2 Formal and informal semantics of basic programs by GL’88

We start by recalling the formal and informal semantics of basic logic programs as they were introduced by Gelfond and Lifschitz (Reference Gelfond, Lifschitz, Kowalski and Bowen1988).

A basic rule is an expression of the form

(1) \begin{equation} A \leftarrow B_1, \dots, B_n,{\mathtt{not}\;} C_1, \dots, {\mathtt{not}\;} C_m, \end{equation}

where $A$ , $B_i$ , and $C_j$ are propositional atoms. The atom $A$ is the head of the rule and the expression $B_1,\ldots, B_n,{\mathtt{not}\;} C_1, \dots, {\mathtt{not}\;} C_m$ is its body. A basic (logic) program is a finite set of such rules. In the sequel, we introduce rules of somewhat different syntactic structure, yet we agree to call the left-hand side of the rule operator/connective, denoted by $\leftarrow$ , head and the right-hand side body. A rule whose body is empty $(n=m=0)$ is called a fact; in such rules connective $\leftarrow$ is often dropped.

For a rule $r$ of the form (1) and a set $X$ of atoms, the reduct $r^X$ is defined whenever there is no atom $C_j$ for $j \in \{1,\ldots, m\}$ such that $C_j \in X$ . If the reduct $r^X$ is defined, then it is the rule

(2) \begin{equation} A \leftarrow B_1,\ldots, B_n. \end{equation}

The reduct $\Pi ^X$ of the program $\Pi$ consists of the rules $r^X$ for all $r\in \Pi$ , for which the reduct is defined. A set $X$ of atoms satisfies rule (2) if $A$ belongs to $X$ or there exists $i\in \{1,\dots, n\}$ such that $B_i\not \in X$ . We say that a set $X$ of atoms is a model of a program consisting of rules of the form (2) when $X$ satisfies all rules of this program. A set $X$ is a stable model/answer set of $\Pi$ , denoted $X \models _{st} \Pi$ , if it is a subset minimal model of $\Pi ^X$ ; A subset minimal model is such that none of its strict subsets is also a model.

Quotes by Gelfond and Lifschitz (Reference Gelfond, Lifschitz, Kowalski and Bowen1988) on Intuitive Meaning of Basic Programs (verbatim modulo names for programs and sets of atoms):

1. Quote 1: The intuitive meaning of stable setsFootnote ¹ can be described in the same way as the intuition behind “stable expansions” in autoepistemic logic: they are “possible sets of beliefs that a rational agent might hold” (Moore, Reference Moore1985) given $\Pi$ as his premises. If $X$ is the set of (ground) atoms that I consider true, then any rule that has a subgoal $not\ C$ with $C\in X$ is, from my point of view, useless; furthermore, any subgoal $not\ C$ with $C\not \in X$ is, from my point of view, trivial. Then I can simplify the premises $\Pi$ and replace them by $\Pi ^X$ . If $X$ happens to be precisely the set of atoms that logically follow from the simplified set of premises $\Pi ^X$ , then I am “rational”.

Later, Gelfond and Lifschitz (Reference Gelfond and Lifschitz1991) say about a basic program the following:

1. Quote 2: A “well-behaved” program has exactly one stable model, and the answer that such a program is supposed to return for a ground query $A$ is yes or no, depending on whether $A$ belongs to the stable model or not. (The existence of several stable models indicates that the program has several possible interpretations).

The historical roots of stable model semantics for logic programs as a formal tool for model-theoretic declarative semantics of Prolog (Kowalski, Reference Kowalski1988) are apparent in these quotes. The expectation is to consider a well-behaved program with a single stable model. Yet, the authors acknowledge the possibility of programs with several stable models that indicates that the program has several possible interpretations or induces several possible sets of beliefs. In this paper, it will prove of value to distinguish between the concepts possible interpretations and possible sets of beliefs. In 1988, these terms were used as synonyms. It is convenient to imagine that the concept of possible interpretation stands behind what we characterize here as ASP, whereas the concept of possible sets of beliefs stands behind ASP-Prolog. We now review these frameworks prior to an attempt to formalize the presented quotes that will result in what we denote as an original informal semantics of basic programs.

2.2 “Formalizing” quotes 1 and 2 or informal semantics of basic programs by GL’88

Before we attempt to make the claims of Quote 1 by Gelfond and Lifschitz (Reference Gelfond, Lifschitz, Kowalski and Bowen1988) precise it is due to discuss three interrelated and yet different concepts and how we understand them within this note:

• • a state of affairs,

• • a belief state, and

• • a set of beliefs/belief set.

