Towards efficient subsumption

Practical Aspects of Declarative Languages, 2018

Proving a theorem in intuitionistic propositional logic, with implication as its single connective, is known as one of the simplest to state PSPACE-complete problem. At the same time, via the Curry-Howard isomorphism, it is instrumental to find lambda terms that may inhabit a given type. However, as hundreds of papers witness it, all starting with Gentzen's LJ calculus, conceptual simplicity has not come in this case with comparable computational counterparts. Implementing such theorem provers faces challenges related not only to soundness and completeness but also too termination and scalability problems. In search for an efficient but minimalist theorem prover, on the two sides of the Curry-Howard isomorphism, we design a combinatorial testing framework using types inferred for lambda terms as well as all-term and random term generators. We choose Prolog as our metalanguage. Being derived from essentially the same formalisms as those we are covering, it reduces the semantic gap and results in surprisingly concise and efficient declarative implementations. Our implementation is available at: https://github.com/ ptarau/TypesAndProofs.

Lecture Notes in Computer Science, 1996

We present a proof procedure for classical and non-classical logics. The proof search is based on the matrix-characterization of validity where an emphasis on paths and connections avoids redundancies occurring in sequent or tableaux calculi. Our uniform path-checking algorithm operates on arbitrary (non-normal form) formulae and generalizes Bibel's connection method for classical logic and formulae in clause-form. It can be applied to intuitionistic and modal logics by modifying the component for testing complementarity of connected atoms. Besides a short and elegant path-checking procedure we present a specialized string-unification algorithm which is necessary for dealing with non-classical logics.

2008

Abstract. Default logics represent an important class of the nonmonotonic formalisms. Using simple by powerful inference rules, called defaults, these logic systems model reasoning patterns of the form ”in the absence of information to the contrary of... ”, and thus formalize the default reasoning, a special type of nonmonotonic reasoning. In this paper we propose an automated system, called DARR, with two components: a propositional theorem prover and a theorem prover for constrained and rational propositional default logics. A modified version of semantic tableaux method is used to implement the propositional prover. Also, this theorem proving method is adapted for computing extensions because one of its purpose is to produce models, and extensions are models of the world described by default theories. 1.

2001

this article we will not attempt to describe all the dierent possible choicesof type theories. Instead we want to discuss the main underlying ideas, with a specialfocus on the use of type theory as the formalism for the description of theoriesincluding proofs

Lecture Notes in Computer Science, 2001

Herbrand's Theorem for £ ¥ ¤ ¦ , i.e., Gödel logic enriched by the projection operator § is proved. As a consequence we obtain a "chain normal form" and a translation of prenex £ ¤ ¦ into (order) clause logic, referring to the classical theory of dense total orders with endpoints. A chaining calculus provides a basis for efficient theorem proving.

In this paper we present KLMLean 1.0, a theorem prover for some logics of default reasoning, namely Preferential logic P and Loop-Cumulative logic CL introduced by Kraus, Lehmann, and Magidor. KLMLean 1.0 implements some tableaux calculi for these logics recently introduced. It is implemented in SICStus Prolog and also comprises a graphical user interface written in Java. KLMLean 1.0 is available for free download at http://www.di.unito.it/spozzato/klmlean1/

2018

Proving a theorem in intuitionistic propositional logic, with implication as its only primitive (also called minimal logic), is known as one of the simplest to state PSPACE-complete problem. At the same time, via the CurryHoward isomorphism, it is instrumental to finding lambda terms that may inhabit a given type. However, as hundreds of papers witness it, all starting with Gentzen’s LJ calculus, conceptual simplicity has not come in this case with comparable computational counterparts. Implementing such theorem provers faces challenges related not only to soundness and completeness but also to termination and scalability problems. Can a simple solution, in the tradition of ”lean theorem provers” be uncovered that matches the conceptual simplicity of the problem, while being able to handle also large randomly generated formulas? In search for an answer, starting from Dyckhoff’s LJT calculus, we derive a sequence of minimalist theorem provers using declarative program transformations...

Automated Deduction—Cade-13, 1996

We use the general scheme of building resolution calculi (also called the inverse method) originating from S.Maslov and G.Mints to design and implement a resolution theorem prover for intuitionistic logic. A number of search strategies is introduced and proved complete. The resolution method is shown to be a decision procedure for a new syntactically described decidable class of intuitionistic logic. We compare the search strategies suitable for the resolution method with strategies suitable for the tableau method. The performance of our prover is compared with the performance of a tableau prover for intuitionistic logic presented in 17].

2008

Background Automatic theorem provers (ATPs) based on the resolution principle, such as SPASS [22] and Vampire [19], have reached a high degree of sophistication. They can often find long proofs even for problems having thousands of axioms. A fundamental limitation, however, is that they reason in first-order logic. Higher-order logic extends first-order logic with λ-notation for functions and with function and predicate variables. It supports reasoning in set theory, using the obvious representation of sets by predicates.

EPiC series in computing, 2018

Studies on type theory have brought numerous important contributions to computer science. In this paper we present a GUI-based proof tool that provides assistance in constructing deductions in type theory and validating implicational intuitionistic-logic formulas. As such, this proof tool is a testbed for learning type theory and implicational intuitionistic-logic. This proof tool focuses on an important variant of type theory named TA λ , especially on its two core algorithms: the principal-type algorithm and the type inhabitant search algorithm. The former algorithm finds a most general type assignable to a given λ-term, while the latter finds inhabitants (closed λ-terms in β-normal form) to which a given type can be assigned. By the Curry-Howard correspondence, the latter algorithm provides provability for implicational formulas in intuitionistic logic. We elaborate on how to implement those two algorithms declaratively in Prolog and the overall GUI-based program architecture. In our implementation, we make some modification to improve the performance of the principal-type algorithm. We have also built a web-based version of the proof tool called λ-Guru.