Splitting a Hybrid ASP Program

We will begin with a brief review of ASP. Let $At$ be a nonempty set of symbols called atoms. A block is an expression of the form

$$b_{1},...,b_{k},\;not\;b_{k+1},\;...,\;not\;b_{k+m}$$

(1)

where $b_{1},\;...,\;b_{k+m}$ are atoms. For a block $B$ as above, let the set of atoms of $B$ be defined as $At\left(B\right)\equiv\{b_{1},\;...,\;b_{k+m}\}$. $B^{+}\equiv b_{1},\;...,\;b_{k}$ is called the positive part of $B$, and $B^{-}\equiv$ $not\;b_{k+1},\;...,\;not\;b_{k+m}$ is called the negative part of $B$. A set operation applied to a block $B$ will indicate the same set operation applied to $At\left(B\right)$ with the block being reconstructed from the result of the set operation. For instance $b_{1},\;b_{2},\;not\;b_{3},\;b_{4}\;\backslash\;\{b_{1},\;b_{4}\}$ will indicate a block $b_{2},\;not\;b_{3}$.

A normal propositional logic programming rule is an expression of the form

$$r\equiv a:-\;B$$

(2)

where $a$ is an atom and $B$ is a block. We define the head of $r$ as $head\left(r\right)\equiv a$, and we define the body of $r$ as $body\left(r\right)\equiv B$. We define $At\left(r\right)\equiv\left\{a\right\}\cup At\left(B\right)$.

Given any set $M\subseteq At$ and a block $B$, we say that $M$ satisfies $B$, written $M\models B$, if $At\left(B^{+}\right)\subseteq M$ and $At\left(B^{-}\right)\cap M=\emptyset$. For a rule $r$, we say that $M$ satisfies $r$, written $M\models r$, if whenever $M$ satisfies the body of $r$, then $M$ satisfies the head of $r$. A normal logic program $P$ is a set of rules. We say that $M\subseteq At$ is a model of $P$, written $M\models P$, if $M$ satisfies every rule of $P$.

A Horn rule is the rule with the empty negative part. A Horn program $P$ is a set of Horn rules. Each Horn program $P$ has a least model under inclusion, $LM_{P}$, which can be defined using the one-step provability operator $T\left[P\right]$ as follows. For any set $A$, let $\mathcal{P}\left(A\right)$ denote the set of all subsets of $A$. The one-step provability operator $T\left[P\right]:\mathcal{P}\left(At\right)\rightarrow\mathcal{P}\left(At\right)$ associated with the Horn program $P$ [11] is defined by setting

$$T\left[P\right](M)=M\cup\{a:\exists r\in P\;(a=head(r)\wedge M\models body(r))\}$$

for any $M\in\mathcal{P}\left(At\right)$. We define $T\left[P\right]^{n}(M)$ by induction by setting $T\left[P\right]^{0}\left(M\right)=M$, $T\left[P\right]^{1}(M)=T\left[P\right](M)$ and $T\left[P\right]^{n+1}(M)=T\left[P\right](T\left[P\right]^{n}(M))$. Then the least model $LM_{P}$ can be computed as $LM_{P}=\bigcup_{n\geq 0}T\left[P\right]^{n}(\emptyset).$

If $P$ is a normal logic program and $M\subseteq At$, then the Gelfond-Lifschitz (GL) reduct of $P$ with respect to $M$ [12] is the Horn program $P^{M}$ which results by eliminating those rules $r$ such that $M\not\models body\left(r\right)^{-}$ and replacing other rules $r$ by $head\left(r\right):-body\left(r\right)^{+}$. We then say that $M$ is a stable model for $P$ if $M$ equals the least model of $P^{M}$.

An answer set programming rule is an expression of the form (2) where $a,b_{1},\ldots,b_{k+m}$ are classical literals, i.e., either positive atoms or atoms preceded by the classical negation sign $\lnot$. The set of literals of $At$ will be denoted $Lit_{At}$. Answer sets are defined in analogy to stable models, but taking into account that atoms may be preceded by classical negation and that atoms $a$ and classically negated atoms $\lnot a$ are mutually exclusive in answer sets.

We will now follow [13] in review of the Splitting Set Theorem and the Splitting Sequence Theorem. A splitting set for a program $P$ is any set $U\subseteq At$ such that for every rule $r\in P$ if $head\left(r\right)\in U$ then $At\left(r\right)\subseteq U$. The set of rules $r\in P$ such that $At\left(r\right)\subseteq U$ is called the bottom of $P$ relative to the splitting set $U$ and is denoted by $b_{U}\left(P\right)$. The set $P\backslash b_{U}\left(P\right)$ is the top of $P$ relative to $U$.

