Connection Tableau Research Articles

Machine Learning Guidance for Connection Tableaux

Connection calculi allow for very compact implementations of goal-directed proof search. We give an overview of our work related to connection tableaux calculi: first, we show optimised functional implementations of connection tableaux proof search, including a consistent Skolemisation procedure for machine learning. Then, we show two guidance methods based on machine learning, namely reordering of proof steps with Naive Bayesian probabilities, and expansion of a proof search tree with Monte Carlo Tree Search.

SOLAR: An automated deduction system for consequence finding

SOLAR (SOL for Advanced Reasoning) is a first-order clausal consequence finding system based on the SOL (Skip Ordered Linear) tableau calculus. The ability to find non-trivial consequences of an axiom set is useful in many applications of Artificial Intelligence such as theorem proving, query answering and nonmonotonic reasoning. SOL is a connection tableau calculus which is complete for finding the non-subsumed consequences of a clausal theory. SOLAR is an efficient implementation of SOL that employs several methods to prune away redundant branches of the search space. This paper introduces some of the key pruning and control strategies implemented in SOLAR and demonstrates their effectiveness on a collection of benchmark problems.

Encoding First Order Proofs in SMT

We present a method for encoding first order proofs in SMT. Our implementation, called ChewTPTP-SMT, transforms a set of first order clauses into a propositional encoding (modulo theories) of the existence of a rigid first order connection tableau and the satisfiability of unification constraints, which is then fed to Yices. For the unification constraints, terms are represented as recursive datatypes, and unification constraints are equations on terms. The finiteness of the tableau is encoded by linear real arithmetic inequalities.We compare our implementation with our previous implementation ChewTPTP-SAT, encoding rigid connection tableau in SAT, and show that for Horn clauses many fewer propositional clauses are generated by ChewTPTP-SMT, and ChewTPTP-SMT is much faster than ChewTPTP-SAT. This is not the case for our non-Horn clause encoding. We explain this, and we conjecture a rule of thumb on when to use theories in encoding a problem.

Connection Tableaux with Lazy Paramodulation

It is well known that the connection refinement of clause tableaux with paramodulation is incomplete (even with weak connections). In this paper, we present a new connection tableau calculus for logic with equality. This calculus is based on a lazy form of paramodulation where parts of the unification step become auxiliary subgoals in a tableau and may be subjected to subsequent paramodulations. Our calculus uses ordering constraints and a certain form of the basicness restriction.

Connection Tableau Calculi with Disjunctive Constraints

Automated theorem proving amounts to solving search problems in usually tremendous search spaces. A lot of research therefore focuses on search space reductions. Our approach reduces the search space which arises when using so-called connection tableau calculi for first-order automated theorem proving. It uses disjunctive constraints over first-order equations to compress certain parts of this search space. We present the basics of our constrained-connection-tableau calculi, a constraint extension of connection tableau calculi, and deal with the efficient handling of constraints during the search process. The new techniques are integrated into the automated connection tableau prover Setheo.

AN AUTOMATED THEOREM PROVER BASED ON CONNECTION TABLEAU CALCULI WITH DISJUNCTIVE CONSTRAINTS

Automated theorem proving with connection tableau calculi imposes search problems in tremendous search spaces. In this paper, we present a new approach to search space reduction in connection tableau calculi. In our approach structurally similar parts of the search space are compressed by means of disjunctive constraints. We describe the necessary changes of the calculus, and we develop elaborate techniques for an efficient constraint processing. Moreover, we present an experimental evaluation of our approach.

Controlled Use of Clausal Lemmas in Connection Tableau Calculi

Proof procedures based on model elimination or the connection tableau calculus have become more and more successful. But these procedures still suffer from long proof lengths as well as from a rather high degree of redundant search effort in comparison with resolution-style search procedures. In order to overcome these problems we investigate the use of clausal lemmas. We develop a method to augment a given set of clauses by a lemma set in a preprocessing phase and discuss the ability of this technique to reduce proof lengths and depths and to provide an appropriate reordering of the search space. We deal with the basic connection tableau calculus as well as with several calculus refinements and extensions. In order to control the use of lemmas we develop techniques for selecting relevant lemmas based on partial evaluation techniques. Experiments with the prover Setheo performed in several domains demonstrate the high potential of our approach.

Controlled integration of the cut rule into connection tableau calculi

In this paper techniques are developed and compared that increase the inferential power of tableau systems for classical first-order logic. The mechanisms are formulated in the framework of connection tableaux, which is an amalgamation of the connection method and the tableau calculus, and a generalization of model elimination. Since connection tableau calculi are among the weakest proof systems with respect to proof compactness, and the (backward) cut rule is not suitable for the first-order case, we study alternative methods for shortening proofs. The techniques we investigate are the folding-up and the folding-down operations. Folding up represents an efficient way of supporting the basic calculus, which is top-down oriented, with lemmata derived in a bottom-up manner. It is shown that both techniques can also be viewed as controlled integrations of the cut rule. To remedy the additional redundancy imported into tableau proof procedures by the new inference rules, we develop and apply an extension of the regularity condition on tableaux and the mechanism of anti-lemmata which realizes a subsumption concept on tableaux. Using the framework of the theorem prover SETHEO, we have implemented three new proof procedures that overcome the deductive weakness of cut-free tableau systems. Experimental results demonstrate the superiority of the systems with folding up over the cut-free variant and the one with folding down.