Integrating A First-order Automatic prover In The HOL Environment

Lecture Notes in Computer Science, 2009

The Thousands of Problems for Theorem Provers (TPTP) problem library is the basis of a well established infrastructure supporting research, development, and deployment of first-order Automated Theorem Proving (ATP) systems. Recently, the TPTP has been extended to include problems in higher-order logic, with corresponding infrastructure and resources. This paper describes the practical progress that has been made towards the goal of TPTP support for higher-order ATP systems.

Journal of Logic and Computation, 1994

Uniform proofs have been presented as a basis for logic programming, and it is known that by restricting the class of formulae it is possible to guarantee that uniform proofs are complete with respect to provability in intuitionistic logic. In this paper we explore the relationship between uniform proofs and classes of formulae more deeply. Firstly we show that uniform proofs arise naturally as a normal form for proofs in rst-order intuitionistic sequent calculus. Next we show that the class of formulae known as hereditary Harrop formulae are intimately related to uniform proofs, and that we may extract such formulae from uniform proofs in two di erent ways. We also give results which may be interpreted as showing that hereditary Harrop formulae are the largest class of formulae for which uniform proofs are guaranteed to be complete. Finally we brie y discuss some possibilities for a slightly more general approach using intermediate and in nitary logics.

LEO-II is a standalone, resolution-based higher-order theorem prover designed for effective cooperation with specialist provers for natural fragments of higher-order logic. At present LEO-II can cooperate with the first-order automated theorem provers E, SPASS, and Vampire. The improved performance of LEO-II, especially in comparison to its predecessor LEO, is due to several novel features including the exploitation of term sharing and term indexing techniques, support for primitive equality reasoning, and improved heuristics at the calculus level. LEO-II is implemented in Objective Caml and its problem representation language is the new TPTP THF language.

2008

LEO-II is a standalone, resolution-based higher-order theorem prover designed for effective cooperation with specialist provers for natural fragments of higher-order logic. At present LEO-II can cooperate with the first-order automated theorem provers E, SPASS, and Vampire.

Lecture Notes in Computer Science, 2007

We present Zenon, an automated theorem prover for first order classical logic (with equality), based on the tableau method. Zenon is intended to be the dedicated prover of the Focal environment, an objectoriented algebraic specification and proof system, which is able to produce OCaml code for execution and Coq code for certification. Zenon can directly generate Coq proofs (proof scripts or proof terms), which can be reinserted in the Coq specifications produced by Focal. Zenon can also be extended, which makes specific (and possibly local) automation possible in Focal.

Rapport Technique

C OQ is a Proof Assistant for a Logical Framework known as the Calculus of Induc- tive Constructions. It allows the interactive construction of formal proofs, and also the manipulation of functional programs consistently with ...

Logic Journal of IGPL, 1997

In this paper we give relational semantics and an accompanying relational proof system for a variety of intuitionistic substructural logics, including (intuitionistic) linear logic with exponentials. Starting with the (Kripke-style) semantics for F L as discussed in [13], we developed, in [11], a relational semantics and a relational proof system for full Lambek calculus. Here, we take this as a base and extend the results to deal with the various structural rules of exchange, contraction, weakening and expansion, and also to deal with an involution operator and with the operators ! and ? of linear logic. To accomplish this, for each extension X of F L we develop a Kripke-style semantics, RelKripke X semantics, as a bridge to relational semantics. The RelKripke X semantics consists of a set with distinguished elements, ternary relations and a list of conditions on the relations. For each extension X, RelKripke X semantics is accompanied by a Kripke-style valuation system analogous to that in [13]. Soundness and completeness theorems with respect to F L X hold for RelKripke X-models. Then, in the spirit of the work of Orlowska [16], [17], and Buszkowski & Orlowska [4], we develop relational logic RF L X for each extension X. The adjective relational is used to emphasize the fact that RF L X has a semantics wherein formulas are interpreted as relations. We prove that a sequent Γ → α in F L X is provable iff, a translation, t(γ 1 • ... • γn ⊃ α)ǫvu, has a cut-complete proof tree which is fundamental. This result is constructive: that is, if a cut-complete proof tree for t(γ 1 • ... • γn ⊃ α)ǫvu is not fundamental, we can use the failed proof search to build a relational countermodel for t(γ 1 • ... • γn ⊃ α) and from this, build a RelKripke X countermodel for γ 1 • ... • γn ⊃ α. 1

2014

Basing program analyses on formal semantics has a long and successful tradition in the logic programming paradigm. These analyses rely on results about the relative correctness of mathematically sophisticated semantics, and authors of such analyses often invest considerable effort into establish-ing these results. The development of interactive theorem provers such as Coq and their recent successes both in the field of program verification as well as in mathematics, poses the question whether these tools can be usefully deployed in logic programming. This paper presents formalisations in Coq of several general results about the correctness of se-mantics in different styles; forward and backward, top-down and bottom-up. The results chosen are paradigmatic of the kind of correctness theorems that semantic analyses rely on and are therefore well-suited to explore the possibilities af-forded by the application of interactive theorem provers to this task, as well as the difficulties like...

2001

Logic programs consist of formulas of mathematical logic and various proof-theoretic techniques can be used to design and analyse execution models for such programs. In this paper we present some initial work on the problem of making systematic the design of logic programming languages. In particular, we identify and discuss several key properties of proofs. A important aspect of this examination is a a more precise specification of sequent calculi inference rules in order to study permutation properties, which are key aspect of the design of logic programming systems. We also show how this specification can be used to manipulate proofs, as well as to establish properties of sets of inference rules. In addition we describe how Boolean expressions can be used to detect unused formulae in a proof, which is important for debugging purposes.