Promise Algebra:
An Algebraic Model of Non-Deterministic Computations

The semantics of the algebra is parameterized by an underlying transition system that specifies structure-to-structure transitions associated with each atomic module symbol. Formally, we have a transition relation $\textbf{Tr}\llbracket{\cdot}\rrbracket$ that maps each atomic module symbol in $\mathcal{M}$ to a binary relation on ${\bf U}$ (the set of all relational $\tau$-structures over the same domain):

$$\textbf{Tr}\llbracket{\cdot}\rrbracket:\mathcal{M}\to{\bf U}\times{\bf U}.$$

In general, the relation $\textbf{Tr}\llbracket{\cdot}\rrbracket$ is specified in any (secondary) logic that constitutes the bottom level of the algebra. In Section 0.6, we give a specific example of such a logic. But, for now, it is sufficient to think of $\textbf{Tr}\llbracket{\cdot}\rrbracket$ as an arbitrary binary relation on structures in ${\bf U}$. The relation $\textbf{Tr}\llbracket{m(\varepsilon)}\rrbracket\subseteq{\bf U}\times{\bf U}$, for $m\in\mathcal{M}$, is not necessarily functional. In this sense, atomic transitions are non-deterministic. But, specific Choice functions (to be defined shortly), that are given semantically only, resolve atomic non-determinism, while taking the history into account.⁵⁵5Notice that concrete Choice functions are not a part of the syntax. The syntax, summarized in (2) and detailed in the table, has only a Choice function variable $\varepsilon$. The terms impose constraints on possible Choice functions. To explain it formally, we first give a definition of a tree.

A tree over alphabet ${\bf U}$ is a (finite or infinite) nonempty set ${\cal T}\subseteq{\bf U}^{*}$ such that for all $x\cdot c\in{\cal T}$, with $x\in{\bf U}^{*}$ and $c\in{\bf U}$, we have $x\in{\cal T}$. The elements of ${\cal T}$ are called nodes, and the empty word $\mathbf{e}$ is the root of ${\cal T}$. For every $x\in{\cal T}$, the nodes $x\cdot c\in{\cal T}$ where $c\in{\bf U}$ are the children of $x$. A node with no children is a leaf.

Each prefix $s_{i}$ of a string $s$ represents a history, i.e., a sequence of states – elements of ${\bf U}$. For each history $s_{i}$, Choice function $h$ selects one possible outcome of an atomic module $m(\varepsilon)$, out of those possible, i.e., those in its interpretation $\textbf{Tr}\llbracket{m(\varepsilon)}\rrbracket$, see Figure 1. It decides $m(\varepsilon)$’s outcome at $s_{i}$, if such an outcome is possible according to the transition system $\textbf{Tr}\llbracket{\cdot}\rrbracket$. The outcome is history-dependent, that is, for the same module $m(\varepsilon)$, we may have different outcomes at different time points $s_{i}$ and $s_{j}$, $i\neq j$. We now present this intuition formally.

The second case, $i=0$, is included so that we can have one tree (cf. Figure 1) per each term, with branches that correspond to different Choice functions, including the choice of allowable input structures. More specifically, when $i=0$, we have $s_{i}=\mathbf{e}$. According to the definition above, if $(\mathfrak{A},\mathfrak{B})\in\textbf{Tr}\llbracket{m(\varepsilon)}\rrbracket$, then $(\mathbf{e}\mapsto\mathfrak{A})\in h(m(\varepsilon))$, where $\mathfrak{A}$ is a possible input structure, see Figure 1. Note that $s(i)=s(i+1)$ is allowed – a “no-change” transition extends the string $s_{i}$ by repeating the same letter in ${\bf U}$.

The set of all Choice functions is denoted ${\sf CH}({\bf U},\mathcal{M})$.

