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.

A note on recognisability and definability

Definitions. An $n$-ary relation is associated with a set of $n$-tuples; an $n$-ary function $f$ is associated with a set of $(n+1)$-tuples $\{(a_1,...,a_n,f(a_1,...,a_n)\}$, which is called the graph of function $f.$ A sequence $s_1,s_2,...$ over $A$ is associated with a function $s:\mathbb N^n\rightarrow A$ such that $s(i)=s_i.$ Given an integer $n$, we use $(n)_p$ to denote its p-ary expansion; given a p-expansion $w\in\{0,...,p-1\}^*$, we use $[w]_p$ to denote its value $\sum_{i=0}^{|w|-1}p^{|w|-1-i}w[i].$ By convention, we set $(0)_p=\epsilon$ and $[\epsilon]_p=0.$

Recognisable Sets

Definitions. A set is called regular or recognisable if it can be recognised by a finite DFA. For $L\subseteq\Sigma^*$, $\sim_L$ is called a right congruence iff $u\sim_L v \iff uw\sim_L vw$ for all $w\in\Sigma^*.$ Note that a right congruence is an equivalence relation.

Myhill-Nerode Theorem. A set $L$ is regular iff $\sim_L$ has finite index over $\Sigma^*.$

Definitions. (p-recognisability) A set of integers is called p-recognisable iff the p-ary expansion of the integers is recognised by a finite automaton with alphabet $\{0,...,p-1\}.$ The automaton would read a word from left to right and ignores the leading zeros, so that if it accepts $w$ then it accepts all words in $0^*w.$
Fact. A sequence $s_1,s_2,...$ over $A$ is p-recognisable if $s^{-1}(a)$ is p-recognisable for each $a\in A.$ When $A$ is finite, it is easier to define a sequence in this way.

Definitions of recognisability generalise to n-ary relations naturally by replacing the alphabet $\{0,...,p-1\}$ to $\{0,...,p-1\}^n.$ All properties and facts listed above still hold in the general case.

Definitions. (p-automata) A p-automaton recognising a sequence $s_1,s_2,...$ over a finite domain is a complete deterministic Moore machine that reads $(n)_p$ and outputs $s_n$ for all $n\ge 1.$ In such case the sequence is called p-recognisable. Note that p-recognisability of sequences generalises p-recognisability of sets, since a set is p-recognisable iff there exists a p-automaton that outputs 1 on $(n)_p$ if $n$ belongs to the set and outputs 0 otherwise. Alternatively, we can describe a set of integers using a characteristic sequence. The characteristic sequence of a set $S$ refers to a sequence over $\{0,1\}$ where the nth bit is 1 iff $n\in S.$ A set is p-recognisable iff its characteristic sequence is p-recognisable.
A p-automaton can recognise an n-ary relation by recognising an n-dimensional sequence $s:\mathbb N^n\rightarrow\{0,1\}.$ The input language $((a_1)_p,...,(a_n)_p)$ is cleverly stipulated as a subset of $(\{0,...,p-1\}^n)^*$ instead of $(\{0,...,p-1\}^*)^n.$ Namely, components of a relation convoluted over alphabet $\{0,...,p-1\}^n$. In this way, theorems such as Myhill-Nerode still hold for $n\ge2.$

Definable Sets

Definitions. A set $L$ of $n$-tuples is called (first-order) definable by a formula $\phi$ in a first-order structure $\mathcal S$ iff $\phi$ has exactly $n$ free variables and $L=\{(a_1,...,a_n)\in D^n :$ $S\models \phi(a_1,...,a_n)\}$, where $D$ is the domain of $\mathcal S.$ A set is definable in a structure $\mathcal S$ if it is definable by a formula in $\mathcal S.$ When the structure is clear in the context we simply called a set definable. A sequence $s_1,s_2,...$ over $A$ is definable if the graph of the function $s:\mathbb N\rightarrow A$ associated with the sequence is definable.
Fact. A sequence $s_1,s_2,...$ over $A$ is definable iff $s^{-1}(a)$ is definable for each $a\in A.$ When $A$ is finite, it is easier to define a sequence in this way.

Definitions. (p-definability) Given $x\in\mathbb N$, define $V_p(x)\equiv$ the largest power of $p$ that divides $x.$ Let $V_p(0)\equiv 1.$ A subset of $\mathbb N$ is called $p$-definable if it is definable in $[\mathbb N,+,V_p].$ Given $x\in\mathbb N$, define $V_p(x)\equiv$ the largest power of $p$ that divides $x$ and $V_p(0)\equiv 1.$ Define a predicate $P_p$ that $P_p(x)$ holds iff $x$ is a power of $p.$ Define a binary relation $B_p$ such that $B_p(x,y)$ holds iff $y$ is a power of $p$ occurring in the p-ary expansion of $x.$ (Thus $B_p(x,y)$ iff $y<x$, $P_p(y)$, and there exits a $0\le z <y$ such that $y|(x-z).$)

