July | 2016 | Discussions of NP-Complete Problems

Today I’m posting 2 problems relating to my student Dan Thornton’s independent study work. I’m out of town this week and next, so I have these posts set to automatically go up on Tuesday afternoon.

Dan’s independent study was based on the “Automaton Identification” problem, but to show that reduction, he needs to use a variant of 3SAT, which he shows here:

The problem: Monotone EQ SAT. This is a specific instance of Monotone SAT.

The description:
We are given a conjunction of clauses $F = \wedge_{i=1}^{l} C_{i}$ where each clause in $F$ contains all negated or non-negated variables $z_{j}$ and the number of clauses and variables are equal, is there an assignment of the variables so that $F$ is satisfied? Our instance will have $l$ clauses and variables.

Example:
Here is an $F$ that has $4$ variables and $4$ clauses.

F = $( \neg x_{1} \vee \neg x_{2} \vee \neg x_{3} \vee \neg x_{4} ) \wedge ( x_{1} \vee x_{2} \vee x_{4} ) \wedge ( \neg x_{2} \vee \neg x_{3} \vee \neg x_{4} ) \wedge ( x_{1} )$

The above $F$ may be satisfied by the following assignment:
$x_{1} \rightarrow True$
$x_{2} \rightarrow False$
$x_{3} \rightarrow False$
$x_{4} \rightarrow True$

The reduction:
We will reduce from Monotone SAT. So we are given an instance of Monotone SAT with the clauses $F = \wedge_{i=1}^{n} C_{i}$ here each clause is of the form $C_{i} = (x_{i1} \vee ... \vee x_{ik_i} )$ where each clause has all negated or non-negated variables. This is different from Monotone EQ SAT as we do not require the number of variables and clauses to be equal.
From this $F$ we must build an instance of Monotone EQ SAT.
We may transform our instance of Monotone SAT, $F$ , into one of Monotone EQ SAT by the following iterative procedure. New variables will be denoted by $z'_{j}$ and new clauses by $C'_{i}$ .

  = ;
 i = 1;
 j = 1;
 While{number of clauses != number of variables}{
   introduce two new variables ;
   If{number of variables  number of clauses}{
     Create the new clause ;
     ;
     ;
   }
   else {
     Create three new clauses:
      
      
      ;
     ;
     ;
   }
 ;
 }

The above algorithm will produce an equation $F'$ that is in Monotone EQ SAT. This may be shown by induction. Notice that before the procedure if $number \ of \ variables > number \ of \ clauses$ that we will add 2 new variables and 3 new clauses.

If $number \ of \ variables < number \ of \ clauses$ then we will add 2 new variables but only a single new clause. Either way the difference between the number of variables and clauses, $dif =| number \ of \ clauses \ - \ number \ of \ variables |$ will decrease by $1$ . So in $O(dif)$ steps we will obtain an formula where $dif$ = 0. Such a formula is an instance of Monotone EQ SAT.

True{Monotone SAT $\Rightarrow$ True{Monotone EQ SAT}
Here we assume that there is a truth assignment function $\theta: z_{j} \rightarrow \lbrace True,False \rbrace$ that maps every variable to a truth value, such that $F$ is satisfied. Then after we preform the above algorithm we have an instance of $F'$ , now our instance of $F'$ will be of the form $F \wedge ( \wedge_{i=1}^{k} C'_{i})$ for some $k \geq 1$ . Now notice that $\theta$ above will satisfy $F$ in $F'$ and we may trivially satisfy $\wedge_{i=1}^{k} C'_{i}$ by simply assigning all new variables $z'_{j}$ to true.
This will give us a new truth assignment function $\theta'$ that will satisfy $F'$

True{Monotone EQ SAT} $\Rightarrow$ True{Monotone SAT}
Here we assume that there is a truth assignment function $theta'$ that will satisfy $F'$ then obviously as $F' = F \wedge ( \wedge_{i=1}^{k} C'_{i})$ then $theta'$ must also satisfy $F$ .

(Back to me)

Difficulty: 3. The logical manipulations aren’t hard, but it is possible to mess them up. For example, it’s important that the algorithm above reduces the difference in variables and clauses by 1 each iteration. If it can reduce by more, you run the risk of skipping over the EQ state.

Here’s the actual problem my student Dan Thornton spent most of his semester trying to understand and prove. It’s pretty crazy.