To summarize, a concrete Choice functions, e.g., $h$, uses the transition relation Tr to assign semantic meaning to an atomic expression in $\mathcal{M}$ as a set of string-to-string transductions, that is, as a (partial) function on ${\bf U}^{+}$. Each $m(\varepsilon)$ corresponds to some non-deterministic transitions in a transition system Tr. A concrete Choice function, as a semantic instantiation of $\varepsilon$, resolves this non-determinism, by taking the history into account. It extends the history (a string in ${\bf U}^{+}$) by one letter in the alphabet ${\bf U}$ that corresponds to one of the possible outcomes of $m(\varepsilon)$ according to Tr. Intuitively, at each time step, $h$ resolves atomic non-determinism, while taking the history, that starts with an input structure, into account. Thus, at different time points, there could be different outcomes.

Each algebraic term imposes constraints on allowable Choice functions, just as any first-order formula imposes constraints on possible instantiations of its free variables. The formal definition of the extension of Choice functions to all terms is given next. Recall that only one free variable $\varepsilon$ is allowed per term.

So far, Choice functions were defined on the domain $\mathcal{M}$ of atomic module symbols. They are now extended to operate on all terms, to define their semantics. Formally,

$$\bar{h}:\it Terms\to({\bf U}^{+}\rightharpoonup{\bf U}^{+}).$$

(3)

The extension $\bar{h}$ of $h$ provides semantics to all algebraic terms as string-to-string transductions.

1. 1.

   $\bar{h}(m(\varepsilon))=h(m(\varepsilon))$ for $m\in\mathcal{M}$.

2. 2.

   $\bar{h}({\rm id})=\{(s_{i}\mapsto s_{i})\mid i\in{\rm Nature}\}$.

3. 3.

   $\bar{h}(\mathord{\curvearrowright}t)=\{(s_{i}\mapsto s_{i})\mid$ there is no string $s^{\prime}$, Choice function $h^{\prime}$ and $k\geq i$ such that $(s_{i}\mapsto s_{k}^{\prime})\in\bar{h}^{\prime}(t)$, where $s^{\prime}_{i}=s_{i}\}$.

4. 4.

   $\bar{h}(t\mathbin{;}t^{\prime})=\{(s_{i}\mapsto s_{j})\mid$ if there exists $l$, with $i\leq l\leq j$, such that $(s_{i}\mapsto s_{l})\in\bar{h}(t)$ and $(s_{l}\mapsto s_{j})\in\bar{h}(t^{\prime})\}$.

5. 5.

   $\bar{h}(t\sqcup t^{\prime})=\{(s_{i}\mapsto s_{j})\mid$ there exists $\bar{h}^{\prime}$ such that $(s_{i}\mapsto s_{j})\in\bar{h}^{\prime}(t)$ and $\bar{h}(t\sqcup t^{\prime})=\bar{h}^{\prime}(t)$ or
   there is no $\bar{h}^{\prime}$ such that $(s_{i}\mapsto s_{k})\in\bar{h}^{\prime}(t)$ for any $k$ and there exists $\bar{h}^{\prime\prime}$ such that $\bar{h}(t\sqcup t^{\prime})=\bar{h}^{\prime\prime}(t^{\prime})\}$.

6. 6.

   $\bar{h}(t^{\uparrow})=\{(s_{i}\mapsto s_{j})\mid$ $i=j$ and $(s_{i}\mapsto s_{j})\in\bar{h}(\mathord{\curvearrowright}t)$    or  $i<j$  and

   (1) there exists $l$ with $i\leq l\leq j$ such that $(s_{i}\mapsto s_{l})\in\bar{h}(t^{\uparrow})$ and $(s_{l}\mapsto s_{j})\in\bar{h}(t)$, and

   (2) there is no $k$ with $l<k\leq j$ and $s(l)=s(k)\}$. Thus, the interpretation of $t^{\uparrow}$ should not induce loops in the transition graph given by the binary relation $\textbf{Tr}\llbracket{\cdot}\rrbracket$.⁶⁶6This is similar to how a transitive closure of a graph is defined. The graph, in our case, is given by the transition system $\textbf{Tr}\llbracket{\cdot}\rrbracket$. In other applications, this condition may be omitted.

