff ¬Φ is not valid.

10. function CNF(Φ) /* pre-condition: Φ implication free and in NNF*/ /* post-condition: CNF(Φ) computes an equivalent CNF for Φ */ begin function case Φ is a literal : returnΦ Φ is Φ1 Φ2: return CNF(Φ1)  CNF(Φ2) Φ is Φ1 Φ2: return DISTR(CNF(Φ1), CNF(Φ2) ) end case end function

11. function DISTR(η1, η2): /* pre-condition: η1 and η2 are in CNF */ /* post-condition: DISTR(η1, η2) computes a CNF for η1η2 */ begin function case η1 is η11  η12 : return DISTR(η11 , η2)  DISTR(η12 , η2) η2 is η21  η22 : return DISTR(η1 , η21)  DISTR(η1 , η22) otherwise (= no conjunction): returnη1η2 end case end function

12. function NNF(Φ) /* pre-condition: Φ is implication free */ /* post-condition: NNF(Φ) computes a NNF for Φ */ begin function case Φ is a literal : returnΦ Φ is ¬¬Φ1 : return NNF(Φ1) Φ is Φ1 Φ2 : return NNF(Φ1)  NNF(Φ2) Φ is Φ1 Φ2 : return NNF(Φ1)  NNF(Φ2) Φ is ¬(Φ1 Φ2) : return NNF(¬Φ1 ¬Φ2) Φ is ¬(Φ1 Φ2): return NNF(¬Φ1 ¬Φ2) end case end function

13. Φ = ¬p  q → p  (r → q) IMPL_FREE Φ = ¬ IMPL_FREE (¬p  q )  IMPL_FREE (p (r → q)) = ¬((IMPL_FREE ¬p )  (IMPL_FREE q ))  IMPL_FREE (p (r → q)) = ¬((¬p )  IMPL_FREE q )  IMPL_FREE (p (r → q)) = ¬ (¬p  q )  IMPL_FREE (p (r → q)) = ¬ (¬p  q )  ((IMPL_FREE (p)  IMPL_FREE (r → q)) = ¬ (¬p  q )  (p  IMPL_FREE (r → q)) = ¬ (¬p  q )  (p  (¬ (IMPL_FREE r)  IMPL_FREE (q))) = ¬ (¬p  q )  (p  (¬ r  IMPL_FREE (q))) = ¬ (¬p  q )  (p  (¬ r  q))

14. IMPL_FREE Φ = ¬ (¬p  q )  (p  (¬ r  q)) NNF (IMPL_FREE Φ ) = NNF (¬ (¬p  q ))  NNF (p  (¬ r  q)) = NNF (¬ (¬p )  ¬q ))  NNF (p  (¬ r  q)) = (NNF (¬¬p ))  (NNF (¬q ))  NNF (p  (¬ r  q)) = (p  (NNF (¬q )))  NNF (p  (¬ r  q)) = (p  ¬q )  NNF (p  (¬ r  q)) = (p  ¬q )  ((NNF p)  (NNF (¬ r  q))) = (p  ¬q )  ( p  (NNF (¬ r  q))) = (p  ¬q )  ( p  ((NNF (¬ r))  (NNF q))) = (p  ¬q )  ( p  (¬ r  (NNF q))) = (p  ¬q )  ( p  (¬ r  q))

15. NNF (IMPL_FREE Φ) = (p  ¬q )  ( p  (¬ r  q))CNF(NNF (IMPL_FREE Φ)) = CNF ((p  ¬q )  ( p  (¬r  q))) = DISTR ( CNF (p  ¬q ), CNF (p  (¬ r  q))) = DISTR (p  ¬q , CNF (p  (¬ r  q))) = DISTR (p  ¬q , p  (¬ r  q)) = DISTR (p  ¬q , p)  DISTR (p  ¬q ,¬ r  q) = (p  ¬q p)  DISTR (p  ¬q ,¬ r  q) = (p  ¬q p)  (p  ¬q ¬ r  q)

16. Horn Formula Φ • is a formula Φ of propositional logic if it is of the form ψ1 ψ2 ...  ψn for some n ≥ 1 such that ψi is of the form p1 p2 ...  pki → qi for some ki ≥ 1, where p1, p1, …, pki, qi are atoms, ┴ or T. We call such ψi a Horn clause.