The problem: Automaton Identification from Given Data. We abbreviate it “AutID”. G&J call this problem “Minimum Inferred Finite State Automaton” and it is problem AL8 in the appendix.

The description: Given a finite set of data, is it possible to build a deterministic finite automata of $n$ or fewer states that agrees with the data?
Prior to giving a description of this problem we must introduce a few concepts and definitions. We want to construct a finite automation, this means a Mealy model deterministic finite automata. Formally a 6-tuple of the form

$FA = \langle U,S,Y,f_{tr},f_{out},s_{0} \rangle$

$U$ is the input alphabet, where $u \in U$ is an individual character, and $\overline{u} \in U^{*}$ is a string.
$S$ the set of states in the model. $s \in S$ refers to an individual state. $S$ may equivalently be referred to as the set $S_{[s]}$ of all equivalence classes $[s]_{\overline{u}}$ , with representative element $\overline{u}$ . Each equivalence class is a set of all input strings $\overline{u} \in U^{*}$ such that $f_{tr}(\overline{u}) = s$ .
$Y$ the output alphabet, where $u \in Y$ is a individual character, $\overline{y} \in Y^{*}$ .
$f_{tr}: S \times U \rightarrow S$ is the transition function.
is the output function.
- Both of the above functions may be extended in a natural way so that
  $f_{tr}: S \times U^{*} \rightarrow S$ and $f_{out}: S \times U^{*} \rightarrow Y^{*}$ respectively. Usage should be clear from context.
$s_{0} \in S$ is the start state.

We use the standard convention that $\phi$ refers to the empty string of no characters.

A black box is similar to an ‘unknown’ Mealy model with the restriction that for any input string $\overline{u}$ we only get to see the last output character $f_{out}$ returns. Consider the following black box. With input alphabet $U = \lbrace 1 , 0 \rbrace$ and output alphabet $Y= \lbrace A,B,C,D \rbrace$ .

Here the edges are labeled $( input / output )$

So if the above known black box was passed the string $0001$ the complete Mealy model’s output function would give us back the string $ADDC$ where as the black box would only give us back $C$ .

Now our above finite automata is constructed using data we are given, here we assume that data is a set $D$ of finite pairs of the form $(input \ string, black \ box \ output)$ .
$D = \lbrace( \overline{u}_{1},y_{1} ) ,...,(\overline{u}_{k},y_{k}) \rbrace$ such that $\overline{u}_{i} \in U^{*}$ is not empty and $\overline{u}_{i} \neq \overline{u}_{j}$ for $i \neq j$ .

An automata $A$ agrees with data $D$ if for every datum $d = (input string, output character) \in D$ , the final output character of $f_{out}(input string) = output character$ .

Formal Problem Statement:
Given input alphabet $U$ , output alphabet $Y$ and data, a set $D$ of finite pairs determine if an automation of $n$ or fewer states exists that agrees with D.

The Reduction:

The reduction by Gold uses Monotone EQ SAT. So we are given a conjunction of clauses $F = \wedge_{i=1}^{l} C_{i}$ where each clause in $F$ contains all negated or non-negated variables $z_{j}$ and the number of clauses and variables is equal, is there an assignment of the variables so that $F$ is satisfied? Our instance will have $l$ clauses and variables. Clauses will be numbered $C_{0} ... C_{l-1}$ and variables will be numbered $z_{1} ... z_{l}$ .

In the above instance of Monotone EQ SAT we do not have a variable $z_{0}$ . Indices in our reduction will sometimes force us to reference $z_{0}$ . We may think of this variable as simply a place holder for $z_{l}$ .

The idea behind this reduction is that we may encode our instance of Monotone EQ SAT into data $D$ in such a way that if we can create a finite automata from our encoding, then we can use it to obtain a solution to the instance of Monotone EQ SAT if one exists.

