A note on useful decidable logics & theories - Part II

Decidable fragments of second-order logic

(Weak) MSO with one successor. The monadic second-order theory of structure $(\mathbb N, <)$ (denoted by S1S) is decidable. Buchi showed in 1962 that given any formula $\phi(x_1,...,x_n)$ over ${\rm MSO}(\mathbb N, <)$, one can effectively compute an $\omega$-automaton $A(x_1,...,x_n)$ defining the same relation modulo convolution. There is a similar connection between weak $MSO(\mathbb N, <)$ (denoted by WS1S) and finite-word automata. See [10] for a historical survey.

MSO over binary trees. The structure of infinite binary trees is $T_2 := (\{1,2\}^*,S_1,S_2)$ with $S_i := \{(v,vi) : v \in \{1,2\}^*\}$. Rabin's Tree Theorem [4] states that the MSO-theory of the infinite binary tree (denoted by S2S) is decidable, or in other terminology: that model-checking the binary tree with respect to MSO-properties is decidable. In [4], Rabin also proved the decidability of a bunch of theories using interpretation in $T_2$, such as MSO over n-ary trees (see below). Note that Rabin's theorem is subsumed by a more general result stating that MSO over the prefix-rewriting systems is decidable [?].

MSO over n-ary trees. The decidability results of MSO-theory over binary trees $T_2$ can be extended to $n$-ary branching trees $T_n$ with vertices in $[1..n]^*$ by encoding a vertex $a_1 a_2 \dots a_k$ of $T_n$ by a vertex $1^{a_1} 2 1^{a_2} 2 \dots 2 1^{a_k} 2$ of $T_2$. The translation of MSO-formulas is straightforward.

MSO over pushdown automata. Let $G := (V,E)$ denote the configuration graph of a pushdown automaton, where $V\subseteq Q\times\Sigma^*$. Let $n := \max(|Q|,|\Sigma|).$ One can MSO-interpret $G$ in $T_n$ by representing the configuration $(q, s_1,...,s_k)$ as the vertex $s_1...s_k\ q$ of $T_n$. Each step between the configurations in $G$ is translated in MSO to a local move from one vertex to another in $T_n$. Decidability of MSO over binary trees hence subsumes the fundamental result of Muller and Schupp [6] that checking MSO properties is decidable for the configuration graph of a pushdown automaton.

Decidable modal logic

Probabilistic modal logic (PML). Let $\Sigma$ be a finite set of actions. The syntax of the logic is defined as follows:$$\phi ::= \top \mid \bot \mid \phi\vee\phi \mid \neg\phi \mid (\phi)_a\ {\rm op}~p,$$where $a \in \Sigma$, ${\rm op}\in \{=,<\}$, and $0\le p\le 1$. The semantics of PML is defined over a discrete non-deterministic probabilistic transition system such that for $s\in S$, e.g.,$$s \models (\phi)_a \le p \quad {\rm iff} \quad \Pr(s' \models \phi \mid s\xrightarrow{a} s') \le p.$$Larsen and Skou [7] show that, when the transition probabilities are rational and bounded above zero, given any PML formula $\phi$ and fixed $0<\epsilon<1$, one can compute an experiment $t_{\epsilon,\phi}$ and an observation $o_{\epsilon,\phi}$ such that $s \models \phi$ iff $\Pr(o_{\epsilon,\phi}$ is observed by running $t_{\epsilon,\phi}$ on $s) \ge 1-\epsilon$, and $s \not\models \phi$ iff $\Pr(o_{\epsilon,\phi}$ is observed by running $t_{\epsilon,\phi}$ on $s) \le \epsilon$. Such experiments can be computed, executed and observed in polynomial time. Larsen and Skou show that PML characterises probabilistic bisimulation in the following sense: two states are probabilistic bisimilar iff they satisfy exactly the same set of PML formulas.

Misc

Reduction of decidability. Suppose that model checking $L$-theory of structure $S$ is known to be decidable. The model checking problem for a structure $S'$ with respect to $L'$-theory is decidable if we can give an $L$-description of $S'$ within structure $S$. That is, we characterise structure $S'$ using $L$-formulas over $S$. In other words, we provide a satisfiability-preserving translation of any $L'$-formulas $\phi$ into $L$-formula $\psi$ such that $S'\models \phi$ iff $S\models \psi$. Hence $L'$-theory over structure $S'$ is also decidable.

References

1. Structural properties of complete problems for exponential time. 1997.
2. Complexity results for classes of quantification formulas. 2001.
3. Verification of randomised security protocols.
4. Decidability of second-order theories and automata on infinite trees. 1969.
5. Complexity results for classes of quantificational formulas. 1980.
6. The theory of ends, pushdown automata, and second-order logic. 1985.
7. Bisimulation through Probabilistic Testing. 1991.
8. The Classical Decision Problem. 2001.
9. A Survival Guide to Presburger Arithmetic. 2018.
10. Automata: From Logics to Algorithms. 2007.
11. Complete instantiation for quantified formulas in satisfiabiliby modulo theories. 2009.