The following example is our key vehicle in this discussion.

Example 1. Consider a toy world with four possible states of affairs that fully describe it:

A belief state is associated with/represented by a conglomeration of states of affairs. In other words, a belief state assumes that multiple states of affairs can be deemed as possible by an agent. Thus, a belief state is often associated with an agent who has partial knowledge of the world.

Returning to our toy world, the powerset of the listed four states of affairs forms the set of belief states for an agent operating in this world. For example, if an agent assumes a belief state consisting of states 1, 2, 3, 4 of affairs – let us denote it as $bs_1$ –, we may conclude that this agent deems everything possible (or knows nothing factual about the world). If an agent assumes a belief state consisting of states 1 and 2 of affairs – let us denote it as $bs_2$ –, we may conclude that this agent is aware of the fact that Mary is a student, whereas John may or may not be a student. In turn, if an agent assumes a belief state consisting of a single state 1 – let us denote it as $bs_3$ –, then we may conclude that this agent is aware of the two facts: Mary is a student and John is a student.

We now connect the concepts of a belief set (or, a set of beliefs) and a belief state. The former is an abstraction of the latter. In other words, we understand belief sets as entities that capture/encode belief states. This encoding may lose some information so that multiple belief states may be “consistent” with a single belief set. For instance, in the context of our toy world, a belief set consisting of a single proposition Mary is a student is consistent with any belief state that contains either state 1 of affairs or state 2 of affairs. Note how this belief set cannot distinguish between these different belief states. Indeed, a belief set consisting of a single proposition Mary is a student cannot distinguish between belief states $bs_1$ , $bs_2$ , and $bs_3$ .

We are now ready to return to the claims of Quote 1 by Gelfond and Lifschitz (Reference Gelfond, Lifschitz, Kowalski and Bowen1988) and attempt to make these precise with the allowance that programs with multiple answer sets that correspond to possible sets of beliefs/possible interpretations are as valid programs as so-called well-behaved programs. In other words, per Quote 1 each answer set represents a set of beliefs of a rational agent (or $I$ ); thus this agent may have multiple sets of beliefs. In the sequel, we drop the reference to “or I” and use “an agent” in the discourse.Footnote ⁴ In the case of basic programs, we take an understanding that

(3)

This understanding is consistent with the view by Gelfond and Lifschitz, which is reiterated in Quote 4 in Section 3.

Claim (3) on the interpretation of answer sets has profound ramifications. Namely, this claim makes the three concepts — a state of affairs, a belief state, and a set of beliefs/belief set — exemplified earlier by highlighting their differences collapse into a single entity. Let us use an example to illustrate this point.

Example 2. Recall the toy world from Example 1. Let us take atoms $student(mary)$ and $student(john)$ to represent propositions Mary is a student and John is a student, respectively. If our signature of discourse is composed only of these two atoms, we can construct four distinct subsets of atoms within this signature, namely,

(4) \begin{equation} \{student(mary),\,\,student(john)\};\,\, \{student(mary)\};\,\, \{student(john)\};\,\, \emptyset . \end{equation}

Each of these sets of atoms can be identified in a natural manner with one of the four states of affairs that capture our toy world from Example 1 Footnote ⁵ ; below we rewrite the table presented in that example by substituting propositions depicting distinct states of affairs with the respective sets of atoms:

Alternatively, let us take sets listed in (4) to represent distinct belief states under the assumption that any atom not listed explicitly in a set under consideration is considered to be false. Thus, the belief state $\{student(mary),\,\,student(john)\}$ represents a belief state consisting of a single state of affairs denoted by 1 and represented by the same set of atoms as illustrated above; belief state $\{student(mary)\}$ represents a belief state consisting of a single state of affairs denoted by 2 and represented by the same set of atoms. The same observation holds for the remaining two belief states depicted by sets of atoms in (4). Thus, given the considered settings we may identify belief states and states of affairs. The same argument applies to the concept of a belief set.

We denote the informal semantics for basic programs as $\mathscr{G}_{\mathbb{I}}$ , where $\mathbb{I}$ stands for intended interpretations of the program’s propositional atoms. It is typical in the informal semantics for classical logic expressions that each atom $A$ has an intended interpretation, $\mathbb{I}(A)$ , which is represented linguistically as a noun phrase about the application domain. The informal semantics $\mathscr{G}_{\mathbb{I}}$ consists of three components:

• • the interpretation of structures – here, answer sets – denoted by $\mathscr{G}_{\mathbb{I}}{\mathbb{S}}$ ,

• • the interpretation of syntactical expressions in a program, denoted by $\mathscr{G}_{\mathbb{I}}^{\mathbb{L}}$ , and

• • the interpretation of the semantic relation – here, satisfaction – denoted by $\mathscr{G}_{\mathbb{I}}^{\models }$ .