Aut ID construction:
Here we transform our instance of Monotone EQ SAT into data $D$ , to do so we need the following functions:

$I_{F}(i,k) = \left\{ \begin{array}{ll} "no-value" & \quad \text{ if }\ z_{l - k} \in C_{i} \\ 0 & \quad otherwise \end{array} \right. \\ \text{where } \\ 0 \leq i \leq l-1 \text{ and } \\ 1 \leq k \leq l$

$\pi_{F}(i) = \left\{ \begin{array}{ll} 1 & \quad \text{if}\ C_{i} \text{ contains uncomplemented variables}\ \\ 0 & \quad \text{if}\ C_{i} \text{ contains complemented variables}\ \end{array} \right. \\ \text{where } \\ 0 \leq i \leq l-1$

The bounds on the above functions are a result of our numbering of clauses $C_{i}$ and $z_{j}$ . Note that $z_{0}$ is really a placeholder for $z_{l}$ .
Given a propositional formula $F$ which is an instance of MonotoneEQ SAT we encode it by the following:

The variables of $F$ are encoded by the set
$D_{L} =\lbrace (1,0),(11,0),....,(1^{l-1},0),(1^{l},1) \rbrace$ . Here each $z_{j}$ corresponds to the element $(1^{j},y)$ where $y$ is an element of the output alphabet, $Y$ .
We encode the clauses of by for , . Here each clause has a \textit{identifying tag}, , then the term is used specify a particular variable, . Importantly, datum in this set for which returns are thrown out.
- Each clause encoding $1^{i}01^{j}$ will be $0$ only if clause $C_{i}$ does not contain variable $z_{l - j}$ , otherwise $I_{F}(i,j)$ will be $"no-value"$ and is thrown out.
We then encode whether each clause is a set of all negated, or non-negated variables. $D_{P} = \lbrace (1^{i}00, \pi_{F}(i)) \rbrace$ for $0 \leq i \leq l-1$ .

As a simple example we convert to data the following formula

$F = (z_{1} \vee z_{2} \vee z_{4}) \wedge (\neg z_{2} \vee \neg z_{3} \neg z_{4})$

So after our conversion we get:

$D_{L} = \lbrace (1,0) , (11,0) , (111,0) , (1111,1) \rbrace$
$D_{C} = \lbrace (01,0),(10111,0) \rbrace$
$D_{F} = \lbrace (00,1),(100,0) \rbrace$

Aut ID instance
Now we create an instance of $Aut \ ID$ by letting $n$ be the number of variables $l$ and $D = D_{L} \cup D_{C} \cup D_{P}$ .

True{Aut ID} $\Rightarrow$ True{Monotone EQ SAT}
Now we may assume that we have a finite automata $A$ that agrees with the data in $D$ . Now we may use the transition function $f_{tr}$ and the output function $f_{out}$ to deduce properties of $A$ .

Let $s_{\overline{u}}$ be $f_{tr}(\overline{u})$ , this is the state that $A$ is in after receiving the input string $\overline{u}$ .

A splitting string of $\overline{u}_{1}$ and $\overline{u}_{2}$ is a string $\overline{u}_{s} \in U^{*}$ so that $f_{out}(\overline{u}_{1} \overline{u}_{s})$ differs from $f_{out}(\overline{u}_{2} \overline{u}_{s})$ . If such a string exists then obviously $s_{\overline{u}_{1}} \neq s_{\overline{u}_{2}}$ . If two states do not have splitting string then they are equivalent

We define variable states as follows, a variable $z_{j}$ has the encoding $1^{j}$ , so $s_{1^{j}}$ is $z_{j}$ ‘s variable state, we will sometimes refer to this as $s_{z_j}$ .

Clause states are defined in a manner similar to variable states. A clause $C_{i}$ corresponds to the identifying tag in the encoding of $C_{i}$ or $1^{i}0$ , so $s_{1^{i}0}$ is $C_{i}$ ‘s clause state, we will sometimes refer to this as $s_{C_i}$ .

In the following theorems we shall talk of assignment of clause state $s_{C_i}$ to variable state $s_{z_j}$ , this means that these two states are equivalent or alternatively that there does not exist a splitting string of $s_{C_i}$ and $s_{z_j}$ .

Theorem 1:All variable states are different.

Proof. We assume two variable states are equal ${s_z}_j$ and ${s_z}_{j'}$ where $j >j'$ . Then in the encoding of $D$ we get $z_{j} = 1^{j}$ and $z_{j'} = 1^{j'}$ respectively. But now we may define the splitting string $1^{l-j}$ . The final element of $f_{out}(1^{j} 1^{l-j}) = f_{out}(1^{l})$ is defined in our data $D$ as $(1^{l},1)$ . By a similar argument we get $f_{out}(1^{j'} 1^{l-j'}) = f_{out}(1^{l'})$ where $l' < l$ , in our data this corresponds to one of the entries $(1^{l'},0)$ . Our automata $A$ must agree with the data, so as a result these states ${s_z}_j$ and ${s_z}_{j'}$ cannot be equal. Thus we have reached a contradiction. $\square$

Theorem 2: Each state in our finite automata $A$ corresponds to exactly one variable.

