2 On the syntax of mini-gringo

There are minor syntactic differences between mini-gringo and the input language of the grounder gringo, explained by the fact that the former is designed for theoretical studies, and the latter for actual programming. For example, the definition of sqrt_b/1 in the introduction, rewritten in the syntax of mini-gringo, becomes

Overlined symbols, such as $\overline 1$ , are “numerals” – syntactic objects representing integers. In examples of rules and programs, we will freely switch between the two styles.

In mini-gringo, precomputed terms are numerals, symbolic constants, and the symbols inf, sup. We assume that a total order on precomputed terms is chosen, such that inf is its least element, sup is its greatest element, and, for all integers m and n, $\overline m < \overline n$ iff $m<n$ . A precomputed atom is an expression of the form $p({\bf t})$ , where p is a symbolic constant and $\bf t$ is a tuple of precomputed terms. A predicate symbol is a pair $p/n$ , where p is a symbolic constant and n is a nonnegative integer. About a rule or another syntactic expression we say that it contains $p/n$ if it contains an atom of the form $p(t_1,\dots,t_n)$ .

3 External behavior

Definition 1. A user guide is a quadruple

(1) \begin{align*}({PH},{In},{Out},{Dom}),\end{align*}

where

• PH is a finite set of symbolic constants, called placeholders,

• In and Out are disjoint finite sets of predicate symbols, called input symbols and output symbols, and

• Dom is a set such that each of its elements is a pair $(v,\mathcal{I})$ , where

  1. (i) v is a function that maps elements of PH to precomputed terms that do not belong to PH, and

  2. (ii) $\mathcal{I}$ is a subset of the set of precomputed atoms that contain an input symbol and do not contain placeholders.

The set Dom is the domain of the user guide, and pairs $(v,\mathcal{I})$ satisfying conditions (i) and (ii) are called inputs. An input $(v,\mathcal{I})$ represents a way to choose the values of placeholders and the extents of input predicates: for every placeholder c, specify v(c) as its value, and add the atoms $\mathcal{I}$ to the rules of the program as facts. If $\Pi$ is a mini-gringo program then $v(\Pi)$ stands for the program obtained from $\Pi$ by replacing every occurrence of every constant c in the domain of v by v(c). Using this notation, we can say that choosing $(v,\mathcal{I})$ as input for $\Pi$ amounts to replacing $\Pi$ by the program $v(\Pi)\cup\mathcal{I}$ .

To use a program in accordance with user guide (1) means to run it for inputs that belong to Dom. The inputs that do not belong to Dom are not related to the external behavior of the program when it is used as intended.

Example 2. We would like to describe the meaning of the word orphan by a logic program (Gelfond and Kahl Reference Gelfond and Kahl2014, Section 4.1.2). The intended use of such a program can be described by user guide (1) with

and with the domain consisting of all inputs. We will denote this user guide by ${UG}_o$ . In the next two sections, we examine two possible definitions of orphan/1 and consider the question of their equivalence with respect to ${UG}_o$ .

User guides are closely related to lp-functions (Gelfond Reference Gelfond2002, Section 2), and also to io-programs (Fandinno et al. Reference Fandinno, Lifschitz, Lühne and Schaub2020, Section 5), reviewed in Appendix C.

An output atom of a user guide UG is a precomputed atom that contains an output symbol of UG.

Example 1, continued If $\Pi$ is one of the three prime number programs from the introduction, and $(v,\mathcal{I})$ is an input in the domain of ${UG}_p$ , then the program $v(\Pi)\cup\mathcal{I}$ is $v(\Pi)$ , and it has a unique stable model. If v is defined by the conditions $v(a)=\overline{10}$ , $v(b)=\overline{15}$ , then that stable model includes the atoms ${prime}(\overline{11})$ , ${prime}(\overline{13})$ , and some atoms containing ${composite}/1$ . The external behavior of each of the programs for this input is $\{\{{prime}(\overline{11}),{prime}(\overline{13})\}\}$ . For the safe and optimized versions, this external behavior can be calculated by instructing clingo to find all answers for the file obtained from the program by appending the directives