The first component determines a function from an answer set/a set of beliefs encoded by a set $X$ of atoms to a belief state of some agent (or, a state of affairs). The second component determines the informal reading of syntactical expressions in a program. The third component determines the informal reading of the satisfaction relation.

In the view of informal semantics $\mathscr{G}_{\mathbb{I}}$ , an answer set encodes a belief state of some agent (or, a state of affairs – as we have seen earlier, these concepts are indistinguishable under claim (3)). To reiterate, An agent in some belief state – represented by a set $X$ of atoms – considers the set of all atoms in $X$ to be the case (believes in them), whereas any atom that does not belong to $X$ is believed to be false by the agent, that is is not the case. Thus, we may explain the meaning of a program in terms of what atoms an agent with its knowledge of the application domain encoded as the program believes as true and what atoms an agent believes as false. Generally, an agent in some belief state considers certain states of affairs as possible and others as impossible. For basic programs, set $X$ of atoms defines a unique state of affairs that the agent regards as possible in a belief state that $X$ represents. Thus, we may identify any belief state captured by $X$ with this state of affairs. We denote a state of affairs captured by a set $X$ of atoms under an intended interpretation $\mathbb{I}$ as $\mathscr{G}_{\mathbb{I}}^{\mathbb{S}}(X)$ . Table 1 summarizes a role of an answer set as a state of affairs in the considered view.

Example 3. Consider a set of beliefs encoded as a set

(5) \begin{equation} X=\{student(mary),\,\,male(john)\} \end{equation}

of atoms under the obvious intended interpretation $\mathbb{I}$ for the propositional atoms in $X$ . This $X$ represents a state of affairs in which the agent considers that both statements Mary is a student and John is a male are true. At the same time, the agent considers any other statements, including John is a student and Mary is a male, false. The $\mathscr{G}_{\mathbb{I}}^{\mathbb{S}}(X)$ component of informal semantics of basic programs provides us with this understanding of set $X$ .

Table 1. The Gelfond-Lifschitz (Reference Gelfond, Lifschitz, Kowalski and Bowen1988) informal semantics of answer sets – sets of atoms

Table 2. The Gelfond-Lifschitz (Reference Gelfond, Lifschitz, Kowalski and Bowen1988) informal semantics for basic logic programs

Table 2 shows the Gelfond-Lifschitz (Reference Gelfond, Lifschitz, Kowalski and Bowen1988) informal semantics $\mathscr{G}_{\mathbb{I}}^{\mathbb{L}}$ of syntactic elements of programs. The term material implication used within the table assumes the conformance to the usual truth table of implication and thus a conditional statement can be identified with a disjunction in which the antecedent of the conditional statement is negated. As it is clear from this table, under $\mathscr{G}_{\mathbb{I}}^{\mathbb{L}}$ , extended logic programs have both classical and non-classical connectives.Footnote ⁶ On the one hand, the comma connective (appearing in the body of rules) is classical conjunction and the rule connective $\leftarrow$ is the classical implication. Note that such reading of the comma connective and the rule connective allows us to identify the empty body of the rule with $\top$ – a propositional constant whose value is always interpreted as true – and a rule with an empty body with a simple propositional statement that contains only head of this rule. Not surprisingly such rules are typically denoted as facts. On the other hand, the implicit composition operator (constructing a program out of individual rules) is non-classical because it performs a closure operation (resulting in the implementation of closed world assumption – presumption that what is not currently known to be true is false): only what is explicitly stated is known. To summarize, Table 2 is devoted to the interpretation of syntactical expressions in a program allowing us to “translate” its syntactic elements and the program itself into natural language expressions. For instance, take $\mathbb{I}$ to be an identity function and consider a simple programFootnote ⁷

The annotations to the right are warranted by $\mathscr{G}_{\mathbb{I}}^{\mathbb{L}}$ presented in Table 2. Table 3 presents the final component $\mathscr{G}_{\mathbb{I}}^{\models _{st}}$ of the $\mathscr{G}_{\mathbb{I}}$ informal semantics. In the context of the simple program used to illustrate the findings presented in Table 2, the findings of Table 3 suggest that the only answer set of this program $\{p(b),\,\, q(a)\}$ is a state of affairs inferred from the knowledge encoded in this program so that both $p(b)$ and $q(a)$ are the case and no other proposition is the case.