Proof. The above follows by the pigeonhole principle, as $A$ has exactly $l$ states, there are $l$ variable states, and by Theorem 1, no two variable states are equal. $\square$

By the above we may talk about the states of $A$ in term of the variable each state corresponds to.

Theorem 3: If clause state ${s_C}_i$ and variable state ${s_z}_j$ are equal, Then clause $C_{i}$ contains variable $z_{l-j}$ .

Proof. Assume ${s_C}_i$ = ${s_z}_j$ and $z_{l-j}$ is not contained in $C_{i}$ . Once again we have the following encodings $C_{i} = 1^{i}0$ and $z_{l-j} = 1^{l-j}$ that were defined in our construction of $D$ . Now we may define a splitting string $u_{s}= 1^{j}$ . When we append the splitting string to each of the above encodings we get $C_{i} + u_{s} = 1^{i}01^{j}$ and $z_{l-j} + u_{s} = 1^{l-j} 1^{j}$ . Both of these strings are defined in our data $D$ as $(1^{i}01^{j},I_{F}(i,j))$ and $(1^{l},1)$ respectively, and by our assumption that $z_{l-j}$ is not contained in $C_{i}$ , $I_{F}(i,j) = 0$ So by the a conclusion identical to that of Theorem 1, ${s_C}_i \neq {s_z}_j$ and our theorem follows. $\square$

Theorem 4:No two clause states ${s_C}_i$ and ${s_C}_k$ may be equal if they are of opposite composition, meaning one has all negated variables, and the other all un-negated variables.

Proof. Once again we proceed by contradiction, assume ${s_C}_{i} = {s_C}_{k}$ and that ${s_C}_{i}$ , ${s_C}_{k}$ are of opposite composition. Once again we get each clauses encoding $1^{i}0$ and $1^{k}0$ respectively. Now we may define the splitting string $0$ . Looking at the final character output by $f_{out}$ we see $f_{out}{1^{i}00} = \pi_{F}(i)$ and $f_{out}{1^{k}00} = \pi_{F}(k)$ . It follows that $\pi_{F}(i) \neq \pi_{F}(k)$ as ${s_C}_i$ and ${s_C}_k$ are of opposite composition. Thus we arrive at a contradiction. $\square$

Theorem 5: $Aut \ ID$ gives us a solution to $F$ our instance of $Monotone EQ SAT$ .

Proof. By Theorem 2 every clause state must be assigned to a single variable state, notice this does not mean only a single assignment is possible but just the given automata $A$ will give a single assignment. By Theorem 1 no clause may be assigned more than one variable state. By Theorem 3 a clause may be assigned to a variable state only if the clause contains the variable. So we see that our automata $A$ assigns clauses to the variables that could potentially satisfy them. Furthermore by Theorem 4, each variable state is given clauses of only one composition, so there is an obvious variable assignment to satisfy all of the clause states equal to a given variable state. $\square$

True{Monotone EQ SAT} $\Rightarrow$ True{Aut ID}
Now we assume there exists a mapping $\theta_{F}: z_{j} \rightarrow \lbrace True(or \ 1), False(or \ 0) \rbrace$ that satisfies our instance of Monotone EQ SAT, $F$ with $l$ clauses and variables. From this mapping we build a finite automata $A_{F}$ that agrees with all data $D$ .

We let $A_{F}$ ‘s 0-transitions refer to transitions on input 0. Similarly 1-transitions refer to the transitions on input 1.

So first we define all 1-transitions as given in the following example automata, obviously such transitions will satisfy all data in $D_{L}$ .

Now for $A_{F}$ to satisfy all of the data in $D_{C}$ we notice by Theorems 3 and 4 above mapping each clause to the variable state that satisfies it seems like a natural thing to do. We may determine this mapping of clauses to the variables that satisfy them from our truth assignment that satisfies $F$ , $\theta_{F}$ .