7. 7.

   $\bar{h}(P=Q)=\{(s_{i}\mapsto s_{i})\mid$ $Q^{s(i)}=P^{s(i)}\}$.

8. 8.

   $\bar{h}(\textbf{\tiny{ BG}}(P\neq Q))=\{(s_{i}\mapsto s_{i})\mid$ there is no $1\leq l<i$ such that $Q^{s(l)}=P^{s(i)}\}$.

   Observe that Back Globally (​$\textbf{\tiny{ BG}}(P\neq Q)$) may “look back” through the entire string, not just a computation of a specific sub-term. ⁷⁷7Note that the intended use of the algebra is to evaluate algebraic expressions with respect to an input structure, that is a one-letter string. In this use, the BG construct never looks into the “pre-history”, since, in that use, all strings of length greater than one are traces of some processes. ⁸⁸8An alternative definition of ​$\textbf{\tiny{ BG}}(P\neq Q)$ is possible, where the scope of the condition is specified explicitly (by e.g., adding parentheses). But, it is not needed since the scope can be controlled by a cleaver use of fresh “registers”, i.e., unary predicate symbols in $\tau_{\rm reg}$.

Thus, the semantics associates, with each algebraic expression $t$, a functional binary relation (i.e., a partial mapping) $\bar{h}(t)$ on strings of relational structures, that depends on a specific Choice function $h$.

The given semantics allows us to view algebraic terms, parameterized by Choice functions, as (partial) functions. Let us make the functional view more explicit. First, we do it for atomic modules, and then generalize to all terms.

Recall that the terms of the algebra take other functions as arguments. The summary of the syntax is presented in the table in Section 0.1. In that table, $\mathcal{F}$ denotes (partial) functions on ${\bf U}^{+}$. Thus, according to (5), the partial functions in the closure of ${\cal F}$ under all algebraic operations (2) map strings of structures to strings of structures, i.e., are string-to-string transductions. This functional view is essential to develop an algebraic formalization of non-deterministic computations.

Tests play a special role in our formalisation of non-deterministic computations as they specify computational decision problems. While tests are, essentially, “yes” or “no” questions, processes specify the actual content of these questions. We now define these notions formally.

Note that it is possible that an atomic module does not change the interpretations of any relations (“registers”), and thus does not make a transition to another state in the transition system Tr. However, such a module would not be considered a test because it still extends the current string by repeating the same letter.

Notice an important difference between tests and processes. A test can “reject” an input string because the domain of the corresponding partial function is empty. For example, $\mathord{\curvearrowright}{\rm id}(x)$ rejects all strings because $\bar{h}(\mathord{\curvearrowright}{\rm id})$ is always undefined. A test can also “accept” a string. For instance, ${\rm id}(x)$ is everywhere defined and thus “accepts” every string. The situation with processes is quite different. Every process defines a string-to-string transduction. It extends a string, provided it is defined in that string.

To interpret the Dynamic Logic (6), we need to provide semantics to both processes and state formulae. Processes are interpreted as before – as partial mappings (parameterized with $h$) on strings in ${\bf U}^{+}$, that can be viewed as string-to-string transductions (5). For state formulas we define, for all concrete Choice functions $h$ and strings $s$,

$$\textbf{Tr},s\models\phi(h/\varepsilon)\ \ \ \text{ iff }\ \ \ \phi^{\textbf{Tr}}(h/\varepsilon)(s)=s.$$

(7)

Notice that (7) is a particular case of (5), the partial mapping specified by a term, where the terms are tests (cf. Definition 2). This is immediate from cases 2,3,7 and 8 of the extended Choice function $\bar{h}$.

In this paper, the relation Tr (that is specified by a secondary logic at the bottom level of the algebra) is fixed, and is omitted from $\textbf{Tr},s\models\phi(h/\varepsilon)$ (cf. (7)), where $\phi$ is a state formula. But, of course, other settings are possible as well.

