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# Lab for Automated Reasoning and Analysis LARA

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sav08:weak_monadic_logic_of_one_successor [2008/05/15 10:02] vkuncak |
sav08:weak_monadic_logic_of_one_successor [2015/04/21 17:30] (current) |
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Let $\mathbb{N} = \{0,1,2,\ldots\}$ denote non-negative integers. Let $D$ be the set of all //finite subsets// of $\mathbb{N}$. We consider the set of interpretations $(D,\alpha)$ where for each variable $v$ we have $\alpha(v) \in D$, where $\subseteq$ is the subset relation | Let $\mathbb{N} = \{0,1,2,\ldots\}$ denote non-negative integers. Let $D$ be the set of all //finite subsets// of $\mathbb{N}$. We consider the set of interpretations $(D,\alpha)$ where for each variable $v$ we have $\alpha(v) \in D$, where $\subseteq$ is the subset relation | ||

- | \[ | + | \begin{equation*} |

\alpha({\subseteq}) = \{ (S_1,S_2) \mid S_1 \subseteq S_2 \} | \alpha({\subseteq}) = \{ (S_1,S_2) \mid S_1 \subseteq S_2 \} | ||

- | \] | + | \end{equation*} |

and the relation $succ(v_1,v_2)$ is the successor relation on integers lifted to singleton sets: | and the relation $succ(v_1,v_2)$ is the successor relation on integers lifted to singleton sets: | ||

- | \[ | + | \begin{equation*} |

\alpha(succ) = \{ (\{k\},\{k+1\}) \mid k \in \mathbb{N} \} | \alpha(succ) = \{ (\{k\},\{k+1\}) \mid k \in \mathbb{N} \} | ||

- | \] | + | \end{equation*} |

The meaning of formulas is given by standard [[First-order logic semantics]]. | The meaning of formulas is given by standard [[First-order logic semantics]]. | ||

Note in particular that quantification is restricted to finite sets (elements of $D$). | Note in particular that quantification is restricted to finite sets (elements of $D$). | ||

+ | |||

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Then we can define addition $N(Z) = N(X) + N(Y)$ by saying that there exists a set of carry bits $C$ such that the rules for binary addition hold: | Then we can define addition $N(Z) = N(X) + N(Y)$ by saying that there exists a set of carry bits $C$ such that the rules for binary addition hold: | ||

\begin{equation*} | \begin{equation*} | ||

- | \exists C.\ 0 \notin i\ \land \forall i.\ | + | \exists C.\ 0 \notin C\ \land \forall i.\ |

\big(\begin{array}[t]{rcl} | \big(\begin{array}[t]{rcl} | ||

((i \in Z) &\leftrightarrow& ((i \in X) \oplus (i \in Y) \oplus (i \in C))\ \land\\ | ((i \in Z) &\leftrightarrow& ((i \in X) \oplus (i \in Y) \oplus (i \in C))\ \land\\ | ||

Line 91: | Line 92: | ||

Relations on singleton sets: | Relations on singleton sets: | ||

- | \[ | + | \begin{equation*} |

r^s_F = \{(p,q) \mid F(\{p\},\{q\}) \} | r^s_F = \{(p,q) \mid F(\{p\},\{q\}) \} | ||

- | \] | + | \end{equation*} |

Relations on binary representations: | Relations on binary representations: | ||

- | \[ | + | \begin{equation*} |

r^b_F = \{(p,q) \mid F(N(p),N(q)) \} | r^b_F = \{(p,q) \mid F(N(p),N(q)) \} | ||

- | \] | + | \end{equation*} |

Addition is not definable as some $r^s_F$, but it is definable as $r^b_F$. | Addition is not definable as some $r^s_F$, but it is definable as $r^b_F$. | ||