Example 2, continued If $\Pi$ is the program

(2)

and $(v,\mathcal{I})$ is an input in the domain of ${UG}_o$ , then the program $v(\Pi)\cup\mathcal{I}$ is $\Pi\cup\mathcal{I}$ , and it has a unique stable model. If $\mathcal{I}$ is

(3) \begin{align*} \{{father}({jacob},{joseph}),{mother}({rachel},{joseph}),\\[6pt]\quad \quad {living}({jacob}),{living}({rachel}),{living}({joseph})\},\end{align*}

then that stable model includes the atoms ${orphan}({jacob})$ , ${orphan}({rachel})$ , and some atoms containing predicate symbols other than orphan/1. The external behavior of this program for ${UG}_o$ and input (3) is

(4)

It can be calculated by instructing clingo to find all answers for the file obtained from program (2) by appending the facts

and the directive #show orphan/1.

In the special case when UG has neither placeholders nor input symbols, and its set of output symbols includes all predicate symbols occurring in $\Pi$ , the external behavior of $\Pi$ with respect to UG and $(\emptyset,\emptyset)$ is the set of stable models of $\Pi$ . In this sense, the concept of external behavior is a generalization of the stable model semantics.

4 Equivalence

Example 1, continued The three programs from the introduction are equivalent to each other with respect to ${UG}_p$ . As discussed in Section 6, this claim can be verified using the automated reasoning tools anthem-p2p and vampire.

Example 2, continued Perhaps surprisingly, the one-rule program

(5)

is not equivalent to (2) with respect to ${UG}_o$ . Indeed, the external behavior of this program with respect to ${UG}_o$ and input (3) is $\{\emptyset\}$ , which is different from (4). We will see that anthem-p2p can help us clarify the relationship between programs (2) and (5).

We understand refactoring a mini-gringo program with respect to a user guide UG as replacing it by a program that is equivalent to it with respect to UG.

This equivalence relation is essentially an example of relativized uniform equivalence with projection (Oetsch and Tompits Reference Oetsch and Tompits2008), except that the language discussed by Oetsch and Tompits includes neither arithmetic operations nor placeholders. It is uniform equivalence, because the programs are extended by adding facts, rather than more complex rules; relativized, because these facts $\mathcal{I}$ are assumed to be atoms containing input symbols, not arbitrary atoms; with projection, because we look at the output atoms in the stable model, not the entire model.

5 Formal notation for user guides

To design software for verifying the equivalence of programs with respect to a user guide, we need to represent user guides in formal notation. The format that we chose for user guide files is similar to the format of specification files, defined by Fandinno et al. (Reference Fandinno, Lifschitz, Lühne and Schaub2020) within their work on the system anthem. Placeholders and input symbols are represented by input statements, for instance:

Output symbols are represented by output statements:

There can be several statements of both kinds in a user guide file, in any order.

The question of representing the domain Dom by a string of characters is more difficult, because the domain is a set of inputs, which is generally infinite. Our approach is to define “assumptions” as sentences of an appropriate first-order language, and characterize the domain by a list of assumptions; an input belongs to the domain iff it satisfies all assumptions on that list.

For any set $\mathcal{P}$ of predicate symbols, by $\sigma_0(\mathcal{P})$ we denote the subsignature of the two-sorted signature $\sigma_0$ , described in Appendix A, in which the set of predicate symbols is limited to the comparison symbols and the symbols from $\mathcal{P}$ . In this paper, an assumption is a sentence over the signature $\sigma_0({In})$ . Besides input and output statements, a user guide file may include one or more statements consisting of the word assume followed by an assumption.