State formulae exhibit implicit quantification over the Choice function variable $\varepsilon$. Indeed, tests of the form $\mathord{\curvearrowright}\mathord{\curvearrowright}t$ represent the domain of $t$ (which is, semantically, is a partial mapping (5)), and, intuitively, mean that there is a Choice function $h$ such that $s\models\mathord{\curvearrowright}\mathord{\curvearrowright}t(h/\varepsilon)$. On the other hand, tests of the form $\mathord{\curvearrowright}t$ are universal, because they claim that there is no Choice function that corresponds to a successful execution of $t$. To summarize, we have the following implicit quantification over Choice functions:

$$\begin{array}[]{rcl}\mathord{\curvearrowright}\mathord{\curvearrowright}t(\varepsilon)(x)&-&\text{ implicitly, }\exists\varepsilon,\\ \mathord{\curvearrowright}t(\varepsilon)(x)&-&\text{ implicitly, }\forall\varepsilon.\end{array}$$

(8)

These are second-order quantifiers, since we quantify over functions. The quantifiers can interleave, since expressions of the form (8) may appear within the term $t$. The two kinds of tests mentioned in (8) are of special interest in the study of computational problems specified by a term, as discussed in Section 0.4.1.

Recall the difference between tests and processes discussed in Section 0.2.5. Notice that, an application of a process-term $t$ can only make a string longer. This is because, in (5), if the extension $\bar{h}(t)$ of $h$ for process $t$ exists, it is always the case that, in $(s_{i}\mapsto s_{j})\in\bar{h}(t)$, we have $i<j$. Thus, the mapping associated with a process-term can be viewed as a set of (non-empty) strings over the alphabet ${\bf U}$. We call the (non-empty) strings in this set the deltas of $t$. In certain contexts, we also call them traces of $t$. The term “delta” is more appropriate when we talk about applying a term, e.g., when we say that we apply a string delta selected by $\bar{h}$. A “trace” is something left after the execution of $t$ from an inputs structure $\mathfrak{A}$. We talk more about traces in Section (0.4.2). Notice that tests never extend a string. Their deltas are the empty strings.

In fact, deltas of the form $s(1,j)$, for some $j$, are associated with certificates for computational decision problems, and are related to some equivalence classes of Choice functions that “agree” on $s(1,j)$, as will be seen shortly, in Section 0.4.2.

In summary, the interpretation of each process-term can be viewed as a set of deltas – non-empty strings – “extensions” applied to strings on the input.

Intuitively, the class $\mathcal{P}_{t}$ contains all structures $\mathfrak{A}$ such that a successful execution of $t$ on input $\mathfrak{A}$ is possible, or $t$ is defined on input $\mathfrak{A}$. We will talk more about defined and undefined terms in Section 0.5.

Many Choice functions witness the main task (9) in the “same way”. We now explain this notion of similarity formally, by introducing equivalence classes on Choice function.

We have seen, in (9), that Choice functions witness the main computational task. But, we don’t want to distinguish between Choice functions that are, in some sense, “similar”, as certificates. This similarity relates to the notion of a trace. We say that string $s$ is a trace of term $t$ from input $\mathfrak{A}$ if there is $h$ such that $(\mathfrak{A}\mapsto s)\in\bar{h}(t(\varepsilon))$. An example of a trace of $t\ :=\ m_{17}\mathbin{;}m_{17}\mathbin{;}m_{17}(\varepsilon)$ is highlighted in Figure 1 by the shaded area.