17. Examples of Horn formulas • (p  q  s → p)  (q  r → p)  (p  s → s) • (p  q  s → ┴)  (q  r → p)  (T → s) • (p2 p3 p5 →p13) (T→p2) (p5 p11 → ┴)

18. Examples of non-Horn formulas • (p  q  s → ¬p)  (q  r → p)  (p  s → s) • (p  q  s → ┴)  (¬q  r → p)  (T → s) • (p2 p3 p5 →p13  p27)  (T→p2) (p5 p11 → ┴) • (p2 p3 p5 →p13 ) (T→p2) (p5 p11  ┴)

19. function HORN(Φ) /* Pre-condition : Φ is a Horn formula*/ /* Post-condition : HORN(Φ) decides the satisfiability for Φ */ begin function mark all atoms p where T → p is a sub-formula of Φ; while there is a sub-formula p1 p2 ...  pki → qi of Φ such that all pj are marked but qi is not do if qi ≡ ┴ then return ‘unsatisfiable’ else mark qi for all such subformulas end while return ‘satisfiable’ end function

20. Theorem • The algorithm HORN is correct for the satisfiability decision problem of Horn formulas and has no more than n cycles in its while-loop if n is the number of atoms in Φ. HORN always terminates on correct input.

21. Kripke structure Let AP be a set of atomic propositions. A Kripke structure M over AP is a four tuple M= (S, S0, R, L) where • S is a finite set of states • S0 S is the set of initial states. • R S × S is a transition relation that must be total, that is for every state s S there is a state s’ S such that R (s, s’). • L: S  2 AP is a function that labels each state with the set of atomic proposition in that state. A path in the structure M from a state s is an infinite sequence of states ω = s0 s1 s2 … such that s0 = s and R (si, si+1) holds for all i ≥ 0.

22. First order representationof Kipke structures • We use interpreted first order formulas to describe concurrent systems. • We use usual logical connectives (and , or , implies , not , and so on) and universal ( ) and existential ( ) quantifications. • Let V = {v1, …, vn} be the set of system variables. We assume that the variables in V range over a finite set D. • A valuation for V is a function that associated a value in D with each variable v in V. Thus, s is a valuation for V when s: V  D. • A state of a concurrent system can be viewed as a valuation for the set of its variables V. • Let V’ = {v’1, …, v’n}. We think of the variables in V as present state variables and the variables in V’ as next state variables.

23. First order representationof Kipke structures Let M = (S, S0, R, L) be a Kripke structure. • S is the set of all valuations for all variables of the system which can be described by a propositionS. Usually, S = True. • The set of initial states S0 can be described by a proposition (on the set of variables) S0. • R can be described by a proposition Rsuch that for any two states s and s’, R(s, s’) holds if R evaluates to True when each variable v is assigned the value s(v) and each variable v’ is assigned the value s(v’). • The labeling function L:S  2AP is defined so that L(s) is the subset of all atomic propositions true in s which can be described by some appropriate proposition.

24. A simple example We consider a simple system with variables x and y that range over D = {0, 1}. Thus, a valuation for the variables x and y is just a pair (d1, d2) D × D where d1 is the value for x and d2 is the value for y. The system consists of one transition x := (x +y) mod 2, Which starts from the state in which x = 1 and y = 1.

