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Synthesis of positive logic programs for checking a class of definitions with infinite quantification. (English) Zbl 1353.68050
The framework is based on typed first-order logic theories such as the following:
Types:
Nat generated by $$\mathrm{zero}:\rightarrow\mathrm{Nat}\quad\mathrm{succ}:\mathrm{Nat}\rightarrow\mathrm{Nat}$$
Seq generated by $$\mathrm{empty}:\rightarrow\mathrm{Seq}\quad\mathrm{succ}:\mathrm{Nat}\times\mathrm{Seq}\rightarrow\mathrm{Seq}$$

Predicates:
$$\mathrm{id}:\mathrm{Nat}\times\mathrm{Nat}$$ axiomatised as:
$$\mathrm{id}(\mathrm{zero},\mathrm{zero})\Leftrightarrow \mathrm{true}$$
$$\forall(\mathrm{id}(\mathrm{succ}(X),\mathrm{zero})\Leftrightarrow\mathrm{false})$$
$$\forall(\mathrm{id}(\mathrm{zero},\mathrm{succ}(Y)\Leftrightarrow\mathrm{false})$$
$$\forall(\mathrm{id}(\mathrm{succ}(X),\mathrm{succ}(Y))\Leftrightarrow\mathrm{id}(X,Y))$$
$$\mathrm{member}:\mathrm{Nat}\times\mathrm{Seq}$$ axiomatised as:
$$\forall(\mathrm{member}(E,\mathrm{empty})\Leftrightarrow\mathrm{false})$$
$$\forall(\mathrm{member}(E,\mathrm{seq}(X,Y ))\Leftrightarrow\mathrm{id}(E,X)\vee\mathrm{member}(E,Y ))$$
The authors consider assertion definitions of the form $$\forall X(r(X)\Leftrightarrow Q Y R(X,Y))$$ where $$r$$ is the defined relation, $$Q$$ is a quantifier, and $$R$$ is the defining relation, namely, a quantifier-free formula in the language of the typed first-order logic theory. For example, the following is an assertion definition for the subset relation: $\forall L,S(\mathrm{subset}(L,S)\Leftrightarrow\forall E(\mathrm{member}(E,L)\Rightarrow\mathrm{member}(E,S)))$ The purpose of the paper is a method to synthesise a positive logic program for checking assertions of the form $$r(X)\theta$$, where $$\theta$$ is a ground substitution for $$X$$, that depends on the quantifier $$Q$$ in such a way that if $$Q$$ is $$\forall$$ then the program searches for refutations, whereas if $$Q$$ is $$\exists$$ then the program searches for proofs. The method satisfies both key requirements:
the synthesis process terminates after a finite amount of transformation steps;
the synthesised program preserves total correctness w.r.t. the class of goals of the form $$\Leftarrow r(X,Y)\theta$$.

With the example of the subset relation, the synthesised program is: \begin{aligned} \forall(\mathrm{subset}(L,S) \Leftarrow \mathrm{subset}_1(E,L,S))\\ \forall(\mathrm{subset}_1(E,\mathrm{seq}(X,Y ),S) \Leftarrow \mathrm{subset}_2(E,S)\wedge \mathrm{subset}_3(E,X))\\ \forall(\mathrm{subset}_1(E,\mathrm{seq}(X,Y ),S)\Leftarrow \mathrm{subset}_1(E,Y ,S)\wedge \mathrm{subset}_4(E,X))\\ \forall(\mathrm{subset}_1(E,\mathrm{empty},S) \Leftarrow \mathrm{subset}_5(E,S))\\ \forall(\mathrm{subset}_2(E,\mathrm{seq}(X,Y )) \Leftarrow \mathrm{subset}_2(E,Y )\wedge \mathrm{subset}_4(E,X))\\ \forall(\mathrm{subset}_2(E,\mathrm{empty}) \Leftarrow \mathrm{subset}_7)\\ \mathrm{subset}_3(\mathrm{zero},\mathrm{zero}) \Leftarrow \mathrm{subset}_9\\ \forall(\mathrm{subset}_3(\mathrm{succ}(X),\mathrm{succ}(Y )) \Leftarrow \mathrm{subset}_3(X,Y ))\\ \forall(\mathrm{subset}_4(\mathrm{zero},\mathrm{succ}(Y )) \Leftarrow \mathrm{subset}_9)\\ \forall(\mathrm{subset}_4(\mathrm{succ}(X),\mathrm{succ}(Y )) \Leftarrow \mathrm{subset}_4(X,Y ))\\ \forall(\mathrm{subset}_4(\mathrm{succ}(X),\mathrm{zero}) \Leftarrow \mathrm{subset}_9)\\ \forall(\mathrm{subset}_5(E,\mathrm{seq}(X,Y )) \Leftarrow \mathrm{subset}_5(E,Y )\wedge \mathrm{subset}_4(E,X))\\ \mathrm{subset}_7 \Leftarrow\\ \mathrm{subset}_9 \Leftarrow\end{aligned}
following a procedure that consists of 6 steps, with intermediate formulas generated on the way, in particular: \begin{aligned}\forall E,L,S(\mathrm{subset}_1(E,L,S)\Leftrightarrow(\mathrm{member}(E,L)\wedge\neg\mathrm{member}(E,S)))\\ \forall(\mathrm{subset}_2(X,Y )\Leftrightarrow\mathrm{true}\wedge\neg\mathrm{member}(X,Y ))\\ \forall(\mathrm{subset}_3(X,Y )\Leftrightarrow\mathrm{id}(X,Y ))\\ \forall(\mathrm{subset}_4(X,Y )\Leftrightarrow\neg\mathrm{id}(X,Y ))\\ \forall(\mathrm{subset}_5(X,Y )\Leftrightarrow\mathrm{false}\wedge\neg\mathrm{member}(X,Y ))\\ \mathrm{subset}_6\Leftrightarrow\mathrm{false}\wedge\neg\mathrm{false}\\ \mathrm{subset}_7\Leftrightarrow\mathrm{true}\wedge\neg\mathrm{false}\\ \mathrm{subset}_8\Leftrightarrow\mathrm{true}\wedge\neg\mathrm{true}\\ \mathrm{subset}_9\Leftrightarrow\mathrm{true}\\ \mathrm{subset}_{10}\Leftrightarrow\neg\mathrm{true}\end{aligned} The method has been tested with many other assertion definitions, listed in the appendix, together with the proofs of termination and correctness.
##### MSC:
 68N30 Mathematical aspects of software engineering (specification, verification, metrics, requirements, etc.) 03B70 Logic in computer science 68N17 Logic programming 68Q60 Specification and verification (program logics, model checking, etc.)
##### Software:
Eiffel; JML; SIMPLIFY; Spec#
Full Text:
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