For the same Choice function, many different terms can produce the same trace, say $\mathfrak{A}\cdot\mathfrak{B}\cdot\mathfrak{C}$, as Example 1 in Section 0.2.3 shows. Dually, for the same term $t$ and an input structure $\mathfrak{A}$, different Choice functions can generate the same trace, but act differently outside of it. We consider two different Choice functions $h$ and $h^{\prime}$ equivalent with respect to $t$ and $\mathfrak{A}$, if for that term, they agree on a trace $s$ of $t$ from $\mathfrak{A}$, i.e., return the same mapping $\mathfrak{A}\mapsto s$:

$$h\cong_{(\mathfrak{A},t)}h^{\prime}\ \ \Leftrightarrow\ \ (\mathfrak{A}\mapsto s)\in\bar{h}(t)\mbox{ iff }(\mathfrak{A}\mapsto s)\in\bar{h^{\prime}}(t),\mbox{ for some string }s\in{\bf U}^{+}.$$

(10)

This equivalence relation gives us equivalence classes $[h]_{(\mathfrak{A},t)}$ of Choice functions. Each equivalence class has a one-to-one correspondence with a trace of $t$ from $\mathfrak{A}$, so we can refer to traces as equivalence classes $[h]_{(\mathfrak{A},t)}$ (i.e., with a string $s$ on which the Choice functions agree). We will see shortly that these equivalence classes (or traces) can be viewed as certificates for the membership of $\mathfrak{A}$ in a computational problem specified by a term.

We define the set of witnesses for $\mathfrak{A}$ in $\mathcal{P}_{t}$ as

$$W^{t}_{\mathfrak{A}}\ :=\ \{[h]_{(\mathfrak{A},t)}\mid\mathfrak{A}\ \models\ |t\rangle{\top}\ (h/\varepsilon)\}\ \ \ \ \text{ and }\ \ \ W^{t}\ :=\ \ \bigcup_{\mathfrak{A}\in\mathcal{P}_{t}}W^{t}_{\mathfrak{A}}.$$

We also call the witnesses “promises”, which gives the name to the algebra (2) – Promise Algebra. If a promise is given, the computation is guaranteed to succeed.

The intended use of the algebra is to study queries of the form (9). In that use, strings in ${\bf U}^{+}$ are partitioned into those that are traces of some algebraic terms that represent programs, and thus can potentially act as “yes”-certificates to computational decision problems, and those that are not traces of any term whatsoever, and thus cannot be “yes”-certificates for any $\mathcal{P}_{t}$. We can consider the Boolean algebra of the set of all potential “yes” certificates as follows.

The Boolean algebra $B(A)$ of a set $A$ is the set of subsets of $A$ that can be obtained by means of a finite number of the set operations union (OR), intersection (AND), and complementation (NOT), see pages 185-186 of Comtet’s 1974 book [15].

Let $\mathcal{H}$ be the set of all promises, i.e., equivalence classes on functions from ${\sf CH}({\bf U},\mathcal{M})$, with respect to the equivalence relation (10).

If $\sigma$ is a signature and $A$, $B$ are $\sigma$-structures (also called $\sigma$-algebras in universal algebra), then a map $h:A\to B$ is a $\sigma$-embedding if all of the following hold:

• •

  $h$ is injective,

• •

  for every $n$-ary function symbol $f\in\sigma$ and $a_{1},\ldots,a_{n}\in A^{n}$, we have

  $$h(f^{A}(a_{1},\ldots,a_{n}))=f^{B}(h(a_{1}),\ldots,h(a_{n})).$$

  (11)

The fact that a map $h:A\to B$ is an embedding is indicated by the use of a “hooked arrow” $h:A\hookrightarrow B$.¹³¹³13Note that this notation is sometimes also used for inclusion maps, but, here, we mean an embedding (which happens to be an inclusion).

The following proposition is straightforward, as it follows immediately from the semantics of terms as string-to-string transductions and the definition of the equivalence relation (10).

Let the state-to-state transition relation Tr (given by the secondary logic), be fixed. Let $s$ be a string in ${\bf U}^{+}$.

Notice that terms that are trace-equivalent on some set $S$ have the same sets of deltas, extending strings in that set $S$.

Note that undefined terms are not considered (strongly) equal. Thus, strong equality is not reflexive. This is an important property for reasoning about computation, as discussed shortly in Section 0.5.4.