Table 3. The Gelfond-Lifschitz (Reference Gelfond, Lifschitz, Kowalski and Bowen1988) informal semantics for the satisfaction relation

3 Formal and informal semantics of extended programs by GL’91

Here, we recall the formal and informal semantics of extended logic programs by Gelfond and Lifschitz (Reference Gelfond and Lifschitz1991). An alternative view of the informal semantics for extended logic programs is provided in (Gelfond and Kahl, Reference Gelfond and Kahl2014, Section 2.2.1) reviewed next.

A literal is either an atom $A$ or an expression $\neg A$ , where $A$ is an atom. An extended rule is an expression of the form (1), where $A$ , $B_i$ , and $C_j$ are propositional literals. An extended program is a finite set of extended rules. Gelfond and Lifschitz (Reference Gelfond and Lifschitz1991) also considered disjunctive rules of the form $D_1\ or\ \dots \ or\ D_l \leftarrow B_1, \dots, B_n,{\mathtt{not}\;} C_1, \dots, {\mathtt{not}\;} C_m,$ where $D_k$ , $B_i$ , and $C_j$ are propositional literals. Yet, the discussion of such rules is outside the scope of this note.

A consistent set of propositional literals is a set that does not contain both $A$ and its complement ${\neg }A$ for any atom $A$ . A believed literal set $X$ is a consistent set of propositional literals. A believed literal set $X$ satisfies an extended rule $r$ of the form (1) if $A$ belongs to $X$ or there exists an $i \in \{1,\ldots, n\}$ such that $B_i \not \in X$ or a $j \in \{1,\ldots, m\}$ such that $C_j \in X$ . A believed literal set is a model of a program $\Pi$ if it satisfies all rules in $\Pi$ . For a rule $r$ of the form (1) and a believed literal set $X$ , the reduct $r^X$ is defined whenever there is no literal $C_j$ for $j \in \{1,\ldots, m\}$ such that $C_j \in X$ . If the reduct $r^X$ is defined, then it is the extended rule of the form (2). The reduct $\Pi ^X$ of the program $\Pi$ consists of the rules $r^X$ for all $r\in \Pi$ , for which the reduct is defined. A believed literal set $X$ is an answer set of $\Pi$ , denoted $X \models _{st} \Pi$ , if it is a subset minimal model of $\Pi ^X$ . (A subset minimal model is such that none of its strict subsets is also a model.)

Quotes by Gelfond and Lifschitz (Reference Gelfond and Lifschitz1991) on Intuitive Meaning of Extended Programs

1. Quote 3: For an extended program, we will define when a set $X$ of ground literals qualifies as its answer set. …A “well-behaved” extended program has exactly one answer set, and this set is consistent. The answer that the program is supposed to return to a ground query $A$ is yes, no, or unknown, depending on whether the answer set contains $A$ , $\neg A$ , or neither. The answer no corresponds to the presence of explicit negative information in the program.

   Consider, for instance, the extended program $\Pi _1$ consisting of just one rule:

   \begin{equation*}\neg q\leftarrow not\ p.\end{equation*}
   Intuitively, this rule means: “ $q$ is false, if there is no evidence that $p$ is true.” We will see that the only answer set of this program is $\{\neg q\}$ . The answers that the program should give to the queries $p$ and $q$ are, respectively unknown and false.

   As another example, compare two programs that do not contain not:

   \begin{equation*} \neg p\leftarrow, \ \ \ p\leftarrow \neg q\quad \hbox { and }\quad \neg p\leftarrow, \ \ \ q\leftarrow \neg p \end{equation*}
   … Thus our semantics is not “contrapositive” with respect to $\leftarrow$ and $\neg$ ; it assigns different meanings to the rules $p\leftarrow \neg q$ and $q\leftarrow \neg p$ . The reason is that it interprets expressions like these as inference rules, rather than conditionals.

This quote echos Quote 2 about basic programs: the notion of a well-behaved program resurfaces. In comparison to basic programs, extended programs provide us with a new possibility to answer queries against a program — namely, unknown. The following quote echos Quote 1 about basic programs:

1. The answer sets of $\Pi$ are, intuitively, possible sets of beliefs that a rational agent may hold on the basis of information expressed by the rules of $\Pi$ . If $X$ is the set of (ground) literals that the agent believes to be true, then any rule that has a subgoal not L with $L\in X$ will be of no use to him, and he will view any subgoal not L with $L\not \in X$ as trivial. Thus he will be able to replace the set of rules $\Pi$ by the simplified set of rules $\Pi ^X$ . If the answer set of $\Pi ^X$ coincides with $X$ , then the choice of $X$ as the set of beliefs is “rational”.