To use assumptions as conditions on an input, we need to relate inputs to interpretations in the sense of first-order logic. If v is a function that maps elements of some set PH of symbolic constants to symbolic constants, and $\mathcal{I}$ is a subset of the set of precomputed atoms that contain a predicate symbol from $\mathcal{P}$ , then there exists a unique interpretation I of $\sigma_0(\mathcal{P})$ such that

1. (a) the domain of the sort general in I is the set of all precomputed terms;

2. (b) the domain of the sort integer in I is the set of all numerals;

3. (c) I interprets every symbolic constant c in PH as v(c);

4. (d) I interprets every precomputed term t that does not belong to PH as t;

5. (e) I interprets the symbols for arithmetic operations as usual in arithmetic;

6. (f) if $p/n$ is a predicate constant from $\mathcal{P}$ , and $\bf c$ is an n-tuple of precomputed atoms, then I interprets $p({\bf c})$ as true iff $p({\bf c})\in\mathcal{I}$ ;

7. (g) I interprets the comparison symbols as in the definition of mini-gringo.

We will denote that interpretation by $I(v,\mathcal{I})$ . The domain of the user guide defined by a set of assumptions is the set of inputs $(v,\mathcal{I})$ such that the interpretation $I(v,\mathcal{I})$ of $\sigma_0({In})$ satisfies all assumptions in that set.

Example 1, continued The user guide ${UG}_p$ can be described by the statements

The first two lines can be written more concisely as

Example 2, continued The user guide ${UG}_o$ can be described by the statements

(6)

The absence of assume statements here shows that the domain is the set of all inputs.

6 Functionality of anthem-p2p

The proof assistant anthem-p2p uses the theorem prover vampire to verify that two mini-gringo programs have the same external behavior with respect to a given user guide. We can verify, for instance, that the first two versions of the prime number program from the introduction are equivalent with respect to the user guide ${UG}_p$ by running anthem-p2p on three files: the unsafe program

(7)

the safe program

(8)

and the user guide

(9)

The system anthem-p2p transforms the task of verifying equivalence with respect to a user guide (1) into the problem of verifying the provability of a formula in a first-order theory over the signature $\sigma_0({In}\cup{Out})$ , and submits that problem to vampire; see Sections 7–9 for details.

The user can help vampire organize search more efficiently by supplying anthem-p2p with “helper” files. Such a file may instruct vampire to prove a series of lemmas before trying to prove the goal formula. A helper file can suggest also instances of the induction schema that may be useful for the job in hand. This kind of help is needed, for instance, for verifying the equivalence of the optimized prime number program to the other two.

The use of anthem-p2p for proving equivalence of programs is, generally, an interactive process. If vampire does not prove the goal formula in the allotted time then one of the options is to provide more lemmas and run anthem-p2p again. Alternatively, the user can look for a counterexample that refutes the equivalence claim, as in Example 2 above.

Sometimes, anthem-p2p can help us clarify the source of a puzzling discrepancy between two versions of a program if we run it in the presence of additional assume statements. If adding an assumption to the user guide makes the programs equivalent then it is possible that perceiving that assumption as self-evident is the reason why the discrepancy is puzzling. For instance, we can observe that the anthem-p2p/vampire combination proves the equivalence of program (2) to program (5) if we extend user guide (6) by two existence and uniqueness assumptions:

The limitations of the anthem-p2p algorithm are inherited from the limitations of anthem and can be described as follows. The predicate dependency graph of a mini-gringo program $\Pi$ (Fandinno et al. Reference Fandinno, Lifschitz, Lühne and Schaub2020, Section 6.3) is the directed graph that

• has the predicate symbols contained in $\Pi$ as its vertices, and

• has an edge from $p/n$ to $q/m$ if some rule of $\Pi$ contains $p/n$ in the head and $q/m$ in the body.

The edge from $p/n$ to $q/m$ is positive if there is a rule R in $\Pi$ such that $p/n$ is contained in the head of R, and $q/m$ is contained in an atom in the body of R that is not in the scope of negation. For example, the predicate dependency graph of program (2) has 6 edges; all of them except for the edge from parent_living/1 to orphan/1 are positive. We say that $\Pi$ is tight if this graph has no cycles consisting of positive edges.

