Informatyka MIMUW

Exercise 1

Let ${\mathfrak A} =\langle{\mathbb N}, p^{\mathfrak A}, q^{\mathfrak A}\rangle$, where:

$\langle a,b\rangle \in p^{\mathfrak A}$ iff $a+b\geq 6$;

$\langle a,b\rangle \in q^{\mathfrak A}$ iff $b=a+2$.

Check if formulas

1. $\forall x p(x,y) \to \exists x q(x,y)$;
2. $\forall x p(x,y) \to \forall x q(x,y)$;
3. $\forall x p(x,y) \to \exists x q(x,z)$;

are satisfied under the valuation $v(y) = 7$, $v(z) = 1$ in structure ${\mathfrak A}$.

Exercise 2

Let ${\mathfrak A} = \langle {\mathbb Z}, f^{\mathfrak A}, r^{\mathfrak A}\rangle$ and ${\mathfrak B} = \langle {\mathbb Z}, f^{\mathfrak B}, r^{\mathfrak B}\rangle$, where $f^{\mathfrak A}(m,n) = \min(m,n)$ for $m,n\in{\mathbb Z}$, and $r^{\mathfrak A}$ is the relation $\geq$;

$f^{\mathfrak B}(m,n) = m^2+n^2$ for $m,n\in{\mathbb Z}$, and $r^{\mathfrak B}$ is the relation $\leq$.

Check if formulas

1. $\forall y(\forall x(r(z,f(x,y))\to r(z,y)))$;
2. $\forall y(\forall x(r(z,f(x,y)))\to r(z,y))$,

are satisfied under the valuation $v(z) =5$, $v(y)=7$ in structures ${\mathfrak A}$ and ${\mathfrak B}$.

Exercise 3

Is formula $\forall x(\lnot r(x,y)\to\exists z(r(f(x,z),g(y))))$ satisfied under the valuation $v(x) =3$, $w(x) = 6$ and $u(x) = 14$

1. in the structure ${\mathfrak A} = \langle {\mathbb N}, r^{\mathfrak A}\rangle$, where $r^{\mathfrak A}$ is the divisibility relation?
2. in the structure ${\mathfrak B} = \langle {\mathbb N}, r^{\mathfrak B}\rangle$, where $r^{\mathfrak B}$ is the congruency modulo 7?

Exercise 4

In which structures is the formula $\exists y (y\neq x)$ satisfied?
And the formula $\exists y (y\neq y)$ obtained by naive substitution of $y$ to $x$?

Exercise 5

Give examples of models and valuations such that the formula
$p(x,f(x)) \to \forall x\exists y\, p(f(y),x)$

is:
a) satisfied;
b) not satisfied.

Exercise 6

Check if the following formulas are tautologies an if they are satisfiable:

1. $\exists x\forall y(p(x) \vee q(y)) \to \forall y(p(f(y))\vee q(y))$;
2. $\forall y(p(f(y))\vee q(y)) \to \exists x\forall y(p(x) \vee q(y))$;
3. $\exists x(\forall y q(y)\to p(x))\to \exists x\forall y(q(y)\to p(x))$;
4. $\exists x(\forall y q(y)\to p(x)) \to\exists x(q(x)\to p(x))$.

Exercise 7

Let $f$ be a function symbol of arity two, that is not used in the formula $\varphi$.
Show that the formula $\forall x\exists y \varphi$ is satisfiable if and only if the formula $\forall x \varphi(f(x)/y)$ is satisfiable.

Exercise 8

Show that the formula $\forall x\exists y\,p(x,y)\wedge \forall x\neg p(x,x) \wedge \forall x\forall y\forall z(p(x,y)\wedge p(y,z)\to p(x,z))$ has only infinite models.

Exercise 9

For each $n$ give a closed formula $\varphi_n$ such that ${\mathfrak A}\models\varphi_n$ iff ${\mathfrak A}$ has exactly $n$ elements.

Exercise 10

Show that for each finite structure ${\mathfrak A}$ over a finite signature there exists a set of first order formulas $\Delta$ such that ${\mathfrak A}\models\Delta$ and for all structures satisfying ${\mathfrak B}\models\Delta$ it holds that ${\mathfrak B}\cong{\mathfrak A}.$

Exercise 11

Is it true that ${\mathfrak A} \models \exists x\,\varphi$ implies existence of a term $t$ such that ${\mathfrak A} \models \varphi[t/x]$?

Exercise 12

Let $\varphi = \forall x\forall y\,(y=f(g(x))\to(\exists u\,(u=f(x)\land y=g(u))))$ and $\psi = \forall x\,[f(g(f(x)))=g(f(f(x)))]$. Does it hold that
$\{\psi\}\models\varphi$?

Exercise 13

Show that the set of axioms of equivalence relation

$\left\{\begin{array}[]{c}\forall x\forall y(Exy\to Eyx)\\ \forall x\ Exx\\ \forall x\forall y\forall z((Exy\land Eyz)\to Exz)\end{array}\right\}$

is independent.

Exercise14

Show that the set of axioms of linear orders

$\left\{\begin{array}[]{c}\forall x\forall y((x\leq y)\lor(y\leq x))\\ \forall x\forall y((x\leq y\land y\leq x)\to x=y)\\ \forall x\forall y\forall z((x\leq y\land y\leq z)\to x\leq z)\end{array}\right\}$

is independent.

Exercise 15

Show that the set of axioms of of group theory (in multiplicative notation, over signature
$\Sigma^{F}_{{2}}=\{*\},\Sigma^{F}_{0}=\{ 1\}$)

$\left\{\begin{array}[]{c}\forall x((1*x=x)\land(x*1=x))\\ \forall x\forall y\forall z((x*y)*z=x*(y*z))\\ \forall x\exists y((x*y=1)\land(y*x=1))\end{array}\right\}$

is independent.

Exercise 16

Show that the formula $(\forall x\exists y\ Exy)\to(\exists x\forall y\ Exy)$ is not a tautology.

Exercise 17

Show that the formula

$(\forall x\forall y((f(x)=f(y))\to(x=y)))\to(\forall x\exists y(f(y)=x))$

is not a tautology. Does its negation have a finite model?

Exercise18

Show that the formula $\exists x\exists y\exists u\exists v((\lnot u=x)\lor(\lnot v=y))\land(f(x,y)=f(u,v))$ is not a tautology. How many non-isomorphic finite models does this formula have?

Exercise 19

Show that the following formulas are tautologies:

• $(\exists y p(y) \to \forall z q(z)) \to \forall y\forall z(p(y)\to q(z))$;
• $(\forall x\exists y r(x,y) \to \exists x\forall y r(y,x))\to \exists x\forall y(r(x,y) \to r(y,x))$;
• $\forall x\exists y((p(x)\to q(y))\to r(y)) \to ((\forall x p(x)\to \forall y q(y))\to \exists y r(y))$;
• $\forall x(p(x)\to \exists y q(y))\to \exists y(\exists x p(x)\to q(y))$.

Exercise 20

Does it hold that
$\{\forall x\underbrace{f\ldots f}_n(x)= x~|~n=2,3,5,7\}\models\forall x \underbrace{f\ldots f}_{11}(x)= x$?