25. A simple example with transition x := (x +y) mod 2 • S = True • S0 (x, y) ≡ x = 1  y = 1 • R (x, y, x’, y’) ≡ x’ = (x +y) mod 2  y’ = y

26. A simple example with transition x := (x +y) mod 2 The Kripke structure M = (S, S0, R, L) for this system is simply: • S = D × D. • S0 = {(1,1)} • R = {((1,1), (0,1)), ((0,1), (1,1)), ((1,0), (1,0)), ((0,0), (0,0))}. • L(1,1) = {x =1, y = 1}, L(0,1) = {x =0, y = 1}, L(1,0) = {x =1, y = 0}, L(0,0) = {x =0, y = 0}. The only path in the Kripke structure that starts in the initial state is (1,1) (0,1) (1,1) (0,1) ….

27. Concurrent systems • A concurrent system consists of a set of components that execute together. • Normally, the components have some means of communicating with each other.

28. Modes of execution We will consider two modes of execution: Asynchronous or interleaved execution, in which only one component makes a step at a time, and synchronous execution, in which all of the components make a step at the same time

29. Modes of communication • We will also distinguish three modes of communication. Components can either communicate by changing the value of shared variables or by exchanging messages using queues or some handshaking protocols.

30. A modulo 8 counter

31. Synchronous circuitA modulo 8 counter The transitions of the circuit are given by • v’0 = v0 • v’1 = v0 v1 • v’2 = (v0 v1) v2 • R0 (v, v’) ≡ (v’0 ↔ v0) • R1 (v, v’) ≡ (v’1 ↔ v0 v1) • R2 (v, v’) ≡ (v’2 ↔ (v0  v1)v2) • R (v, v’) ≡ R0 (v, v’)  R1 (v, v’)  R2 (v, v’)

32. Synchronous circuitGeneral case • Let V = {v0, …., vn-1} and V’ = {v’0, …., v’n-1} • Let v’i = fi (V), 1= 0, …, n-1. • Define Ri (v, v’) ≡ ( v’i ↔ fi (V)). • Then, the transition relation can be described as R (v, v’) ≡ R0 (v, v’)  …  Rn-1 (v, v’).

33. Asynchronous circuitGeneral case • In this case, the transition relation can be described as R (v, v’) ≡ R0 (v, v’)  …  Rn-1 (v, v’), Where Ri (v, v’) ≡ ( v’i ↔ fi (V)) j ≠ i (v’j ↔ vj )).

34. Example • Let V = {v0, v1}, v’0 = v0 v1 and v’1 = v0 v1. • Let s be a state with v0 = 1  v1 = 1. • For the synchronous model, the only successor of s is the state v0 = 0  v1 = 0. • For the asynchronous model, the state s has two successors: • 1. v0 = 0  v1 = 1 ( the assignment to v0 is taken first). • 2. v0 = 1  v1 = 0 ( the assignment to v1 is taken first).

35. Labeled program Given a statement P, the labeled statement PL is defined as follows: • If P is not a composite statement then P = PL. . • If P = P1; P2 then PL = P1L ; l’’ : P2L. • If P = if b then P1else P2end if, then PL = if b then l1 : P1Lelse l2 : P2Lend if. • If P = while b do P1 end while, then PL = while b do l1 : P1Lend while.

36. Some assumptions • We assume that P is a labeled statement and that the entry and exit points of P are labeled by m and m’, respectively. • Let pc be a special variable called the program counter that ranges over the set of program labels and an additional value ┴ called the undefined value. • Let V denote the set of program variables, V’ the set of primed variables for V, and pc’ the primed variables for pc. • Let same (Y) = y ε Y (y’ = y).

37. The set of initial states of P • Given some condition pre (V) on the initial variables for P, S0 (V, pc) ≡pre (V)  pc = m.

