Analysis of Logic Programs Using Regular Tree Languages - (Extended Abstract)

The field of finite tree automata provides fundamental notations and tools for reasoning about set of terms called regular or recognizable tree languages. We consider two kinds of analysis using regular tree languages, applied to logic programs. The first approach is to try to discover automatically a tree automaton from a logic program, approximating its minimal Herbrand model. In this case the input for the analysis is a program, and the output is a tree automaton. The second approach is to expose or check properties of the program that can be expressed by a given tree automaton. The input to the analysis is a program and a tree automaton, and the output is an abstract model of the program. These two contrasting abstract interpretations can be used in a wide range of analysis and verification problems. Finite Tree Automata A tree language is a set of trees, commonly represented as terms. Terms are ubiquitous in computing, representing entities as diverse as data structures, computation trees, and computation states. Here we consider only finite terms. Informally, a finite tree automaton (FTA) is a mathematical machine for recognising terms. FTAs provide a means of finitely specifying possibly infinite sets of ground terms, just as finite state automata specify sets of strings. Detailed definitions, algorithms and complexity results can be found in the literature [1]. An FTA includes a grammar for trees given by transition rules of the form f(q1, . . . , qn)→ q, where f is a function symbol from a given signature Σ and q, q1, . . . , qn are states of the automaton. This rules states that a term f(t1, . . . , tn) is accepted by at state q of the FTA if t1, . . . , tn are accepted at states q1, . . . , qn respectively. If q is a final state, then the term f(t1, . . . , tn) is recognised by the FTA. A notational variant of FTAs is given by alternating tree automata, though the class of recognisable terms is the same as for FTAs. For the purposes of static analysis, a subset of alternating tree automata we call conjunctive FTA (CFTA) is useful. In a CFTA, transition rules have the form f(C1, . . . , Cn)→ q, which is like an FTA transition except that C1, . . . , Cn are nonempty sets of automata states. Such a rule states that a term f(t1, . . . , tn) is accepted at state ? Supported by Danish Research Council grants FNU 272-06-0574, FNU 10-084290.