A vertex $p/n$ of the graph is private for a user guide UG if it is neither an input symbol nor an output symbol of UG. We say that $\Pi$ uses private recursion for UG if

• the predicate dependency graph of $\Pi$ has a cycle such that every vertex in it is a private symbol, or

• $\Pi$ includes a choice rule with the head containing a private symbol.

As discussed in the next two sections, the applicability of the algorithm implemented in anthem-p2p to a pair of mini-gringo programs and a user guide UG is guaranteed whenever the programs are tight and do not use private recursion with respect to UG. We expect that it will be possible to replace the tightness requirement by a significantly weaker condition using the ideas of a recent paper on “locally tight” programs (Fandinno and Lifschitz Reference Fandinno and Lifschitz2021); this is a topic for future work.

7 Equivalence of tight programs

The theorem stated below relates equivalence of tight programs to the satisfaction relation of second-order logic. Its statement refers to the concept of second-order completion, reviewed in Appendix B, and also to the concept of standard interpretations, defined as follows. An interpretation I of $\sigma_0(\mathcal{P})$ is standard for a set PH of symbolic constants if it satisfies conditions (a), (b), (d), (e), (g) from Section 5 and the condition

(cʹ) interprets every symbolic constant in PH as a term that does not belong to PH.

Theorem Let UG be a user guide (PH,In,Out,Dom) such that its domain is described by a finite set of assumptions, and let Asm be the conjunction of these assumptions. For any tight mini-gringo programs $\Pi_1$ , $\Pi_2$ such that the heads of their rules do not contain the input symbols of UG, $\Pi_1$ is equivalent to $\Pi_2$ with respect to UG iff the sentence

(10) \begin{align*}{Asm}\to(\hbox{COMP}(\Pi_1,{In},{Out}) \leftrightarrow\hbox{COMP}(\Pi_2,{In},{Out})),\end{align*}

is satisfied by all interpretations of the signature $\sigma_0$ (In $\cup$ Out) that are standard for PH.

This theorem shows that the equivalence of tight programs may be established by choosing a first-order theory T over the signature $\sigma_0({In}\cup{Out})$ such that its axioms are satisfied by all interpretations that are standard for PH, and then exhibiting a derivation of formula (10) from the axioms of T in classical second-order logic. For programs that do not use private recursion, the problem of constructing such a derivation can be reduced to proof search in first-order logic (see Section 8 below), for which many automated reasoning tools are available. This is the core of the procedure used by anthem-p2p.

The proof of the theorem, including the lemma below, uses terminology related to io-programs, which is reviewed in C.

are equivalent to each other. Indeed, since the heads of rules of $v(\Pi)$ do not contain input atoms, the set of input atoms in $\mathcal{M}$ is $\mathcal{I}$ .

Proof of the Theorem. The condition

(11) \begin{align*}\Pi_1\hbox{ is equivalent to }\Pi_2\hbox{ with respect to }{UG,}\end{align*}

means that for any input $(v,\mathcal{I})$ such that $I(v,\mathcal{I})\models{Asm}$ and any set $\mathcal{J}$ of output atoms,

(12) \begin{align*}\begin{array} c\mathcal{J}\hbox{ is an element of the external behavior of $\Pi_1$ for {UG} and }(v,\mathcal{I})\\\hbox{iff}\\\mathcal{J}\hbox{ is an element of the external behavior of $\Pi_2$ for {UG} and }(v,\mathcal{I}).\end{array}\end{align*}

By the lemma, condition (12) can be reformulated as follows:

By the theorem quoted at the end of Appendix C, this can be further reformulated as

(13)

Hence condition (11) is equivalent to requiring that (13) hold for all inputs $(v,\mathcal{I})$ such that $I(v,\mathcal{I})\models{Asm}$ and all set $\mathcal{J}$ of output atoms.