To define the notion of a Before-After equivalence of terms, we need the following notion of a projection onto ${\bf U}$, which is simply the set of all pairs (start, end) of a term’s deltas.

A projection ${\sf Pr}$ of term $t$ under Choice function $h$ onto ${\bf U}$ is defined as:

$$(\,s(i),s(j)\,)\in{\sf Pr}(t,h)\ \ \ \Leftrightarrow\ \ \ (s_{i}\mapsto s_{j})\in\bar{h}(t).$$

Both equalities we have just introduced can be uses to specify the existence of a certificate, that invisible elephant in the room, as we show next.

An important consequence of the definitions of Strong equality and Before-After equality is that, in both cases, such an equality is not necessarily reflexive. As a consequence, self-equality is identified with definedness. On the other hand, undefinedness, in $x$, denoted $t(x)\!\not\downarrow$, is associated with $\mathord{\curvearrowright}t(x)\doteq x$ (here, $\fallingdotseq$ can be considered as well, depending on specific application needs).

To summarize,

$$\begin{array}[]{llll}t(x)\!\!\downarrow&\text{ abbreviates }&t(x)\doteq t(x),&\text{ a ``yes'' certificate for $x$ in $\mathcal{P}_{t}$ exists},\\ t(x)\!\not\downarrow&\text{ abbreviates }&\mathord{\curvearrowright}t(x)\doteq x,&\text{ no ``yes'' certificates for $x$ in $\mathcal{P}_{t}$ exist}.\end{array}$$

Since terms denote partial mappings, strong equalities such as $t(x)\doteq g(x)$ may hold for some strings $x$, but not for all $x$, as in Example 6. For that reason, the algebra (2) cannot be axiomatized by universally quantified equalities between terms. Instead of equational theories that axiomatize algebraic structures such as groups, we need to consider quasi-equational theories, i.e., those with conditional equations in the Horn form.

The fact that the existence of a certificate is associated with equality of certain terms opens up a possibility developing a proof system where deriving the existence of a certificate is possible. Developing such a quasi-equational theory and a proof system is outside of the scope of this paper.

Recall that we have used the relation $\textbf{Tr}\llbracket{m(\varepsilon)}\rrbracket$ that represents a transition system in Definition 1 of the semantics of atomic modules, via Choice functions. We have, temporarily, treated it as an arbitrary binary relation. However, this relation has an important property that always holds for any update – everything that is not explicitly modified in the update, must remain the same. This property is commonly called the Law of Inertia, starting from McCarthy and Hayes 1969 [36]. Figure 6 illustrates the Law of Inertia for atomic modules. Recall that the vocabulary symbols $\tau$ are partitioned into EDB relations and “registers”, $\tau\ :=\ \tau_{\rm EDB}\uplus\tau_{\rm reg}$, see (1).

We restrict atomic transductions so that the only relations that change from structure to structure are unary and contain one domain element at a time, similarly to registers in a Register Machine [46]. Please see Section 0.6.6 for an explicit discussion of the machine model used.

Register values, i.e., the content of unary singleton-set relations, get updated by state-to-state transitions. Updating the registers can be thought of re-interpreting a fixed set (a vector) of $k$ constants, interpreted by domain elements. Such an update can be specified in any way. For example, it can be a linear algebra transformation $\bar{b}=L\bar{a}$, or any other transformation $\bar{b}=T(\bar{a},{\rm EDB})$, that takes into account the input EDB relations, in general.

A vector $\bar{a}=R_{1}\cdots R_{k}$ of registers (one-dimensional array) can encode an $l$-dimensional matrix by providing appropriate indices such as $i,j,l$, e.g., $R_{i,j,l}$ for a 3-dimensional matrix. These indices correspond to keys $i,j,l$ in $([i,j,l];\ R)$, a Graph Normal Form (GNF) representation of the 3-dimensional matrix.

Note that EDB relations given on the input are never updated. All computational work happens in the registers.