The following quote states the precise relationship between basic and extended programs:Footnote ⁸

1. Quote 4: the semantics of extended programs applied to basic programs turns into the stable model semantics. But there is one essential difference: The absence of an atom $A$ in a stable model of a basic program represents the fact that $A$ is false; the absence of $A$ and $\neg A$ in an answer set of an extended program is taken to mean that nothing is known about $A$ .

In the section on Representing Knowledge Using Classical Negation, Gelfond and Lifschitz (Reference Gelfond and Lifschitz1991) say

1. The difference between not p and $\neg p$ in a logic program is essential whenever we cannot assume that the available positive information about $p$ is complete, that is when the “closed world assumption” [Reiter, 1978] is not applicable to $p$ . The closed world assumption for a predicate $p$ can be expressed in the language of extended programs by the rule

   \begin{equation*} \neg p\leftarrow {\mathtt {not}\;} p \end{equation*}
   When this rule is included in the program, $not\ p$ and $\neg p$ can be used interchangeably in the bodies of other rules. Otherwise, we use not p to express that $p$ is not known to be true, and $\neg p$ to express that $p$ is false.

To summarize, Gelfond and Lifschitz describe informal semantics for extended programs based on epistemic notions of default and autoepistemic reasoning. We now present the informal semantics $\mathscr{GL}_{\mathbb{I}}$ for extended programs just as we presented $\mathscr{G}_{\mathbb{I}}$ for basic programs. This presentation at times (in particular, Example4) follows the lines by Denecker et al. (Reference Denecker, Lierler, Truszczynski and Vennekens2019).

We begin by discussing a crucial difference between $\mathscr{G}_{\mathbb{I}}$ and $\mathscr{GL}_{\mathbb{I}}$ . Informal semantics $\mathscr{GL}_{\mathbb{I}}$ views a believed literal set $X$ as an abstraction of a belief state (in fact, of a class of belief states that it cannot distinguish) of some agent; $\mathscr{G}_{\mathbb{I}}$ views a set $X$ of atoms as a state of affairs. The change from “sets of atoms” to “sets of literals” and the elimination of Assumption (3) are crucial. Recall how an agent in some belief state considers certain states of affairs as possible and others as impossible. Within $\mathscr{G}_{\mathbb{I}}$ , set $X$ of atoms ends up representing a unique possible state of affairs associated with a belief state so that we may identify these two concepts. Yet, believed literal set $X$ is the set of all literals $L$ that the agent believes in, that is, those that are true in all states of affairs that the agent regards as possible. Importantly, it is not the case that a literal $L$ that does not belong to $X$ is believed to be false by the agent. Rather, it is not believed by the agent or as stated in Quote 4 nothing is known about $L$ to the agent. Denecker et al. (Reference Denecker, Lierler, Truszczynski and Vennekens2019) take the following interpretation of a statement literal $L$ is not believed by an agent/nothing is known about $L$ : literal $L$ is false in some states of affairs the agent holds possible, and $L$ must be true in at least one of the agent’s possible states of affairs (unless the agent believes the complement of $L$ ). This note adopts such an interpretation. We denote the class of informal belief states that are abstracted to a given formal believed literal set $X$ under an intended interpretation $\mathbb{I}$ as $\mathscr{GL}_{\mathbb{I}}^{\mathbb{S}}(X)$ . Table 4 summarizes a view on a believed literal set as an abstraction of a belief state of some agent.

Table 4. The Gelfond-Lifschitz (1991) informal semantics of answer sets – sets of literals

The component $\mathscr{GL}_{\mathbb{I}}^{\mathbb{L}}$ captures the informal readings of the connectives of the informal semantics of extended programs by Gelfond-Lifschitz (Reference Gelfond and Lifschitz1991). We summarize it by (i) the entries in rows 1, 3–5 of Table 2, where we replace $\mathscr{G}_{\mathbb{I}}^{\mathbb{L}}$ by $\mathscr{GL}_{\mathbb{I}}^{\mathbb{L}}$ , and (ii) the entries in Table 5. The definition of $\mathscr{GL}_{\mathbb{I}}^{\mathbb{L}}$ suggests that of the two negation operators, symbol $\neg$ is classical negation, whereas not is a non-classical negation. It is commonly called default negation. The component $\mathscr{GL}_{\mathbb{I}}^{\models _{st}}$ explains what it means for a believed literal set $X$ to be an answer set/stable model of an extended program. Table 6 presents its definition.