Since assumptions do not contain output symbols, $I(v,\mathcal{I})\models{Asm}$ is equivalent to $I(v,\mathcal{I}\cup\mathcal{J})\models{Asm}$ . It follows that (11) is equivalent to asserting that implication (10) is satisfied by $I(v,\mathcal{I}\cup\mathcal{J})$ for all inputs $(v,\mathcal{I})$ and all sets $\mathcal{J}$ of output atoms. It remains to observe that an interpretation of the signature $\sigma_0({In}\cup{Out})$ can be represented in the form $I(v,\mathcal{I}\cup\mathcal{J})$ if and only if it is standard for PH.

8 Reduction to first-order logic

If $\Pi_1$ and $\Pi_2$ do not use private recursion then the reference to second-order consequences of the axioms of T in Section 7 can be eliminated in the following way. Represent the formula COMP $(\Pi_1,{In},{Out})$ in the form

$$\exists {\bf P}\left(\bigwedge_i F_i({\bf P}) \land F'({\bf P})\right),$$

where $\bf P$ is a list of distinct predicate variables corresponding to the private symbols $p_1,p_2,\dots$ of $\Pi_1$ , and $F_i({\bf P})$ is the formula obtained from the completed definition of $p_i$ in $\Pi_1$ by replacing each of $p_1,p_2,\dots$ by the corresponding member of $\bf P$ . (Thus the conjunctive members of $F'({\bf P})$ correspond to the completed definitions of the output symbols and to the constraints of $\Pi_1$ .) Similarly, write COMP $(\Pi_2,{In},{Out})$ as

(14) \begin{align*}\exists {\bf Q}\left(\bigwedge_j G_j({\bf Q}) \land G'({\bf Q})\right),\end{align*}

where $\bf Q$ is a list of distinct predicate variables corresponding to the private symbols $q_1,q_2,\dots$ of $\Pi_2$ , and the formulas $G_j({\bf Q})$ are obtained from the completed definitions of these symbols in $\Pi_2$ by replacing them with corresponding variables. Take one half

(15) \begin{align*}{Asm}\to(\hbox{COMP}(\Pi_1,{In},{Out}) \to\hbox{COMP}(\Pi_2,{In},{Out})),\end{align*}

of condition (10). Since $\Pi_2$ does not use private recursion, formula (14) is equivalent to

$$\forall {\bf Q}\left(\bigwedge_j G_j({\bf Q}) \to G'({\bf Q})\right).$$

(Fandinno et al. Reference Fandinno, Lifschitz, Lühne and Schaub2020, Theorem 3). It follows that formula (15) is equivalent to

$${Asm}\to \left(\exists {\bf P}\left(\bigwedge_i F_i({\bf P}) \land F'({\bf P})\right) \to\forall {\bf Q}\left(\bigwedge_j G_j({\bf Q}) \to G'({\bf Q})\right)\right),$$

and consequently to

(16) \begin{align*}\forall {\bf PQ}\left(\left( {Asm}\land\bigwedge_i F_i({\bf P})\land\bigwedge_j G_j({\bf Q}) \right) \to (F'({\bf P})\to G'({\bf Q}))\right),\end{align*}

(with the bound variables in P, Q renamed, if necessary, to ensure that they are pairwise disjoint). Similarly, the second half

$${Asm}\to(\hbox{COMP}(\Pi_2,{In},{Out}) \to\hbox{COMP}(\Pi_1,{In},{Out})),$$

of condition (10) is equivalent to the formula obtained from (16) by swapping $F'({\bf P})$ with $G'({\bf Q})$ . Thus (10) can be rewritten as

$$\forall {\bf PQ}\left(\left( {Asm}\land\bigwedge_i F_i({\bf P}) \land\bigwedge_j G_j({\bf Q}) \right) \to (F'({\bf P})\leftrightarrow G'({\bf Q}))\right).$$

Finally, observe that this formula is entailed by the axioms of T if and only if the axioms entail the first-order formula

(17) \begin{align*}\left( {Asm}\land\bigwedge_i F_i({\bf p})\land\bigwedge_j G_j({\bf q}) \right) \to (F'({\bf p})\leftrightarrow G'({\bf q})),\end{align*}

where p, q are lists of fresh predicate constants.