Consider $X\subseteq At$. For each rule $r\in P$ such that $At(body\left(r\right)^{+})\cap U\subseteq X$ and $At(body\left(r\right)^{-})\cap U\cap X=\emptyset$ take the rule $r^{\prime}$ defined by

$$head\left(r\right)\;:-\;body\left(r\right)\;\backslash\;U$$

The program consisting of all rules $r^{\prime}$ obtained in this way will be denoted by $\epsilon_{U}\left(P,X\right)$.

A solution to $P$ with respect to $U$ is a pair $\left(X,Y\right)$ of sets of literals such that

• •

  $X$ is an answer set for $b_{U}\left(P\right)$

• •

  $Y$ is an answer set for $\epsilon_{U}\left(P\;\backslash\;b_{U}\left(P\right),\;X\right)$

• •

  $X\cup Y$ is consistent (a set is consistent if for any atom $a$ it does not contain both $a$ and classically negated atom $-a$)

Splitting Set Theorem. Let $U$ be a splitting set for a program $P$. A set $A$ of literals is a consistent answer set for $P$ if and only if $A=X\cup Y$ for some solution $\left(X,Y\right)$ to $P$ with respect to $U$.

We will now review extending the definition of a splitting set to a splitting sequence. A sequence is a family whose index set is an initial segment of ordinals, $\{\alpha:\alpha<\mu\}$. The ordinal $\mu$ is the length of the sequence. A sequence $\left\langle U_{\alpha}\right\rangle_{\alpha<\mu}$ of sets is monotone if $U_{\alpha}\subset U_{\beta}$ whenever $\alpha<\beta$, and continuous if, for each limit ordinal $\alpha<\mu$, $U_{\alpha}={\displaystyle\bigcup\limits_{\beta<\alpha}}U_{\beta}$.

A splitting sequence for a program $P$ is a monotone, continuous sequence $\left\langle U_{\alpha}\right\rangle_{\alpha<\mu}$ of splitting sets for $P$ such that ${\displaystyle\bigcup\limits_{\alpha<\mu}}U_{\alpha}=Lit_{At}$. The definition of a solution with respect to a splitting set is extended to splitting sequence as follows. A solution to $P$ with respect to $\left\langle U_{\alpha}\right\rangle_{\alpha<\mu}$ is a sequence $\left\langle X_{\alpha}\right\rangle_{\alpha<\mu}$ of sets of literals such that

• •

  $X_{0}$ is an answer set for $b_{U_{0}}\left(P\right)$,

• •

  for any $\alpha$ such that $\alpha+1<\mu$, $X_{\alpha+1}$ is an answer set for $\epsilon_{U_{\alpha}}(b_{U_{\alpha+1}}\left(P\right)\backslash b_{U_{\alpha}}\left(P\right),\;{\displaystyle\bigcup\limits_{\beta\leq\alpha}}X_{\beta})$,

• •

  for any limit ordinal $\alpha<\mu$, $X_{\alpha}=\emptyset$,

• •

  ${\displaystyle\bigcup\limits_{\alpha<\mu}}X_{\alpha}$ is consistent.

Splitting Sequence Theorem. Let $U\equiv\left\langle U_{\alpha}\right\rangle_{\alpha<\mu}$ be a splitting sequence for a program $P$. A set $A$ of literals is a consistent answer set for $P$ if and only if $A={\displaystyle\bigcup\limits_{\alpha<\mu}}X_{\alpha}$ for some solution $\left\langle X_{\alpha}\right\rangle_{\alpha<\mu}$ to $P$ with respect to $U$.

We will now proceed with the review of Hybrid ASP. A Hybrid ASP program $P$ has an underlying parameter space $S$. Elements of $S$ are of the form $\mathbf{p}=(t,x_{1},\ldots,x_{l})$ where $t$ is time and $x_{i}$ are arbitrary parameter values. We shall let $t(\mathbf{p})$ denote $t$ and $x_{i}(\mathbf{p})$ denote $x_{i}$ for $i=1,\ldots,l$. We refer to the elements of $S$ as generalized positions. Let $At$ be a set of atoms of $P$. Then the universe of $P$ is $At\times S$. Let $B$ be a block. We will define

$$B\times\mathbf{p}\equiv\{\left(x,\mathbf{p}\right):\;x\in B\}.$$

If $M\subseteq At\times S$, we let $GP(M)=\{\mathbf{p}\in S:(\exists a\in At)((a,\mathbf{p})\in M)\}$. Given an initial condition, defined as a subset $I\subseteq S$ let $GP_{I}\left(M\right)=GP\left(M\right)\cup I$. Given $M\subseteq At\times S$ and $\mathbf{p}\in S$, we say that $M$ and initial condition $I$ satisfy a block $B$ of the form (1) at the generalized position $\mathbf{p}$, written $M\models_{I}\left(B,\mathbf{p}\right)$, if the following holds:

