[Paper Review] Propositional Logics Complexity and the Sub-Formula Property

Edward Hermann Hæusler|Jan 31, 2014

Logic, Reasoning, and Knowledge15 references9 citations

TL;DR

This paper establishes that any propositional (possibly modal) structural logic with a Natural Deduction system satisfying the sub-formula property is in PSPACE; if the logic includes the purely implicational fragment of minimal logic ($\mathbf{M}_{\rightarrow}$), it is PSPACE-complete. The key contribution is a proof-theoretic method linking the sub-formula principle to computational complexity bounds, extending Statman’s result and showing that finitely many-valued logics also have their tautology sets in PSPACE.

ABSTRACT

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.

Motivation & Objective

• To establish a general proof-theoretic criterion for PSPACE membership in propositional logics based on the sub-formula property.
• To extend Statman’s PSPACE-completeness result for the implicational fragment of intuitionistic logic to broader classes of structural logics.
• To show that any finitely many-valued logic has its set of tautologies in PSPACE using proof-theoretic techniques.
• To investigate the conditions under which EXPTIME-complete logics, such as PDL and common-knowledge epistemic logic, could satisfy the sub-formula principle, linking it to the PSPACE vs. EXPTIME question.
• To explore the feasibility of applying the method to labeled Natural Deduction systems and first-order fragments of logic.

Proposed method

• Formalizing a general schema for introduction and elimination rules in Natural Deduction, allowing for arbitrary logical constants defined via such rules.
• Defining the sub-formula principle as a constraint on derivations: all formula occurrences in a proof must be sub-formulas of the conclusion or premises.
• Using a polynomial-space bounded search over derivations that only involve sub-formulas, leveraging the sub-formula property to bound the search space.
• Applying this technique to show that the validity problem for any logic with the sub-formula property is in PSPACE.
• Demonstrating that if the logic includes $\mathbf{M}_{\rightarrow}$, then the validity problem is PSPACE-complete, via reduction from known PSPACE-complete problems.
• Extending the method to finitely many-valued logics by showing that truth evaluation over all valuations can be performed in polynomial space, reusing space across evaluations.

Experimental results

Research questions

• RQ1 Under what conditions on a propositional logic’s Natural Deduction system is the validity problem in PSPACE?
• RQ2 Can the sub-formula property alone guarantee PSPACE membership for a logic, even without full harmony between introduction and elimination rules?
• RQ3 Is the PSPACE-completeness of $\mathbf{M}_{\rightarrow}$ sufficient to imply PSPACE-completeness for any logic extending it?
• RQ4 What is the relationship between the sub-formula property and the complexity class of EXPTIME-complete logics such as PDL and common-knowledge epistemic logic with two or more agents?
• RQ5 Can the proof-theoretic method be extended to labeled Natural Deduction systems used in modal and epistemic logics?

Key findings

• Any propositional logic with a Natural Deduction system satisfying the sub-formula principle has its validity problem in PSPACE.
• If a logic includes the purely implicational fragment of minimal logic ($\mathbf{M}_{\rightarrow}$), then its validity problem is PSPACE-complete.
• The validity problem for any finitely many-valued logic is in PSPACE, as the truth evaluation of formulas can be computed in polynomial space with space reuse.
• EXPTIME-complete logics such as PDL and common-knowledge epistemic logic with at least two agents satisfy the sub-formula principle if and only if PSPACE = EXPTIME.
• The method does not require the inversion principle or full harmony between introduction and elimination rules, relying solely on the sub-formula property and a general rule schema.