We return to this formula in the description of the design of anthem-p2p below. Note that its subformulas $F_i({\bf p})$ , $G_j({\bf q})$ , $F'({\bf p})$ , $G'({\bf q})$ are parts of the first-order completion formulas of $\Pi_1$ and $\Pi_2$ , modified by replacing their private symbols $p_1,p_2,\dots$ , $q_1,q_2,\dots$ by members of the lists p and q.

9 Design of anthem-p2p

The system anthem-p2p is a Python program that operates by converting a claim about the equivalence of two mini-gringo programs into an input for anthem. The system anthem verifies the correctness of an io-program with respect to a formal specification. The file describing a specification includes lists of placeholders, input symbols, output symbols, and assumptions, and also a list of “specs” that describe the intended behavior of the future program by sentences over the signature $\sigma_0({In}\cup{Out})$ .

Given programs $\Pi_1$ and $\Pi_2$ and a user guide (PH,In,Out,Dom) with the domain described by assumptions Asm, anthem-p2p constructs the following specification Sp:

1. (i) the placeholders of Sp are the placeholders PH of the given user guide;

2. (ii) the input symbols of Sp are the input symbols In of the user guide and the predicate symbols p corresponding to the private symbols $p_1,p_2,\dots$ of the program $\Pi_1$ ;

3. (iii) the output symbols of Sp are the output symbols Out of the user guide;

4. (iv) the assumptions of Sp are the assumptions Asm of the user guide and the modified completed definitions $F_i({\bf p})$ of the private symbols of $\Pi_1$ ;

5. (v) the specs of Sp are the remaining conjunctive terms $F'({\bf p})$ of the modified first-order completion formula of $\Pi_1$ .

Then anthem-p2p instructs anthem to prove the claim that the io-program $(\Pi_2,{PH},{In},{Out})$ implements Sp. Providing anthem with such an instruction makes it look for a derivation of the formula

(18) \begin{align*}\left( {Asm}\land\bigwedge_i F_i({\bf p})\land\bigwedge_j G_j({\bf q}) \right) \to (G'({\bf q})\leftrightarrow F'({\bf p})),\end{align*}

from the axioms of T by invoking the theorem prover vampire (Fandinno et al. Reference Fandinno, Lifschitz, Lühne and Schaub2020,

Section 6.4). This formula is equivalent to (17). Thus instructing anthem to verify that the io-program $(\Pi_2,{PH},{In},{Out})$ implements the specification Sp amounts to verifying the provability of formula (17) in T.

As an example, consider the operation of the anthem-p2p algorithm on programs (7) and (8) and user guide (9). In each of the programs, the only private predicate is composite/1; it corresponds to both $p_1$ and $q_1$ in the notation of Section 8. The symbols composite_1/1 and composite_2/1, generated by anthem-p2p, play the parts of $\bf p$ and $\bf q$ in formula (17). The file describing the specification Sp is obtained in this case from user guide (9) by adding three statements. First, in accordance with clause (ii) in the description of Sp above, anthem-p2p adds the statement

Second, in accordance with clause (iv), a definition of composite_1/1 is assumed:

Finally, in accordance with clause (v), a definition of prime/1 in terms of composite_1/1 is added as a spec:

Once Sp is generated, anthem calls vampire to prove formula (18) in the theory T, first by deriving the specs $F'({\bf p})$ from the antecedent of (18) and $G'({\bf q})$ (“verification of specification from translated program”), and then by deriving $G'({\bf q})$ from the antecedent of (18) and the specs $F'({\bf p})$ (“verification of translated program from specification”). In this example, the runtime of vampire will be significantly reduced (a few seconds instead of a few minutes) if we instruct it to start by proving two lemmas:

The anthem-p2p website ^{Footnote 2} allows users to experiment with the system in their web browser. The proof search is conducted on a University of Nebraska Omaha server (Oracle Linux 8, 4 Intel(R) Xeon(R) Gold 6248 CPUs, 4 GB RAM) subject to a 10 minute timeout. For smaller problems, this is the recommended introduction to the system.