• •

  if $B^{+}\neq\emptyset$ then $B^{+}\times\mathbf{p}\subseteq M$ and $B^{-}\times\mathbf{p\cap M=\emptyset}$

• •

  if $B^{+}=\emptyset$ then $B^{-}\times\mathbf{p\cap M=\emptyset}$ and $\mathbf{p}\in GP_{I}\left(M\right)$.

We say that $M$ satisfies a n-tuple of blocks written as $B_{1};\;...;\;B_{n}$ with the initial condition $I$ at the n-tuple of generalized positions $(\mathbf{p}_{1},\;...,\;\mathbf{p}_{n})$, written $M\models_{I}(B_{1};\;...;B_{n},\;(\mathbf{p}_{1},\;...,\mathbf{p}_{n}))$, if $M\models_{I}(B_{i},\mathbf{p}_{i})$ for $i=1,...,n$.

There are two types of rules in Hybrid ASP. Advancing rules are of the form

$$r\equiv a:-B_{1};B_{2};\ldots;B_{n}:A,O$$

(3)

where $A$ is a function returning a set of generalized positions, $body\left(r\right)\equiv B_{1},\;...,\;B_{n}$ are blocks, $head\left(r\right)\equiv a$ is a literal, and $O$ is a subset of $S^{n}$ such that if $(\mathbf{p}_{1},\ldots,\mathbf{p}_{n})\in O$, then $t(\mathbf{p}_{1})<\cdots<t(\mathbf{p}_{n})$ and $A\left(\mathbf{p}_{1},\ldots,\mathbf{p}_{n}\right)$ ($A$ applied to $\mathbf{p}_{1},\ldots,\mathbf{p}_{n}$) is a subset of $S$ such that for all $\mathbf{q}\in A\left(\mathbf{p}_{1},\ldots,\mathbf{p}_{n}\right)$, $t(\mathbf{q})>t(\mathbf{p}_{n})$. Here and in the next rule, we allow blocks to be empty for any $i$. $O$ is called the constraint set of the rule $r$ and will be denoted by $CS(r)$. $A$ is called the advancing algorithm of the rule $r$ and is denoted by $\operatorname*{Adv}(r)$. The arity of rule $r$, $N\left(r\right)$, is equal to $n$.

The idea is that if $(\mathbf{p}_{1},\ldots,\mathbf{p}_{n})\in O$ and for each $i$, $B_{i}$ is satisfied at the generalized position $\mathbf{p}_{i}$, then the function $A$ can be applied to $(\mathbf{p}_{1},\ldots,\mathbf{p}_{n})$ to produce a set of generalized positions $O^{\prime}$ such that if $\mathbf{q}\in O^{\prime}$, then $t(\mathbf{q})>t(\mathbf{p}_{n})$ and $(a,\mathbf{q})$ holds. Thus advancing rules are like input-output devices in that the function $A$ allows the user to derive possible successor generalized positions as well as certain atoms $a$ which are to hold at such positions. The advancing algorithm $A$ can access outside sources quite arbitrarily in that it may involve functions for solving differential or integral equations, solving a set of linear equations or linear programming equations, solving an optimization problem, etc. (as for example in [6]).

Stationary rules are of the form

$$r\equiv a:-B_{1};B_{2};\ldots;B_{n}:H,O$$

(4)

where $body\left(r\right)\equiv B_{1},...,B_{n}$ are blocks, $head\left(r\right)\equiv a$ is a literal, $H$ is called a boolean algorithm of the rule $r$ and will be denoted by $Bool\left(r\right)$, and $O\subseteq S^{k}$ is the constraint set of the rule $r$ denoted $CS(r)$. A boolean algorithm is a function returning either true or false. We will sometimes treat a boolean algorithm of the rule as a set. For instance $H\cap O$ will indicate all the n-tuples of generalized positions $(\mathbf{p}_{1},\ldots,\mathbf{p}_{n})$ such that $H\left(\mathbf{p}_{1},\ldots,\mathbf{p}_{n}\right)$ is true and $(\mathbf{p}_{1},\ldots,\mathbf{p}_{n})\in O$. The arity of rule $r$, $N\left(r\right)$, is equal to $n$.

