283 research outputs found
Range-Restricted Interpolation through Clausal Tableaux
We show how variations of range-restriction and also the Horn property can be
passed from inputs to outputs of Craig interpolation in first-order logic. The
proof system is clausal tableaux, which stems from first-order ATP. Our results
are induced by a restriction of the clausal tableau structure, which can be
achieved in general by a proof transformation, also if the source proof is by
resolution/paramodulation. Primarily addressed applications are query synthesis
and reformulation with interpolation. Our methodical approach combines
operations on proof structures with the immediate perspective of feasible
implementation through incorporating highly optimized first-order provers
On multiple conclusion deductions in classical logic
Kneale observed that Gentzen’s calculus of natural deductions NK for classical
logic is not symmetric and has unnecessarily complicated hypothetical inference rules.
Kneale proposed inference rules with multiple conclusions as a basis for a symmetric natural
deduction calculus for classical logic. However, Kneale’s informally presented calculus is
not complete. In this paper, we define a calculus of multiple conclusion natural deductions
(MCD) for classical propositional logic based on Kneale’s multiple conclusion inference
rules. For MCD we present elementary proof search that produces proofs in normal form.
MCD proof search is motivated and explained as being a notational variant of Smullyan’s
analytic tableaux method in its initial part and a notational variant of refutation proofs
based on Robinson’s resolution in its final part. We consider MCD to have semantic motivation
of both its inference rules and its proof search. This is unusual for the natural
deduction calculi as they are syntactically motivated. Syntactic motivation is adequate for
intuitionistic logic but not a natural fit for truth-functional classical propositional logic
Integrating a Global Induction Mechanism into a Sequent Calculus
Most interesting proofs in mathematics contain an inductive argument which
requires an extension of the LK-calculus to formalize. The most commonly used
calculi for induction contain a separate rule or axiom which reduces the valid
proof theoretic properties of the calculus. To the best of our knowledge, there
are no such calculi which allow cut-elimination to a normal form with the
subformula property, i.e. every formula occurring in the proof is a subformula
of the end sequent. Proof schemata are a variant of LK-proofs able to simulate
induction by linking proofs together. There exists a schematic normal form
which has comparable proof theoretic behaviour to normal forms with the
subformula property. However, a calculus for the construction of proof schemata
does not exist. In this paper, we introduce a calculus for proof schemata and
prove soundness and completeness with respect to a fragment of the inductive
arguments formalizable in Peano arithmetic.Comment: 16 page
Converting ALC Connection Proofs into ALC Sequents
The connection method has earned good reputation in the area of automated
theorem proving, due to its simplicity, efficiency and rational use of memory.
This method has been applied recently in automatic provers that reason over
ontologies written in the description logic ALC. However, proofs generated by
connection calculi are difficult to understand. Proof readability is largely
lost by the transformations to disjunctive normal form applied over the
formulae to be proven. Such a proof model, albeit efficient, prevents inference
systems based on it from effectively providing justifications and/or
descriptions of the steps used in inferences. To address this problem, in this
paper we propose a method for converting matricial proofs generated by the ALC
connection method to ALC sequent proofs, which are much easier to understand,
and whose translation to natural language is more straightforward. We also
describe a calculus that accepts the input formula in a non-clausal ALC format,
what simplifies the translation.Comment: In Proceedings PxTP 2019, arXiv:1908.08639. Thanks to CAPES:
Coordination for the Improvement of Higher Level Personne
Satisfiability Games for Branching-Time Logics
The satisfiability problem for branching-time temporal logics like CTL*, CTL
and CTL+ has important applications in program specification and verification.
Their computational complexities are known: CTL* and CTL+ are complete for
doubly exponential time, CTL is complete for single exponential time. Some
decision procedures for these logics are known; they use tree automata,
tableaux or axiom systems. In this paper we present a uniform game-theoretic
framework for the satisfiability problem of these branching-time temporal
logics. We define satisfiability games for the full branching-time temporal
logic CTL* using a high-level definition of winning condition that captures the
essence of well-foundedness of least fixpoint unfoldings. These winning
conditions form formal languages of \omega-words. We analyse which kinds of
deterministic {\omega}-automata are needed in which case in order to recognise
these languages. We then obtain a reduction to the problem of solving parity or
B\"uchi games. The worst-case complexity of the obtained algorithms matches the
known lower bounds for these logics. This approach provides a uniform, yet
complexity-theoretically optimal treatment of satisfiability for branching-time
temporal logics. It separates the use of temporal logic machinery from the use
of automata thus preserving a syntactical relationship between the input
formula and the object that represents satisfiability, i.e. a winning strategy
in a parity or B\"uchi game. The games presented here work on a Fischer-Ladner
closure of the input formula only. Last but not least, the games presented here
come with an attempt at providing tool support for the satisfiability problem
of complex branching-time logics like CTL* and CTL+
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