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I have several related questions, rooted in the fact that I don't think I fully understand the definition of the complement of a decision problem. First, some definitions from my Discrete Mathematics lecture notes:

  • Given a decision problem $X$, its complement $\bar X$ is the same decision problem with the yes and no answers reversed.

  • SATISFIABLE: the set of all Boolean formulae that are satisfiable (a satisfiable formula evaluates to TRUE for some assignment of truth values to its literals).   Given a Boolean formula $B$, if there's an assignment of truth values to the literals in $B$ such that $B$ evaluates to TRUE, then $B$ results in a yes answer; else $B$ results in a no answer.

  • NOT-SATISFIABLE: the set of all Boolean formulae that are not satisfiable (a formula that is not satisfiable evaluates to FALSE for all assignments of truth values to the literals).   Given a Boolean formula $B$, if there's an assignment of truth values to the literals in $B$ such that $B$ evaluates to TRUE, then $B$ results in a no answer; else $B$ results in a yes answer.

  • TAUTOLOGY: the set of all Boolean formulae that are tautologies (a tautology evaluates to TRUE for all assignments of truth values to its literals).   Given a Boolean formula $B$, if there's an assignment of truth values to the literals in $B$ such that $B$ evaluates to FALSE, then $B$ results in a no answer; else $B$ results in a yes answer.

  • NOT-TAUTOLOGY: the set of all Boolean formulae that are not tautologies (a formula that is not a tautology evaluates to FALSE for some assignment of truth values to its literals).   Given a Boolean formula $B$, if there's an assignment of truth values to the literals in $B$ such that $B$ evaluates to FALSE, then $B$ results in a yes answer; else $B$ results in a no answer.

  1. I understand that $A$ is a tautology $\iff \neg A$ is unsatisfiable.

  2. I am trying to understand the claim that SATISFIABLE is the complement of TAUTOLOGY. If so, then NOT-SATISFIABLE should be equivalent to TAUTOLOGY. Yet this is not the case: in the above definitions, their FALSE and TRUE are reversed! Why is this so, please?

  3. Is my following reasoning is sound?

    If the yes answers for SATISFIABLE are changed to no, then SATISFIABLE is transformed to NOT-TAUTOLOGY, and vice versa. If the yes answers for TAUTOLOGY are changed to no, then TAUTOLOGY is transformed to NOT-SATISFIABLE, and vice versa.

    Hence SATISFIABLE is equivalent to NOT-TAUTOLOGY, and TAUTOLOGY is equivalent to NOT-SATISFIABLE.

ryang
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Caleb Owusu-Yianoma
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    A formula $A$ is a tautology iff its negation : $\lnot A$ is not satisfiable. – Mauro ALLEGRANZA Oct 12 '16 at 19:18
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    @MauroALLEGRANZA (and CKKOY): I recommend not using any when you mean every in this kind of context (as any can be read as meaning some, turning the universal quantifier you intended into an existential quantifier). This is possibly the source of CKKOY's difficulty. – Rob Arthan Oct 12 '16 at 20:17
  • @RobArthan That is a helpful comment which I will keep in mind. However, it was not the source of my difficulty. – Caleb Owusu-Yianoma Oct 13 '16 at 17:20
  • @MauroALLEGRANZA Why does it follow that $\neg A$ is false for every assignment? – Caleb Owusu-Yianoma Oct 13 '16 at 17:22
  • @MauroALLEGRANZA I now understand why a formula $A$ is a tautology iff its negation $\neg A$ is unsatisfiable. However, I don't see the link between this and the statement: SAT is the complement of TAUTOLOGY. 'Why don't I see this link?' you might ask. The reason is that I am really struggling to understand the definition of the complement of a decision problem. Help would be much appreciated. – Caleb Owusu-Yianoma Oct 13 '16 at 18:09
  • If $A$ is true for the assignment $v$, then the negation of $A$, i.e. $\lnot A$, is not true, i.e. false, for that assignment. Ergo, if $A$ is true for every assignment, then $\lnot A$ is false for every assignment. – Mauro ALLEGRANZA Oct 13 '16 at 18:13
  • @MauroALLEGRANZA Yes, thank you. I understand your most recent comment. However, I'm still trying to make the link between this and why SAT (the set of all satisfiable Boolean formulae) is the complement of TAUTOLOGY (the set of all tautologies). These sets don't seem to complement each other in my mind! – Caleb Owusu-Yianoma Oct 13 '16 at 18:15
  • Let us continue this discussion in chat. – Caleb Owusu-Yianoma Oct 13 '16 at 18:16
  • Indeed. I had that realisation myself. So, why do my lecture notes state that SAT is the complement of TAUT?? – Caleb Owusu-Yianoma Oct 13 '16 at 18:19
  • We have to consider "complement", as you said, with yes and no swapped... Is $A$ in TAUT ? If the answer is NO, then $\lnot A$ is in SAT. – Mauro ALLEGRANZA Oct 13 '16 at 18:31
  • Hmm...I thought we consider the same decision problem with the yes and no answers swapped (not consider the complement with the yes and no answers swapped). Am I becoming unnecessarily confused, or does my reasoning make sense? – Caleb Owusu-Yianoma Oct 13 '16 at 18:35

2 Answers2

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A formula $A$ is a tautology if it is true with every assignment.