Such mappings may be added to $A_{F}$ by noticing the following, as soon as we encounter a 0 in input we must transition to a clause state. So we may define a 0-transition for every state in $A_{F}$ that maps the clause state ${s_C}_{i}$ (which recall corresponds to the string $1^{i}0$ ) to the variable state ${s_z}_j$ that satisfies it. When adding 0-transitions we must be careful as if clause $C_{i}$ is satisfied by variable $z_{j}$ then in order to agree with the data in $D_{C}$ we really should map $C_{i}$ to variable $z_{l - j}$ due to the definition of $I_{F}$ .

Now we must assign an output to each of these 0-transitions. To do so we iterate through every clause state ${s_C}_{i}$ by sequentially considering $s_{1^{i}0}$ . At each clause state we assign the output of the 0-transition starting at this state to be $\pi(i)$ . Notice that if two clause states ${s_C}_{i}$ and ${s_C}_{k}$ are identical, then they must both be satisfied by the same variable and so therefore $\pi(i) = \pi(j)$ .

Below is an example of adding such a 0-transition and assigning its output:

The Blue edge is the zero transition added that maps the clause state $C_{2}$ ,which corresponds to the string $110$ , to the variable state $z_{l - 2}$ . Semantically this means that the variable $z_{l - (l-2)} = z_{2}$ satisfies this clause.
The Red edge shows us adding the output value of a 0-transition, as from the above blue edge we know ${s_z}_{l-2}$ corresponds to the clause $C_{2}$ ‘s clause state, so to the zero transition from this clause state will have output $\pi(2)$ .

Theorem 6: Our constructed automata $A_{F}$ agrees with all $D_{L}$

Proof. This is quite obvious from our example of $A_{F}$ in Figure 1. We have defined all the $1$ transitions, and notice that the final output of all $f_{out}(1^{l}) = 0$ if $l < n$ and $f_{out}(1^{n}) = 1$ this exactly matches our data $D_{L}$ . $\square$

Theorem 7: Our constructed automata $A_{F}$ agrees with all $D_{C}$
Proof:Here we consider the final character output of $f_{out}$ for all encodings of fixed clause $C_{i}$ , these encodings are precisely the $1^{i}01^{j}$ for $1 \leq j \leq l$ . Now with the addition of the $0$ -transitions, we transition into $s_{1^{i}0}$ , and from this state we only worry about $1$ -transitions. Recall that in our definition of $D_{C}$ , all datum are of the form $(input \ string, 0)$ . In $A_{F}$ there is a single $1$ -transition that outputs the non-zero value $1$ . To agree with $D_{C}$ it is sufficient to make sure that this non-zero output occurs for an input string not in $D_{C}$ .

From our assignment of $0$ -transitions defined above this singular $1$ -transition that outputs $1$ will occur for the variable $z_{l-j}$ that satisfies $C_{i}$ , so then $z_{l-j}$ must be in $C_{i}$ . So $A_{F}$ will give the datum ( $1^{i}01^{l-j}$ ,1). Now notice the corresponding entry in $D_{C}$ as defined by $I_{F}(i,j)$ would be ( $1^{i}01^{l-j}$ , $"no-value"$ ), but all of these datum were removed from $D_{C}$ . So the sufficiency condition above is satisfied. $\square$

Theorem 8 :Our constructed automata $A_{F}$ agrees with all $D_{P}$

Proof: The manner in which we defined our 0 transitions explicitly involved a $\pi_{i}$ term, so for any clause state ${s_C}_{i} = {s_{1^{i}0}}$ , now a 0 transition as defined above will output $\pi(i)$ which will obviously agree with $D_{P}$

By Theorem 6, Theorem 7, and Theorem 8. We conclude that $A$ will agree with data $D$

(Back to me)

Difficulty: 9. This is basically the reduction in the Gold paper, though Dan reorganized it a bunch to make it fit our format, and be more understandable. But the formatting of the various variables, the fact that you need to define the I_F function on variable k to talk about z_l-k instead of z_k, and the three different kinds of test data all make this super hard to understand. In some ways, I wonder if this should be a 10, given the work we did on it. But I’m saving the 10’s for things that are truly impenetrable, and we did manage to figure this one out.

Discussions of NP-Complete Problems

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Monthly Archives: July 2016

Monotone EQSat

Minimum Inferred Finite State Automaton

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