Table 5. The Gelfond-Lifschitz (Reference Gelfond and Lifschitz1991) informal semantics for some expressions in extended programs

Table 6. The Gelfond-Lifschitz (Reference Gelfond and Lifschitz1991) informal semantics for the satisfaction relation

We are now ready to comment on the meaning of querying a well-behaved extended program – a program that has exactly one answer set. Take $X$ to be the unique answer set/believed literal set of a well-behaved extended program. In accordance with Table 6, $X$ is the set of literals the agent believes. In accordance with $\mathscr{GL}_{\mathbb{I}}$ summarized in Table 4, set $X$ is an abstraction of a belief state $B \in \mathscr{GL}_{\mathbb{I}}^{\mathbb{S}}(X)$ , where $B$ belongs to the unique class $C$ of belief states associated with $X$ . In turn, all members of $C$ are indistinguishable by their abstraction $X$ , which characterizes some properties of possible states of affairs associated with all elements in $C$ . Due to the uniqueness of the believed literal set, for the case of well-behaved extended programs, we may simplify the reading of the unique believed literal set $X$ as the abstraction of all possible states of affairs (from the perspective of a considered agent). In other words, what an agent believes coincides with factual information about the world. Take an atom $A$ to be a query. The following table summarizes the interpretation of possible query responses.

Provided account of informal semantics of extended logic programs echos the interpretation of an answer set of an extended program as a possible “set of beliefs” and can be seen as informal semantics for the syntactic constructs that are fundamental in ASP-Prolog.

4 Informal semantics of extended logic programs by GK’14

Gelfond and Kahl (Reference Gelfond and Kahl2014) consider a language of extended logic programs with the addition of (i) disjunctive rules and (ii) rules called constraints that have the form

(6) \begin{equation} \leftarrow B_1, \dots, B_n,{\mathtt{not}\;} C_1, \dots, {\mathtt{not}\;} C_m, \end{equation}

where $B_i$ , and $C_j$ are propositional literals (empty head can be identified with $\bot$ ). It is due to note that constraints have been in the prominent use of ASP/ASP-Prolog for some time. In particular, they are the kinds of rules that populate the test group of generate-define-test programs mentioned earlier. We come back to this point in the next section. To generalize the concept of an answer set to extended programs with constraints, it is sufficient to provide a definition of rule satisfaction when the head of the rule is empty: A believed literal set $X$ satisfies a constraint (6), if there exists an $i \in \{1,\ldots, n\}$ such that $B_i \not \in X$ or a $j \in \{1,\ldots, m\}$ such that $C_j \in X$ . As before, we do not present definitions for programs with disjunctive rules.

Quote by Gelfond and Kahl (Reference Gelfond and Kahl2014) on Intuitive Meaning of Extended Programs with Constraints

1. Informally, program $\Pi$ can be viewed as a specification for answer sets — sets of beliefs that could be held by a rational reasoner associated with $\Pi$ . Answer sets are represented by collections of ground literals. In forming such sets the reasoner must be guided by the following informal principles:

   1. Satisfy the rules of $\Pi$ . In other words, believe in the head of a rule if you believe in its body.

   2. Do not believe in contradictions.

   3. Adhere to the “Rationality Principle” that says, “Believe nothing you are not forced to believe.”

   Let’s look at some examples. $\dots$

   Example 2.2.1.

   
   Note that the second rule is a fact. Its body is empty. Clearly, any set of literals satisfies an empty collection, and hence, according to our first principle, we must believe $q(a)$ . The same principle applied to the first rule forces us to believe $p(b)$ . The resulting set $S1 = \{q(a), p(b)\}$ is consistent and satisfies the rules of the program. Moreover, we had to believe in each of its elements. Therefore, it is an answer set of our program. Now consider set $S2 = \{q(a), p(b), q(b)\}$ . It is consistent and satisfies the rules of the program, but contains the literal $q(b)$ , which we were not forced to believe in by our rules. Therefore, $S2$ is not an answer set of the program.