38. The transition relation for P • C (l, P, l’) describes the set of transitions in P as a disjunction of all transitions in the set. • Assignment: C ( l, v ← e, l’) ≡ pc =l  pc’ = l’  v’ = e  same (V \ {v}) • Skip: C ( l, skip, l’) ≡ pc =l  pc’ = l’  same (V) • Sequential composition: C ( l, P1; l’’ : P2, l’) ≡ C ( l, P1, l’’)  C ( l’’, P2, l’)

39. The transition relation for P (continued) • Conditional: C (l, if b then l1: P1else l2 : P2end if, l’) is the disjunction of the following formulas: • pc = l  pc’ = l1  b  same (V) • pc = l  pc’ = l2  b  same (V) • C (l1, P1, l’) • C (l2, P2, l’)

40. The transition relation for P (continued) • While: C (l, while b do l1 : P1 end while, l’) is the disjunction of the following formulas: • pc = l  pc’ = l1  b  same (V) • pc = l  pc’ = l’  b  same (V) • C (l1, P1, l)

41. Concurrent programs • A concurrent program consists of a set of processes that can be executed in parallel. A process is a sequential program. • Let Vi be the set of variables that can be changed by process Pi. V is the set of all program variables. • pci is the program counter of process Pi. PC is the set of all program counters. • A concurrent program has the form cobegin P1 || P2 … || Pncoend where P1, …, Pn are processes.

42. Labeling transformation • We assume that no two labels are identical and that the entry and exit points of P are labeled m and m’, respectively. • If P = cobegin P1 || P2 … || Pncoend, then PL = cobegin l1 : P1L l’1 || l2 : P2L l’2 || … || ln : PnL l’ncoend.

43. The set of initial states of P S0 (V, pc) ≡pre (V)  pc = m  i= 1, … n (pci = ┴ )

44. The transition relation for P C (l, cobegin l1 : P1Ll’1 || … || ln : PnL l’ncoend,l’) Is the disjunction of three formulas: • pc = l  pc’1 = l1  …  pc’n = ln  pc’ = ┴ • pc = ┴  pc1 = l’1  …  pcn = l’n  pc’ = l’  i= 1, … n (pc’i = ┴) • i= 1, … n (C (li, Pi, l’i)  (same (V \ Vi)  same (PC \ {pci}))

45. Shared variables • Wait: C (l, wait (b), l’) is a disjunction of the following two formulas: • (pci = l  pc’i = l  b  same (Vi)) • (pci = l  pc’i = l’  b  same (Vi)) • Lock (v) (= wait (v = 0)): C (l, lock (v), l’) is a disjunction of the following two formulas: • (pci = l  pc’i = l  v = 1  same (Vi)) • (pci = l  pc’i = l’  v = 0  v’ = 1  same (Vi \ {v})) • Unlock (v): C (l, unlock (v), l’) ≡ pci = l  pc’i = l’  v’ = 0  same (Vi \ {v})

46. A simple mutual exclusion programP = m: cobegin P0 || P1 coend m’ P0 :: l0 : while Truedo NC0 : wait (turn = 0); CR0 : turn :=1; end while; l’0 P1 :: l1 : whileTrue do NC1 : wait (turn = 1); CR1 : turn := 0; end while; l’1

47. Kripke structure • pc takes values in the set { m, m’, ┴ }. • pci takes values in the set { li, l’i, NCi, CRi, ┴ }. • V = V0 = V1 = {turn}. • PC = {pc, pc0, pc1}.

48. The set of initial states of P • S0 (V, PC) ≡ pc = m  pc0 = ┴  pc1 = ┴.

49. The transition relation for P • R (V, PC, V’, PC’) is the disjunction of the following four formulas: • pc = m  pc’0 = l0  pc’1 = l1  pc’ = ┴. • pc0 = l’0  pc1 = l’1  pc’ = m’  pc’0 = ┴  pc’1 = ┴. • C (l0, P0, l’0)  same (pc, pc1). • C (l1, P1, l’1)  same (pc, pc0).

50. The transition relation of Pi For each process Pi, C (li, Pi, l’i) is the disjunction of: • pci = li pc’i = NCi  True  same (turn) • pci = NCi pc’i = CRi  turn = i same (turn) • pci = CRi pc’i = li  turn’ = (i+1) mod 2 • pci = NCi pc’i = NCi  turn ≠ i same (turn) • pci = li pc’i = l’i  False  same (turn)