Propositional Logics Complexity and the Sub-Formula Property

In 1979 Richard Statman proved, using proof-theory, that the purely implicational fragment of Intuitionistic Logic (M-imply) is PSPACE-complete. He showed a polynomially bounded translation from full Intuitionistic Propositional Logic into its implicational fragment. By the PSPACE-completeness of S4, proved by Ladner, and the Goedel translation from S4 into Intuitionistic Logic, the PSPACE- completeness of M-imply is drawn. The sub-formula principle for a deductive system for a logic L states that whenever F1,...,Fk proves A, there is a proof in which each formula occurrence is either a sub-formula of A or of some of Fi. In this work we extend Statman result and show that any propositional (possibly modal) structural logic satisfying a particular formulation of the sub-formula principle is in PSPACE. If the logic includes the minimal purely implicational logic then it is PSPACE-complete. As a consequence, EXPTIME-complete propositional logics, such as PDL and the common-knowledge epistemic logic with at least 2 agents satisfy this particular sub-formula principle, if and only if, PSPACE=EXPTIME. We also show how our technique can be used to prove that any finitely many-valued logic has the set of its tautologies in PSPACE.Comment: In Proceedings DCM 2014, arXiv:1504.0192

Proof-graphs for Minimal Implicational Logic

It is well-known that the size of propositional classical proofs can be huge. Proof theoretical studies discovered exponential gaps between normal or cut free proofs and their respective non-normal proofs. The aim of this work is to study how to reduce the weight of propositional deductions. We present the formalism of proof-graphs for purely implicational logic, which are graphs of a specific shape that are intended to capture the logical structure of a deduction. The advantage of this formalism is that formulas can be shared in the reduced proof. In the present paper we give a precise definition of proof-graphs for the minimal implicational logic, together with a normalization procedure for these proof-graphs. In contrast to standard tree-like formalisms, our normalization does not increase the number of nodes, when applied to the corresponding minimal proof-graph representations.Comment: In Proceedings DCM 2013, arXiv:1403.768

Two basic results on translations between logics

The aim of the present paper is to show two basic results concerning translation between logics:[1] The first result establishes that given two logics S1 and S2 with languages L1 and L2, and a translation F of L1 into L2 that interprets S1 into S2, then, given any intermediate logic S3 between S1 and S2, the same translation F interprets S1 into S3.[2] The second result establishes that the translation F cannot interpret S3 into S2.Dois resultados básicos sobre traduções entre lógicasO objetivo do presente trabalho é mostrar dois resultados básicos relativos à tradução entre lógicas:[1] O primeiro resultado estabelece que, dadas duas lógicas S1 e S2, com linguagens L1 e L2, e uma tradução F de L1 em L2 que interpreta S1 em S2, então, dada qualquer lógica intermediária S3 entre S1 e S2, a mesma tradução F Interpreta S1 em S3.[2] O segundo resultado estabelece que a tradução F não pode interpretar S3 em S2.---Artigo em inglês

On the horizontal compression of dag-derivations in minimal purely implicational logic

In this report, we define (plain) Dag-like derivations in the purely implicational fragment of minimal logic M_{\imply}. Introduce the horizontal collapsing set of rules and the algorithm {\bf HC}. Explain why {\bf HC} can transform any polynomial height-bounded tree-like proof of a M_{\imply} tautology into a smaller dag-like proof. Sketch a proof that {\bf HC} preserves the soundness of any tree-like ND in M_{\imply} in its dag-like version after the horizontal collapsing application. We show some experimental results about applying the compression method to a class of (huge) propositional proofs and an example, with non-hamiltonian graphs, for qualitative analysis. The contributions include the comprehensive presentation of the set of horizontal compression (HC), the (sketch) of a proof that HC rules preserve soundness and the demonstration that the compressed dag-like proofs are polynomially upper-bounded when the submitted tree-like proof is height and foundation poly-bounded. Finally, in the appendix, we show an algorithm that verifies in polynomial time on the size of the dag-like proofs whether they are valid proofs of their conclusions.Comment: This is a comprehensive report with the set of rules and the algorithm for compressing Natural Deduction proofs in the purely implicational minimal logic.It reports experiments with implementation applied to a class of huge proofs. It has new references, new section 5 with subsec 5.1, and updated the acknowledgements. This report has a much more detailed proof of soundnes