Stationary rules are much like normal logic programming rules in that they allow us to derive new atoms at a given generalized position $\mathbf{p}_{n}$. The idea is that if $(\mathbf{p}_{1},\ldots,\mathbf{p}_{n})\in O\cap H$ and for each $i$, $B_{i}$ is satisfied at the generalized position $\mathbf{p}_{i}$, then $(a,\mathbf{p}_{n})$ holds. The difference is that a derivation with our stationary rules can depend on what happens in the multiple past time points and the boolean algorithm $H$ can be any sort of a function which returns either true or false.  

For an advancing rule or a stationary rule $r$ as above we define the positive part of the body of $r$, denoted $body\left(r\right)^{+}\equiv B_{1}^{+};...;B_{n}^{+}$ and we define the negative part of the body of $r$, denoted $body\left(r\right)^{-}\equiv B_{1}^{-};...;B_{n}^{-}$. For the rest of the paper, we denote by $n$ the arity of a hybrid ASP rule when the rule is clear from the context.

A Hybrid ASP program $P$ is a collection of Hybrid ASP advancing and stationary rules. To define the notion of a stable model of $P$, we first must define the notion of a Hybrid ASP Horn program and the one-step provability operator for Hybrid ASP Horn programs.

A Hybrid ASP Horn program is a Hybrid ASP program which does not contain any negated atoms. Let $P$ be a Horn Hybrid ASP program and $I\subseteq S$ be an initial condition. Then the one-step provability operator $T\left[P,I\right]$ is defined so that given $M\subseteq At\times S$, $T\left[P,I\right](M)$ consists of $M$ together with the set of all $(a,J)\in At\times S$ such that

1. 1.

   there exists a stationary rule $r$ and $(\mathbf{p}_{1},\ldots,\mathbf{p}_{n})\in CS\left(r\right)\cap Bool\left(r\right)\cap\left(GP_{I}(M)\right)^{n}$ such that $(head\left(r\right),J)=(a,\mathbf{p}_{n})$ and $M\models(body\left(r\right),\;(\mathbf{p}_{1},...,\mathbf{p}_{n}))$ or

2. 2.

   there exists an advancing rule $r$ and $(\mathbf{p}_{1},\ldots,\mathbf{p}_{n})\in CS\left(r\right)\cap\left(GP_{I}(M)\right)^{n}$ such that $J\in\operatorname*{Adv}\left(r\right)(\mathbf{p}_{1},\ldots,\mathbf{p}_{n})$ and $M\models(body\left(r\right),\;(\mathbf{p}_{1},...,\mathbf{p}_{n}))$ and $a=head\left(r\right)$.

The stable model semantics for Hybrid ASP programs is defined as follows. Let $M\subseteq At\times S$ and $I$ be an initial condition in $S$. An Hybrid ASP rule $r\equiv a:-B_{1};\ldots,B_{n}:A,O$ is inapplicable for $(M,I)$ if for all $(\mathbf{p}_{1},\ldots,\mathbf{p}_{n})\in O\cap\left(GP_{I}(M)\right)^{n}$, either (i) there is an $i$ such that $M\not\models(B_{i}^{-},\mathbf{p}_{i})$, (ii) $A\left(\mathbf{p}_{1},\ldots,\mathbf{p}_{n}\right)\cap GP_{I}(M)=\emptyset\,$ if $A$ is an advancing algorithm, or (iii) $A(\mathbf{p}_{1},\ldots,\mathbf{p}_{n})=0$ if $A$ is a boolean algorithm.

If $r$ is not inapplicable for $\left(M,I\right)$ then we define the GL reduct of $r$ over $M$ and $I$, denoted by $r^{M,I}$ as follows:

1. 1.

   If $r$ is an advancing rule $r\equiv a:-B_{1};...;B_{n}:A,O$ then $r^{M,I}\equiv B_{1}^{+};\ldots,B_{n}^{+}:A^{M,I},O^{M,I}$ where $O^{M,I}$ is equal to the set of $(\mathbf{p}_{1},\ldots,\mathbf{p}_{n})$ in $O\cap\left(GP_{I}(M)\right)^{n}$ such that  $M\models_{I}(body\left(r\right)^{-},\;(\mathbf{p}_{1},\ldots,\mathbf{p}_{n}))$ and $A(\mathbf{p}_{1},\ldots,\mathbf{p}_{n})\cap GP_{I}(M)\neq\emptyset$, and $A^{M,I}(\mathbf{p}_{1},\ldots,\mathbf{p}_{n})\equiv A(\mathbf{p}_{1},\ldots,\mathbf{p}_{n})\cap GP_{I}(M)$.

2. 2.