As mentioned above, the bottom level of the algebra can be specified in any formalism. In the sections that follow, we specify a particular secondary logic for axiomatizing atomic transductions, that is similar to Conjunctive Queries (CQs).

Intuitively, we take a monadic primitive positive (M-PP) relation, i.e., a unary relation definable by a unary CQ, and output, arbitrarily, only a single element contained in the relation, instead of the whole relation. There could be several such CQs in the same modules, applied ot once. A formal definition will be given shortly.

For readers familiar with Datalog (such a reader is invited to jump ahead to see the general form (16)), we mention informally that, at the bottom level of the algebra, we have a set of Datalog-like “programs”, one per each atomic module symbol $m\in\mathcal{M}$. The “programs” are similar to Conjunctive Queries (non-recursive Datalog programs). The rules in such programs have a unary predicate symbol in the head of each rule. Each (simultaneous) application of the rules puts only one domain element into each unary IDB relation in the head, out of several possible. This creates non-determinism in the atomic module applications. The applications of the modules are controlled by algebraic expressions in (2) at the top level of the algebra.

This logic is the second parameter that affects the expressive power of the language, in addition to the selection of the algebraic operations presented earlier. The logic is carefully constructed to ensure complexity bounds presented towards the end of the paper.

We start our formal exposition by reviewing queries, Conjunctive Queries(CQs), and PP-definable relations.

Let $C$ be a class of relational structures of some vocabulary $\tau$. Following Gurevich [25], we say that an $r$-ary $C$-global relation $q$ (a query) assigns to each structure $\mathfrak{A}$ in $C$ an $r$-ary relation $q(\mathfrak{A})$ on $\mathfrak{A}$; the relation $q(\mathfrak{A})$ is the specialization of $q$ to $\mathfrak{A}$. The vocabulary $\tau$ is the vocabulary of $q$. If $C$ is the class of all $\tau$-structures, we say that $q$ is $\tau$-global. Let $\mathcal{L}$ be a logic. A $k$-ary query $q$ on $C$ is $\mathcal{L}$-definable if there is an $\mathcal{L}$-formula $\psi(x_{1},\dots,x_{k})$ with $x_{1},\dots,x_{k}$ as free variables such that for every $\mathfrak{A}\in C$,

$$q(\mathfrak{A})=\{(a_{1},\dots,a_{k})\in\mathcal{D}^{k}\mid\mathfrak{A}\models\psi(a_{1},\dots,a_{k})\}.$$

(12)

In this case, $q(\mathfrak{A})$ is called the output of $q$ on $\mathfrak{A}$. Query $q$ is unary if $k=1$. A conjunctive query (CQ) is a query definable by a FO formula in prenex normal form built from atomic formulas, $\land$, and $\exists$ only. A relation is Primitive Positive (PP) if it is definable by a CA:

$$\forall x_{1}\dots\forall x_{k}\ \big{(}R(x_{1},\dots,x_{k})\leftrightarrow\underbrace{\exists z_{1}\dots\exists z_{m}\ (B_{1}\land\cdots\land B_{m})}_{\Phi(x_{1},\dots,x_{k})}\big{)},$$

and each atomic formula $B_{i}$ has object variables from $x_{1},\dots,x_{k},z_{1},\dots,z_{m}$. We will use $\bar{x}$, $\bar{u}$, etc., to denote tuples of variables, and use the broadly used Datalog notation for the sentence above:

$$R(\bar{x})\leftarrow\underbrace{B_{1}(\bar{u}_{1}),\dots,B_{n}(\bar{u}_{n})}_{\Phi_{\rm body}},$$

(13)

where the part on the right of $\leftarrow$ is called the body, and the part to the left of $\leftarrow$ the head of the rule or the answer. Notice that an answer symbol is always different from the symbols in the body (13), since, by definition, CQs are not recursive. We say that $R(x_{1},\dots,x_{k})$ is monadic if $k=1$, and apply this term to the corresponding PP-relations as well.