A formula $A$ is satisfiable if there is at least an assignemnt $v$ such that $A$ is true for $v$.

If $A$ is true for the assignment $v$, then its negation, $¬A$, is false for that assignment.

A formula $A$ is a tautology iff its negation, $¬A$, is not satisfiable.

The complement of a decision problem :

is the decision problem resulting from reversing the yes and no answers.

Thus, in a nutshell, if the answer to the problem "is $A$ in TAUT ?" is NO, then $¬A$ is in SAT.

More precisely, the problem of determining if some formula $A$ is not a tautology is thus equivalent to the problem of determining if the negation of the formula, $¬A$, is satisfiable.

It seems to me that it is only a terminological issue. Compare with :

Now we define some additional complexity classes related to $\text {P}$ and $\text {NP}$.

If $L ⊆ \{ 0, 1 \}^∗$ is a language, then we denote by $\overline L$ the complement of $L$.

We make the following definition: $\text {coNP} = \{ L \mid \overline L ∈ \text {NP} \}$.

$\text {coNP}$ is not the complement of the class $\text {NP}$. The following is an example of a $\text {coNP}$ language: $\overline {\text {SAT} } = \{ \varphi \mid \varphi \text { is not satisfiable} \}$.

The decision problems (or languages) are complementary : not the corresponding classes of formulae.

Community
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Mauro ALLEGRANZA
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    The answer is helpful. Should the last line of the excerpt from Sanjeev Arora & Boaz Barak read '$\text{SAT}={\varphi \mid \varphi \text{ is satisfiable}}$'? – Caleb Owusu-Yianoma Oct 14 '16 at 18:35
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    @CalebOwusu-Yianoma It's correct the way it is now; there's a line above SAT, and that line is common notation for the complement of a set. – G. Bach Sep 15 '18 at 08:33
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  • SATISFIABLE: a satisfiable formula evaluates to TRUE for some assignment of truth values to its literals
  • NOT-SATISFIABLE: a formula that is not satisfiable evaluates to FALSE for all assignments of truth values to the literals
  • TAUTOLOGY: a tautology evaluates to TRUE for all assignments of truth values to its literals
  • NOT-TAUTOLOGY: a formula that is not a tautology evaluates to FALSE for some assignment of truth values to its literals
  1. I understand that $A$ is a tautology $\iff \neg A$ is unsatisfiable.

Yes: negating a valid formula (which our current discussion is calling a tautology), e.g., $x=x,$ gives an unsatisfiable formula (and vice versa).

To add: negating a invalid formula gives a satisfiable formula (and vice versa).

(These two facts are straightforward to verify using the above definitions.)

  1. I am trying to understand the claim that SATISFIABLE is the complement of TAUTOLOGY. If so, then NOT-SATISFIABLE should be equivalent to TAUTOLOGY. Yet this is not the case: in the above definitions, their FALSE and TRUE are reversed! Why is this so, please?

Who is falsely claiming that VALID (here also known as TAUTOLOGY) and SATISFIABLE are complementary sets? The formula $(P\land \lnot P)$ belongs to neither set.

VALID is in fact a proper subset of SATISFIABLE.

  1. If the yes answers for SATISFIABLE are changed to no, then SATISFIABLE is transformed to NOT-TAUTOLOGY, and vice versa. If the yes answers for TAUTOLOGY are changed to no, then TAUTOLOGY is transformed to NOT-SATISFIABLE, and vice versa.

Firstly, the name NOT-TAUTOLOGY is confusing, and is better changed to NON-TAUTOLOGY or simply INVALID; this is because the negations of tautologies form only a proper subset of this set (for instance, the atomic formula $P$ is a non-tautology but isn't the negation of a tautology).

Secondly, no: UNSATISFIABLE (here also known as NOT-SATISFIABLE) is a proper subset of INVALID (here also known as NOT-TAUTOLOGY).

Hence SATISFIABLE is equivalent to NOT-TAUTOLOGY,

No, SATISFIABLE and INVALID aren't equivalent sets, since the formula $(P\lor \lnot P)$ belongs only to the former and the formula $(P\land\lnot P)$ belongs only to the latter.

INVALID and SATISFIABLE aren't complementary sets either, since the atomic formula $P$ belongs to both sets.

and TAUTOLOGY is equivalent to NOT-SATISFIABLE.

No, VALID and UNSATISFIABLE aren't equivalent sets.

VALID and UNSATISFIABLE aren't complementary sets either, since the atomic formula $P$ belongs to neither set.

Summary (Tautology, Valid, Contingent, Unsatisfiable, Contradiction: relationship?):

  1. VALID is a proper subset of SATISFIABLE.
  2. UNSATISFIABLE is a proper subset of INVALID.
  3. Although negating a valid formula gives an unsatisfiable formula (and vice versa),
    VALID and UNSATISFIABLE aren't complementary sets.
  4. Although negating a invalid formula gives a satisfiable formula (and vice versa),
    INVALID and SATISFIABLE aren't complementary sets.

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ryang
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