Great efforts were made in the history to characterise 2-recognisable sets in an FO structure. Three candidate structures were considered for this purpose in order: $[\mathbb N, +, P_2]$, $[\mathbb N, +, B_2]$, and $[\mathbb N, +, V_2].$ We note two facts here about these structures:
Fact. $[\mathbb N, +, B_p]$ and $[\mathbb N, +, V_p]$ are (FO-)interpretable in each other.
Fact. $[\mathbb N, +, P_p]$ is strictly weaker than $[\mathbb N, +, V_p].$
To see the first, note that for $x>0$, $V_p(x)=y$ iff $B_2(x,y)\wedge \forall 0\le z<y.\ \neg B_p(x,z)$, and that $B_p(x,y)$ holds iff $\exists 0\le z <y.\ V_p(x-z, y).$ The second fact follows from an interesting decidability result. Let $p^x$ denote the exponential function. It is known that $[\mathbb N,+,V_p]$ and $[\mathbb N,+,p^x]$ are decidable, but $[\mathbb N,+,p^x,V_p]$ is undecidable. Clearly $P_p$ can be defined by $V_p$ and $p^x.$ If $[\mathbb N, +, P_p]$ is equal to $[\mathbb N, +, V_p]$, then $V_p$ will be definable by $p_x$, which implies $[\mathbb N,+,V_p]$ and $[\mathbb N,+,p^x,V_p]$ are mutually interpretable. This is impossible since the first is decidable and the second is not. The ultimate characterisation of 2-recognisable sets is given by the following theorem.
Theorem I. For any prime number p, a set is p-definable iff it is p-recognisable.
Proof. ($\Rightarrow$) It suffices to show that given any formula $\phi$ from a structure $\mathcal S$ equivalent to $[\mathbb N,+,V_p]$, we can construct a p-automaton $A_\phi$ recognising the set definable by $\phi$, i.e., $\mathcal S \models \phi(a_1,...,a_n)$ implies that $A_\phi$ accepts $([a_1]_p,...,[a_n]_p).$ To simply our proof, we choose $\mathcal S=[\mathbb N,R_1,R_2]$, where $R_1(x,y,z)$ holds iff $x+y=z$ and $R_2(x,y)$ holds iff $V_p(x)=y.$ Now we prove the claim by structural induction. First note that the three atom formulas $x=y$, $R_1(x,y,z)$, and $R_2(x,y)$ in $\mathcal S$ are p-recognisable (Exercise!). To complete the induction, it suffices to show that we can construct p-automata $A_{\neg\phi}$, $A_{\phi\vee\psi}$ and $A_{\exists x\phi}$ from p-automata $A_\phi$ and $A_\psi.$ Negation construction is immediate since p-automata is closed under complementation. For union, we modify the transition functions of the two p-automata so that they operate over the same set of input variables. For instance, suppose the free variables in $\phi$ are $x_1,...,x_m,y_1,...,y_n$ and the free variables in $\psi$ are $y_1,...,y_n,z_1,...,z_r.$ Then after the modification, the transition functions of the p-automata have the same domain $(x_1,...,x_m,$ $y_1,...,y_n,$ $z_1,...,z_r).$ The union of the two modified automata gives $A_{\phi\vee\psi}.$ For projection, to construct $\exists x.\phi(x,x_1,...,x_n)$ we just relabel the transitions of the p-automaton by replacing $(a,a_1,...,a_n)$ with $(a_1,...,a_n).$ Note that we may need to add new loops after the relabelling to recognise leading zeros caused by the projection. See [3] for details of this proof.

Corollary. The theory of $[\mathbb N,+,V_p]$ is decidable.
Proof. Given a sentence in $[\mathbb N,+,V_p]$, we can rewrite it to the form of either $\exists x.\phi(x)$ or $\neg\exists x.\phi(x).$ By Theorem I, $\phi(x)$ is p-recognisable. Hence we can construct an finite automaton $A$ that accepts a word $w$ iff $[\mathbb N,+,V_p]\models\phi([w]_p).$ Hence the truth of the sentence can be checked by checking emptiness of a recognisable language, which is decidable.

Corollary. The set of squares $\{n^2:n\in\mathbb N\}$ is not p-recognisable.
Proof. Suppose to the contrary that the set of squares is p-recognisable. By Theorem I, there exists a formula $\phi$ in $[\mathbb N,+,V_p]$ such that $\phi(x)$ holds iff $x$ is a square. Define $$\psi(c,d):=\phi(d)\wedge\forall d'.(d'>c\wedge\phi(d')\implies d'\ge d),$$ which asserts that $d$ is the smallest square larger than $c$. Observe that the set $\{(x,y)\in\mathbb N^2: y=x^2\}$ is p-definable by $$\lambda(x,y):=\phi(y)\wedge\exists a.\ \psi(y,a)\wedge a=y+2x+1.$$ Also, the set $\{(x,y,z)\in\mathbb N^3: z=xy\}$ is p-definable by formula
$\exists a,b,c.\ \lambda(a,x)\wedge\lambda(b,y)\wedge\phi(c)\wedge c=a+b+2z \wedge\exists d.\ \psi(c,d)\wedge d=c+2x+2y+1$
The validity of this formula exploits the fact that $c=(x+y)^2$ if $c$ is a square and the smallest square greater than $c$ is $c+2x+2y+1$ for some $x,y\in\mathbb N$. It then follows that multiplication is p-definable, which is impossible since $[\mathbb N,+,\cdot]$ is not decidable but $[\mathbb N,+,V_p]$ is.
(To be continued)

References and further reading

1. Recognizable sets of integers, Michel Rigo.
2. Languages, Automata, and Logic, Wolfgang Thomas.
3. Logic and p-recognizable sets of integers, V. Bruyère et al.