Towards Deciding Second-order Unification Problems Using Regular Tree Automata

International audienceThe second-order unification problem is undecidable [5]. While unification procedures, like Huet's pre-unification, terminate with success on unifiable problems, they might not terminate on non-unifiable ones. There are several decidability results for unification problems with infinitely-many pre-unifiers, such as for monadic second-order problems [3]. These results are based on the regular structure of the solutions of these problems and by computing minimal unifiers. Beyond the importance of the knowledge that searching for unifiers of decidable problems always terminates, one can also use this information in order to optimize unification algorithms, such as in the case for pattern unification [10]. Nevertheless, being able to prove that the unification problem of a certain class of unification constraints is decidable is far from easy. Some results were obtained for certain syntactic restrictions on the problems (see Levy [8] for some results and references) or on the unifiers (see Schmidt-Schauß [11], Schmidt-Schauß and Schulz [12, 13] and Je˙ z [7] for some results). Infinitary unification problems, like the ones we are considering, might suggest that known tools for dealing with the infinite might be useful. One such tool is the regular tree automaton. The drawback of using regular automata for unification is, of course, their inability to deal with variables. In this paper we try to overcome this obstacle and describe an ongoing work about using regular tree automata [1] in order to decide more general second-order unification problems. The second-order unification problems we will consider are of the form λz n .x 0 t. = λz n .C(x 0 s) where C is a non-empty context [2] and x 0 does not occur in t or s. We will call such problems cyclic problems. An important result in second-order unification was obtained by Ganzinger et al. [4] and stated that second-order unification is undecidable already when there is only one second-order variable occurring twice. The unification problem they used for proving the undecidability result was an instance of the following cyclic problem. Note that we chose to use in the definition only unary second-order variables but that this restriction should not be essential. x 0 (w 1 , g(y 1 , a)) = g(y 2 , x 0 (w 2 , a)) (1) Our decidability result is obtained by posing one further restriction over cyclic problems which is based on the existence and location of variables other than the cyclic one. A sufficient condition for the decidability of second-order unification problems was given by Levy [8]. This condition states that if we can never encounter, when applying Huet's pre-unification procedure [6] to a problem, a cyclic equation, then the procedure terminates. It follows from this result that deciding second-order unification problems depends on the ability to decide cyclic problems. The rules of Huet's procedure (PUA) are given in Fig. 1. Imitation partial bindings and projection partial bindings are defined in [14] and are denoted, respectively, by PB(f, α) and PB(i, α) where α is a type, Σ a signature f ∈ Σ and i > 0

Advanced Proof Viewing in ProofTool

Sequent calculus is widely used for formalizing proofs. However, due to the proliferation of data, understanding the proofs of even simple mathematical arguments soon becomes impossible. Graphical user interfaces help in this matter, but since they normally utilize Gentzen's original notation, some of the problems persist. In this paper, we introduce a number of criteria for proof visualization which we have found out to be crucial for analyzing proofs. We then evaluate recent developments in tree visualization with regard to these criteria and propose the Sunburst Tree layout as a complement to the traditional tree structure. This layout constructs inferences as concentric circle arcs around the root inference, allowing the user to focus on the proof's structural content. Finally, we describe its integration into ProofTool and explain how it interacts with the Gentzen layout.Comment: In Proceedings UITP 2014, arXiv:1410.785

Towards Transparent Legal Formalization

A key challenge in making a transparent formalization of a legal text is the dependency on two domain experts. While a legal expert is needed in order to interpret the legal text, a logician or a programmer is needed for encoding it into a program or a formula. Various existing methods are trying to solve this challenge by improving or automating the communication between the two experts. In this paper, we follow a different direction and attempt to eliminate the dependency on the target domain expert. This is achieved by inverting the translation back into the original text. By skipping over the logical translation, a legal expert can now both interpret and evaluate a translation

Functions-as-Constructors Higher-Order Unification

Unification is a central operation in the construction of a range of computational logic systems based on first-order and higher-order logics. First-order unification has a number of properties that dominates the way it is incorporated within such systems. In particular, first-order unification is decidable, unary, and can be performed on untyped term structures. None of these three properties hold for full higher-order unification: unification is undecidable, unifiers can be incomparable, and term-level typing can dominate the search for unifiers. The so-called pattern subset of higher-order unification was designed to be a small extension to first-order unification that respected the basic laws governing lambda-binding (the equalities of alpha, beta, and eta-conversion) but which also satisfied those three properties. While the pattern fragment of higher-order unification has been popular in various implemented systems and in various theoretical considerations, it is too weak for a number of applications. In this paper, we define an extension of pattern unification that is motivated by some existing applications and which satisfies these three properties. The main idea behind this extension is that the arguments to a higher-order, free variable can be more than just distinct bound variables: they can also be terms constructed from (sufficient numbers of) such variables using term constructors and where no argument is a subterm of any other argument. We show that this extension to pattern unification satisfies the three properties mentioned above

Certification of Prefixed Tableau Proofs for Modal Logic

International audienceDifferent theorem provers tend to produce proof objects in different formats and this is especially the case for modal logics, where several deductive formalisms (and provers based on them) have been presented. This work falls within the general project of establishing a common specification language in order to certify proofs given in a wide range of deductive formalisms. In particular, by using a translation from the modal language into a first-order polarized language and a checker whose small kernel is based on a classical focused sequent calculus, we are able to certify modal proofs given in labeled sequent calculi, prefixed tableaux and free-variable prefixed tableaux. We describe the general method for the logic K, present its implementation in a Prolog-like language, provide some examples and discuss how to extend the approach to other normal modal logics

Legal linguistic templates and the tension between legal knowledge representation and reasoning

There is an inherent tension between knowledge representation and reasoning. For an optimal representation and validation, an expressive language should be used. For an optimal automated reasoning, a simple one is preferred. Which language should we choose for our legal knowledge representation if our goal is to apply automated legal reasoning? In this paper, we investigate the properties and requirements of each of these two applications. We suggest that by using Legal Linguistic Templates, one can solve the above tension in some practical situations

The Proof Certifier Checkers

International audienceDifferent theorem provers work within different formalisms and paradigms, and therefore produce various incompatible proof objects. Currently there is a big effort to establish foundational proof certificates (FPC), which would serve as a common " specification language " for all these formats. Such framework enables the uniform checking of proof objects from many different theorem provers while relying on a small and trusted kernel to do so. Checkers is an implementation of a proof checker using foundational proof certificates. By trusting a small kernel based on (focused) sequent calculus on the one hand and by supporting FPC specifications in a prolog-like language on the other hand, it can be used for checking proofs of a wide range of theorem provers. The focus of this paper is on the output of equational resolution theorem provers and for this end, we specify the paramodulation rule. We describe the architecture of Checkers and demonstrate how it can be used to check proof objects by supplying the FPC specification for a subset of the inferences used by E-prover and checking proofs using these inferences