   Example 2.2.2. (Classical Negation)

   
   There is no difference in reasoning about negative literals. In this case, the only answer set of the program is $\{\neg p(b), \neg q(a)\}$ . $\dots$

   Example 2.2.4. (Constraints)Footnote ⁹

   
   The first rule forces us to believe $p(a)$ or to believe $p(b)$ . The second rule is a constraint that prohibits the reasoner’s belief in $p(a)$ . Therefore, the first possibility is eliminated, which leaves $\{p(b)\}$ as the only answer set of the program. In this example you can see that the constraint limits the sets of beliefs an agent can have, but does not serve to derive any new information. Later we show that this is always the case. $\dots$

   Example 2.2.5. (Default Negation) Sometimes agents can make conclusions based on the absence of information. For example, an agent might assume that with the absence of evidence to the contrary, a class has not been canceled. …. Such reasoning is captured by default negation. Here are two examples.

   
   No rule of the program has $q(a)$ in its head, and hence, nothing forces the reasoner, which uses the program as its knowledge base, to believe $q(a)$ . So, by the rationality principle, he does not. To satisfy the only rule of the program, the reasoner must believe $p(a)$ ; thus, $\{p(a)\}$ is the only answer set of the program. $\dots$

We now state the informal semantics hinted by the quoted examples in unifying terms of this paper. We denote it by $\mathscr{GK}_{\mathbb{I}}$ and detail its three components $\mathscr{GK}_{\mathbb{I}}^{\mathbb{S}}$ , $\mathscr{GK}_{\mathbb{I}}^{\mathbb{L}}$ , and $\mathscr{GK}_{\mathbb{I}}^{\models }$ . To begin with $\mathscr{GK}_{\mathbb{I}}^{\mathbb{S}}$ coincides with $\mathscr{GL}_{\mathbb{I}}^{\mathbb{S}}$ .

Table 7 presents $\mathscr{GK}_{\mathbb{I}}^{\mathbb{L}}$ . In this presentation, we take the liberty to identify an expression

\begin{equation*} \textit {(proposition)} p \textit {does not belong to your set of beliefs} \end{equation*}

used in the examples of the quote listed last with the expression

\begin{equation*} \textit {the agent is not made to believe (proposition)}\, p. \end{equation*}

We summarize $\mathscr{GK}_{\mathbb{I}}^{\models }$ by the entries in Table 6, where we replace $\mathscr{GL}_{\mathbb{I}}^{\models }$ by $\mathscr{GK}_{\mathbb{I}}^{\models }$ .

Table 7. The Gelfond-Kahl (Reference Gelfond and Kahl2014) informal semantics for extended programs with constraints

5 Informal semantics of GDT theories by D-V’12

As discussed earlier, Lifschitz (Reference Lifschitz2002) coined a term generate-define-test for the commonly used methodology when applying ASP toward solving difficult combinatorial search problems. Under this methodology, a program typically consists of three parts: the generate, define, and test groups of rules.

The role of generate is to generate the search space. In modern dialects of ASP choice rules of the form

(7) \begin{equation} \{A\} \leftarrow B_1, \dots, B_n,{\mathtt{not}\;} C_1, \dots, {\mathtt{not}\;} C_m, \end{equation}

are typically used within this part of the program. Symbols $A$ , $B_i$ , and $C_j$ in (7) are propositional atoms. The define part consists of basic rules (1). This part defines concepts required to state necessary conditions in the generate and test parts of the program. The test part is usually modeled by constraints of the form (6), where $B_i$ and $C_j$ are propositional atoms.

Denecker et al. (Reference Denecker, Lierler, Truszczyński and Vennekens2012) defined the logic ASP-FO, where they took the generate, define, and test parts to be the first-class citizens of the formalism. In particular, the ASP-FO language consists of three kinds of expressions: G-modules, D-modules, and T-modules. The authors then present formal and informal semantics of the formalism that can be used in practicing ASP. Here we simplify the language ASP-FO by focusing on its propositional counterpart. We call this language GDT. Focusing on the propositional case of ASP-FO helps us in highlighting the key contribution by Denecker et al. (Reference Denecker, Lierler, Truszczyński and Vennekens2012) – the development of objective informal semantics for logic programs used within ASP or generate-define-test approach.