   If $r$ is a stationary rule $r\equiv a:-B_{1};...;B_{n}:A,O$ then $r^{M,I}\equiv a:-B_{1}^{+};\ldots,B_{n}^{+}:H|_{O^{M,I}},O^{M,I}$ where $O^{M,I}$ is equal to the set of all $(\mathbf{p}_{1},\ldots,\mathbf{p}_{n})$ in $O\cap\left(GP_{I}(M)\right)^{n}$ such that $\ M\models_{I}(body\left(r\right)^{-},\;(\mathbf{p}_{1},\ldots,\mathbf{p}_{n}))$ and $H(\mathbf{p}_{1},\ldots,\mathbf{p}_{n})$ is true.

One note to make about the definition above is that GL reduct cannot derive generalized positions that are not in $GP_{I}\left(M\right)$. This is because the range of $A^{M,I}$ in the definition is restricted to $GP_{I}(M)$.

We form a GL reduct of $P$ over $M$ and $I$, $P^{M,I}$ as follows.

1. 1.

   Eliminate all rules which are inapplicable for $(M,I)$.

2. 2.

   If a rule $r\in P$ is not eliminated in step 1, then replace it by the rule $r^{M,I}$.

We then say that $M$ is a stable model of $P$ with initial condition $I$ if ${\displaystyle\bigcup\limits_{k=0}^{\infty}}T\left[P^{M,I},I\right]^{k}\left(\emptyset\right)=M.$

Answer sets are defined in analogy to stable models, but taking into account that atoms may be preceded by classical negation and that $(a,\mathbf{p)}$ and $\left(-a,\mathbf{p}\right)$ are mutually exclusive in answer sets.

We will now introduce additional notation that will be used throughout the rest of the paper.

Without loss of generality assume that all advancing rules are of the form

$$a:-B_{1};\;...;\;B_{n}:O,A$$

and all of stationary rules are of the form

$$a:-B_{1};\;...;\;B_{n}:O,H$$

where $a$ is a literal, $B_{1}$, …, $B_{n}$ are blocks, $O$ is a constraint set, $A$ is an advancing algorithm, and $H$ is a boolean algorithm.

Let $M$ be a set of literals and generalized position pairs, and let $\mathbf{p}$ be a generalized position. Define

$$M|_{\mathbf{p}}\equiv\{\left(a,\mathbf{q}\right)\in M\;:\;\mathbf{q=p}\}$$

$$At\left(M\right)\equiv\{a\;:\;\left(a,\mathbf{p}\right)\in M\}$$

Let $U\subseteq Lit_{At}\times S$. We say that $U$ is a splitting set of $P$ with initial condition (w.i.c.) $J$ if for all $r\in P$

1. 1.

   if $r$ is advancing and $(\mathbf{p}_{1},\;...,\;\mathbf{p}_{n})\in CS\left(r\right)$ and $\mathbf{p}\in Adv\left(r\right)\left(\mathbf{p}_{1},\;...,\;\mathbf{p}_{n}\right)$ and $\left(a,\mathbf{p}\right)\in U$ then both for $i=1,...,n$, $B_{i}\times\mathbf{p}_{i}\subseteq U$ and $\{\mathbf{p}_{1},\;...,\;\mathbf{p}_{n}\}\subseteq GP_{J}\left(U\right)$.

2. 2.

   if $r$ is stationary and $(\mathbf{p}_{1},\;...,\;\mathbf{p}_{n})\in CS\left(r\right)$ and $\left(a,\mathbf{p}_{n}\right)\in U$ then both for $i=1,...,n$, $B_{i}\times\mathbf{p}_{i}\subseteq U$ and $\{\mathbf{p}_{1},\;...,\;\mathbf{p}_{n}\}\subseteq GP_{J}\left(U\right)$.

As in the case of the original splitting set theorem [13] the splitting set $U$ acts to split Hybrid ASP program $P$ into the part that can derive $U$ or one of its subsets, and the remaining part of $P$, which can derive $At\times S\backslash U$ or one of its subsets. The difference, however, is that for a given rule the conclusion of the rule may be in $U$ for some n-tuples of generalized positions ($\mathbf{p}_{1},\;...,\;\mathbf{p}_{n}$) and not for others. So, the splitting set splits not only the program, but the rules themselves. This will be elaborated below.

As in the case of the original splitting set theorem we identify by $b_{U}\left(P\right)$ a set of new rules that capture the rules and generalized positions that may contribute to generating $U$.

