Proof complexity of intuitionistic implicational formulas

Abstract We study implicational formulas in the context of proof complexity of intuitionistic propositional logic (IPC). On the one hand, we give an efficient transformation of tautologies to implicational tautologies that preserves the lengths of intuitionistic extended Frege (EF ) or substitution Frege (SF ) proofs up to a polynomial. On the other hand, EF proofs in the implicational fragment of IPC polynomially simulate full intuitionistic logic for implicational tautologies. The results also apply to other fragments of other superintuitionistic logics under certain conditions. In particular, the exponential lower bounds on the length of intuitionistic EF proofs by Hrubeš [9], generalized to exponential separation between EF and SF systems in superintuitionistic logics of unbounded branching by Jeřábe

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

Computer-Aided Discovery and Categorisation of Personality Axioms

We propose a computer-algebraic, order-theoretic framework based on intuitionistic logic for the computer-aided discovery of personality axioms from personality-test data and their mathematical categorisation into formal personality theories in the spirit of F.~Klein's Erlanger Programm for geometrical theories. As a result, formal personality theories can be automatically generated, diagrammatically visualised, and mathematically characterised in terms of categories of invariant-preserving transformations in the sense of Klein and category theory. Our personality theories and categories are induced by implicational invariants that are ground instances of intuitionistic implication, which we postulate as axioms. In our mindset, the essence of personality, and thus mental health and illness, is its invariance. The truth of these axioms is algorithmically extracted from histories of partially-ordered, symbolic data of observed behaviour. The personality-test data and the personality theories are related by a Galois-connection in our framework. As data format, we adopt the format of the symbolic values generated by the Szondi-test, a personality test based on L.~Szondi's unifying, depth-psychological theory of fate analysis.Comment: related to arXiv:1403.200

Intuitionistic implication makes model checking hard

We investigate the complexity of the model checking problem for intuitionistic and modal propositional logics over transitive Kripke models. More specific, we consider intuitionistic logic IPC, basic propositional logic BPL, formal propositional logic FPL, and Jankov's logic KC. We show that the model checking problem is P-complete for the implicational fragments of all these intuitionistic logics. For BPL and FPL we reach P-hardness even on the implicational fragment with only one variable. The same hardness results are obtained for the strictly implicational fragments of their modal companions. Moreover, we investigate whether formulas with less variables and additional connectives make model checking easier. Whereas for variable free formulas outside of the implicational fragment, FPL model checking is shown to be in LOGCFL, the problem remains P-complete for BPL.Comment: 29 pages, 10 figure

A simplified lower bound for implicational logic

We present a streamlined and simplified exponential lower bound on the length of proofs in intuitionistic implicational logic, adapted to Gordeev and Haeusler's dag-like natural deduction.Comment: 31 page

The model checking problem for intuitionistic propositional logic with one variable is AC1-complete

We show that the model checking problem for intuitionistic propositional logic with one variable is complete for logspace-uniform AC1. As basic tool we use the connection between intuitionistic logic and Heyting algebra, and investigate its complexity theoretical aspects. For superintuitionistic logics with one variable, we obtain NC1-completeness for the model checking problem.Comment: A preliminary version of this work was presented at STACS 2011. 19 pages, 3 figure