A G-module is a set of choice rules with the same atom in the head; this atom is called open. A D-module is a basic logic program whose atoms appearing in the heads of the rules are called defined or output. A T-module is a constraint. A GDT theory is a set of G-modules, D-modules, and T-modules so that no G-modules or D-modules coincide on open or defined atoms. To define the semantics for GDT theory, we introduce several auxiliary concepts including that of an input answer set (Lierler and Truszczyński, Reference Lierler and Truszczyński2011) and G-completion. For a basic program $\Pi$ , we call a set $X$ of atoms an input answer set of $\Pi$ if $X$ is an answer set of a program $\Pi \cup (X\setminus{H\!eads(\Pi )})$ , where $H\!eads(\Pi )$ denotes the set of atoms that occur in the heads of the rules in $\Pi$ .

Rules occurring in modules of GDT theory are such that their bodies have the form

(8) \begin{equation} B_1, \dots, B_n,{\mathtt{not}\;} C_1, \dots, {\mathtt{not}\;} C_m. \end{equation}

Given $Body$ of the form (8) by $Body^{cl}$ , we denote a classical formula of the form

\begin{equation*}B_1\wedge \cdots \wedge B_n\wedge \neg C_1 \wedge \cdots \wedge \neg C_m.\end{equation*}

For a G-module $G$ of the form

\begin{equation*}\{\{A\}\leftarrow Body_1, \dots, \{A\}\leftarrow Body_n\}\end{equation*}

by G-completion, $\mathit{Gcomp}(G)$ we denote the classical formula

\begin{equation*} A\rightarrow Body_1^{cl}\vee \cdots \vee Body_n^{cl}.\end{equation*}

For a GDT theory $P$ composed of G-modules $G_1,\dots, G_i$ , D-modules $D_1,\dots, D_j$ , T-modules

\begin{equation*}\leftarrow Body_1,\dots, \leftarrow Body_k,\end{equation*}

we say that set $X$ of atoms is an answer set of $\Pi$ , denoted $X \models _{st} \Pi$ , if

• • $X$ satisfies formulas $\mathit{Gcomp}(G_1)$ , $\dots$ , $\mathit{Gcomp}(G_i)$ (we associate a set $X$ of atoms with an interpretation of classical logic that maps propositional atoms in $X$ to truth value true and propositional atoms outside of $X$ to truth value false; we then understand the concept of satisfaction in usual terms of classical logic.);

• • $X$ is an input answer set of D-modules $D_1\dots D_j$ ; and

• • $X$ satisfies formulas $Body_1^{cl}\rightarrow \bot$ , $\dots$ , $Body_k^{cl}\rightarrow \bot$ .

We refer the reader to Denecker et al. (Reference Denecker, Lierler, Truszczynski and Vennekens2019) to the discussion of Splitting Theorem results that often allows us to identify ASP logic programs with GDT theories.

We now provide the informal semantics for GDT theory $\Pi$ by Denecker et al., (Reference Denecker, Lierler, Truszczyński and Vennekens2012; 2019). We denote it by $\mathscr{DV}_{\mathbb{I}}$ and detail its three components $\mathscr{DV}_{\mathbb{I}}^{\mathbb{L}}$ , $\mathscr{DV}_{\mathbb{I}}^{\mathbb{S}}$ , and $\mathscr{DV}_{\mathbb{I}}^{\models }$ . To begin with $\mathscr{DV}_{\mathbb{I}}^{\mathbb{S}}$ coincides with $\mathscr{G}_{\mathbb{I}}^{\mathbb{S}}$ . We summarize $\mathscr{DV}_{\mathbb{I}}^{\mathbb{L}}$ by (i) the entries in rows 1–3 of Table 2, where we replace $\mathscr{G}_{\mathbb{I}}^{\mathbb{L}}$ by $\mathscr{DV}_{\mathbb{I}}^{\mathbb{L}}$ and (ii) the entries in Table 8. Table 9 presents $\mathscr{DV}_{\mathbb{I}}^{\models }$ . Note how an entry in the right column of Table 9 gives us clues on how to simplify the parallel entry in the right column of Table 3. We can rewrite it as follows: For basic program $\Pi$ , property $\mathscr{G}_{\mathbb{I}}^{\mathbb{L}}(\Pi )$ holds in the state $\mathscr{G}_{\mathbb{I}}^{\mathbb{S}}(X)$ of affairs.

Table 8. The Denecker et al. (Reference Denecker, Lierler, Truszczyński and Vennekens2012) informal semantics for some expressions in GDT theories

Table 9. The Denecker et al. (Reference Denecker, Lierler, Truszczyński and Vennekens2012) informal semantics for the satisfaction relation

Provided account of informal semantics of GDT theories echos the interpretation of an answer set of a basic program as a possible “interpretation” and can be seen as an informal semantics for the syntactic constructs that are fundamental in ASP practice nowadays.