Define $Rules_{b}\left(U,P\right)$ as

$$\{\;r\in P\;:\;\text{if }r\text{ is advancing and there exists }(\mathbf{p}_{1},\;...,\;\mathbf{p}_{n})\in CS\left(r\right)$$

$$\text{and }\mathbf{p}\in Adv\left(r\right)(\mathbf{p}_{1},\;...,\;\mathbf{p}_{n})\text{ such that }\left(a,\mathbf{p}\right)\in U$$

$$\text{if }r\text{ is stationary and there exists }(\mathbf{p}_{1},\;...,\;\mathbf{p}_{n})\in CS\left(r\right)\cap Bool\left(r\right)$$

$$\text{such that }\left(a,\mathbf{p}_{n}\right)\in U\;\}$$

In other words, $Rules_{b}\left(U,P\right)$ is the set of all rules of $P$ that could contribute to $U$ for some tuple of generalized positions.

For an advancing rule $r$ let

$$CS_{b}\left(U,r\right)\equiv\{\;(\mathbf{p}_{1},\;...,\;\mathbf{p}_{n})\in CS\left(r\right)\;:$$

$$\text{there exists }p\in Adv\left(r\right)(\mathbf{p}_{1},\;...,\;\mathbf{p}_{n})\text{ such that }\left(a,\mathbf{p}\right)\in U\;\}$$

For a stationary rule $r$ let

$$CS_{b}\left(U,r\right)\equiv\{\;(\mathbf{p}_{1},\;...,\;\mathbf{p}_{n})\in CS\left(r\right)\cap Bool\left(r\right)\;:\;\left(a,\mathbf{p}_{n}\right)\in U\;\}$$

That is $CS_{b}\left(U,r\right)$ are all the generalized position tuples for which $r$ could contribute to $U$.

For an advancing rule $r\in Rules_{b}\left(U,P\right)$ define $Adv_{b}\left(U,r\right)$ by

$$Adv_{b}\left(U,r\right)(\mathbf{p}_{1},\;...,\;\mathbf{p}_{n})\equiv\{\;\mathbf{p}\;:\ \mathbf{p}\in Adv\left(r\right)(\mathbf{p}_{1},\;...,\;\mathbf{p}_{n})$$

$$\text{such that }\left(a,\mathbf{p}\right)\in U\text{ if }(\mathbf{p}_{1},\;...,\;\mathbf{p}_{n})\in CS_{b}\left(U,r\right)\;\}$$

$Adv_{b}\left(U,r\right)$ is an advancing algorithm that for any tuple of generalized positions will only generate those $\mathbf{p}$ that contribute to $U$.

For an advancing rule $r$ let

$$b_{U}\left(r\right)\equiv head\left(r\right):-\;body\left(r\right):CS_{b}\left(U,r\right),\;Adv_{b}\left(U,r\right)$$

For a stationary rule $r$ let

$$b_{U}\left(r\right)\equiv head\left(r\right):-\;body\left(r\right):CS_{b}\left(U,r\right),\;Bool\left(r\right)$$

Define the bottom of $P$ with respect to $U$, $b_{U}\left(P\right)$ as

$$b_{U}\left(P\right)\equiv\{\;b_{U}\left(r\right)\;:\;r\in Rules_{b}\left(U,P\right)\;\}$$

The idea is that just like in [13], $b_{U}\left(P\right)$ forms only those rules that could contribute to $U$, and so $X$ will be an answer set of $b_{U}\left(P\right)$ w.i.c. $J$ iff $M\cap U=X$ for some answer set $M$ of $P$ w.i.c. $J$.

We will now proceed to define $\epsilon_{U}\left(P,X\right)$ with the understanding that the same rule may contribute to $U$ for some generalized position tuples and contribute to $Lit_{At}\times S\;\backslash U$ for others.

First, we need to identify remainder $\operatorname*{Rem}\left(U,P\right)$ of $Rules_{b}\left(U,P\right)$ not captured by $b_{U}\left(P\right)$. That is we need to identify the parts contributing to the complement of $U$ of those rules that have other parts contributing to $U$. This is due to an important difference between Hybrid ASP and ASP. In ASP a rule contributes a single conclusion. Thus if ASP rule contributes to the splitting set then it must be in the bottom of the program. In Hybrid ASP, however, a rule acts more like a collection of rules contributing different conclusions for different generalized position tuples. Consequently, the parts of the rules that contribute to the complement of the splitting set need to be separated from those that contribute to the splitting set itself. We will now proceed with the definition.

For an advancing rule $r$ define

$$CS_{\operatorname*{Rem}}\left(U,r\right)\equiv\{\ (\mathbf{p}_{1},\;...,\;\mathbf{p}_{n})\in CS\left(r\right):$$

$$\text{there exists }\mathbf{p}\in Adv\left(r\right)(\mathbf{p}_{1},\;...,\;\mathbf{p}_{n})\text{ }\left(a,\mathbf{p}\right)\notin U\;\}$$

For a stationary $r$ define

$$CS_{\operatorname*{Rem}}\left(U,r\right)\equiv\{\;(\mathbf{p}_{1},\;...,\;\mathbf{p}_{n})\in CS\left(r\right)\cap Bool\left(r\right):\;\left(a,\mathbf{p}_{n}\right)\notin U\;\}$$

That is, $CS_{\operatorname*{Rem}}\left(U,r\right)$ contains those generalized position tuples such that for them the rule $r$ contributes to the complement of $U$.

For an advancing rule $r\in Rules_{b}\left(U,P\right)$ and $(\mathbf{p}_{1},\;...,\;\mathbf{p}_{n})$ define

$$Adv_{\operatorname*{Rem}}(U,r)(\mathbf{p}_{1},\;...,\;\mathbf{p}_{n})\equiv\left\{\begin{array}[c]{l}\{\mathbf{p}\;:\;\mathbf{p}\in Adv\left(r\right)(\mathbf{p}_{1},\;...,\;\mathbf{p}_{n})\text{ s.t. }\left(a,\mathbf{p}\right)\notin U\;\}\\ \text{ \ \ \ if }(\mathbf{p}_{1},\;...,\;\mathbf{p}_{n})\in CS_{\operatorname*{Rem}}\left(U,r\right)\\ \emptyset\text{ if }(\mathbf{p}_{1},\;...,\;\mathbf{p}_{n})\notin CS_{\operatorname*{Rem}}(U,r)\end{array}\right.$$

That is $Adv_{\operatorname*{Rem}}(U,r)$ is a restriction of $Adv\left(r\right)$ to those generalized positions such that for them $r$ contributes to the complement of $U$.

When $CS_{\operatorname*{Rem}}\left(U,r\right)\neq\emptyset$ define

$$\operatorname*{Rem}\left(U,r\right)\equiv\left\{\begin{array}[c]{c}head\left(r\right):-\;body\left(r\right):CS_{\operatorname*{Rem}}\left(U,r\right),\;Adv_{\operatorname*{Rem}}\left(U,r\right)\text{ if }r\text{ is advancing}\\ head\left(r\right):-\;body\left(r\right):CS_{\operatorname*{Rem}}\left(U,r\right),\;Bool\left(r\right)\text{ if }r\text{ is stationary}\end{array}\right.$$

In other words, $\operatorname*{Rem}\left(U,r\right)$ is the part of $r$ that contributes to the complement of $U$.

Define

$$\operatorname*{Rem}\left(U,P\right)\equiv\{\;\operatorname*{Rem}\left(U,r\right):\;r\in Rules_{b}\left(U,P\right)\text{ and }CS_{\operatorname*{Rem}}\left(U,r\right)\neq\emptyset\;\}$$

That is $\operatorname*{Rem}\left(U,P\right)$ contain those parts of the rules in $Rules_{b}\left(U,P\right)$ that contribute to the complement of $U$.

Let $X\subseteq U$. For a rule $r$ define

$$CS_{\epsilon}\left(U,r,X\right)\equiv\{\;(\mathbf{p}_{1},\;...,\;\mathbf{p}_{n})\in CS\left(r\right):$$

$$\text{for }i=1,\;...,\;n\text{ }\{B_{i}^{+}\times\mathbf{p}_{i}\}\cap U\subseteq X\text{ and }\{B_{i}^{-}\times\mathbf{p}_{i}\}\cap X=\emptyset\;\}$$

That is $CS_{\epsilon}\left(U,r,X\right)$ is the set of those generalized position tuples such that for them the ”projection” of $body\left(r\right)$ onto $U$ is satisfied by $X$.

Finally

$$\epsilon_{U}\left(P,X\right)\equiv\{$$

$$r^{\prime}\equiv a:-\;B_{1}\backslash At\left(U|_{\mathbf{p}_{1}}\right);\;...;\;B_{n}\backslash At\left(U|_{\mathbf{p}_{n}}\right):\;\{(\mathbf{p}_{1},\;...,\;\mathbf{p}_{n})\},\;Q\text{ }|$$

$$r\equiv a:-\;B_{1};\;...;\;B_{n}:O,Q\;\in\;\{\;r\in P:\;CS\left(\epsilon_{U},r,X\right)\neq\emptyset\;\}\text{ and}$$

$$(\mathbf{p}_{1},\;...,\;\mathbf{p}_{n})\in CS_{\epsilon}\left(U,r,X\right)\;\}$$

In other words, for every rule $r\in P$ such that$\;CS\left(\epsilon_{U},r,X\right)\neq\emptyset$ and for every $(\mathbf{p}_{1},\;...,\;\mathbf{p}_{n})\in CS_{\epsilon}\left(U,r,X\right)$ where the ”projection” of $body\left(r\right)$ onto $U$ is satisfied by $X$ at $(\mathbf{p}_{1},\;...,\;\mathbf{p}_{n})$, we add to $\epsilon_{U}\left(P,X\right)$ a rule $r^{\prime}$, which is a part of rule $r$ that will be active only for that $(\mathbf{p}_{1},\;...,\;\mathbf{p}_{n})$ with the ”projection” part removed.

Theorem 1. (The Splitting Set Theorem for Hybrid ASP). Let $P$ be a Hybrid ASP program over $Lit_{At}\times S$. Let $U\subseteq Lit_{At}\times S$ be a splitting set of $P$ w.i.c. $J\subseteq S$. A set $M$ is a answer set of $P$ w.i.c. $J$ iff $X\equiv M\cap U$ is a answer set of $b_{U}\left(P\right)$ w.i.c. $J$ and $M\backslash U$ is a answer set of $\epsilon_{U}(P\backslash Rules_{b}\left(U,P\right)\cup\operatorname*{Rem}\left(U,P\right),\;X)$ w.i.c. $GP_{J}\left(X\right)$.

Sketch of a proof. We first prove that if $M$ is an answer set of $P$ w.i.c. $J$ then $X\equiv M\cap U$ is an answer set of $b_{U}\left(P\right)$ w.i.c. $J$. That is, we want to show that $X={\displaystyle\bigcup\limits_{k=0}^{\infty}}T\left[b_{U}\left(P\right)^{X,J},J\right]^{k}\left(\emptyset\right)$. In $\supseteq$ direction we show by induction on $k$ in one-step provability operator $T\left[b_{U}\left(P\right)^{X,J},J\right]^{k}$ that if a rule $b_{U}\left(r\right)^{X,J}$ in $b_{U}\left(P\right)^{X,J}$ derives $\left(a,\mathbf{p}\right)$ in $T\left[b_{U}\left(P\right)^{X,J},J\right]^{k+1}\left(\emptyset\right)$, then the rule $r^{M,J}$ must derive $\left(a,\mathbf{p}\right)$ in $T\left[P^{M,J},J\right]^{m+1}\left(\emptyset\right)$ for some $m$. In $\subseteq$ direction we show by induction on $k$ in $T\left[P^{M,J},J\right]^{k}\left(\emptyset\right)$ that if $r^{M,J}$ derives $\left(a,\mathbf{p}\right)$ in $T\left[P^{M,J},J\right]^{k+1}\left(\emptyset\right)$ where $\left(a,\mathbf{p}\right)\in U$, then $b_{U}\left(r\right)^{X,J}$ derives $\left(a,\mathbf{p}\right)$ in $T\left[b_{U}\left(P\right)^{X,J},J\right]^{m+1}\left(\emptyset\right)$ for some $m$.

We then proceed to prove that if $M$ is an answer set of $P$ w.i.c. $J$, and $Y\equiv M\backslash U$ then $Y$ is an answer set of $Q\equiv\epsilon_{U}(P\backslash Rules_{b}\left(U,P\right)\cup\operatorname*{Rem}\left(U,P\right),X)$ w.i.c. $L\equiv GP_{J}\left(X\right)$. That is, we want to show that $Y={\displaystyle\bigcup\limits_{k=0}^{\infty}}T\left[Q^{Y,L},L\right]^{k}\left(\emptyset\right)$. In $\supseteq$ direction we prove by induction that if $r^{Y,L}$ derives $\left(a,\mathbf{p}\right)$ in $T\left[Q^{Y,L},L\right]^{k+1}\left(\emptyset\right)$ then there is a corresponding rule $q^{M,J}$ in $P^{M,J}$ that derives $\left(a,\mathbf{p}\right)$ in $T\left[P^{M,J},J\right]^{m+1}\left(\emptyset\right)$ for some $m$. In $\subseteq$ direction we prove by induction on $k$ in $T\left[P^{M,J},J\right]^{k}\left(\emptyset\right)$ that if $q^{M,J}$ derives $\left(a,\mathbf{p}\right)$ in $T\left[P^{M,J},J\right]^{k+1}\left(\emptyset\right)$ where $\left(a,\mathbf{p}\right)\in M\backslash U$ then there is a corresponding $r$ in $Q^{Y,L}$ that derives $\left(a,\mathbf{p}\right)$ in $T\left[Q^{Y,L},L\right]^{m+1}\left(\emptyset\right)$ for some $m$.