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Register a new userOn Quantified Propositional Logics and the Exponential Time Hierarchy
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Publication date
2016
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Informatics Engineering
and research's language is
English
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We study quantified propositional logics from the complexity theoretic point of view. First we introduce alternating dependency quantified boolean formulae (ADQBF) which generalize both quantified and dependency quantified boolean formulae. We show that the truth evaluation for ADQBF is AEXPTIME(poly)-complete. We also identify fragments for which the problem is complete for the levels of the exponential hierarchy. Second we study propositional team-based logics. We show that DQBF formulae correspond naturally to quantified propositional dependence logic and present a general NEXPTIME upper bound for quantified propositional logic with a large class of generalized dependence atoms. Moreover we show AEXPTIME(poly)-completeness for extensions of propositional team logic with generalized dependence atoms.
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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.
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Eric Alsmann (School of Electrical Engineering
, Computer Science
, n University of Kassel
2021
We investigate the expressive power of the two main kinds of program logics for complex, non-regular program properties found in the literature: those extending propositional dynamic logic (PDL), and those extending the modal mu-calculus. This is inspired by the recent discovery of a decidable program logic called Visibly Pushdown Fixpoint Logic with Chop which extends both the modal mu-calculus and PDL over visibly pushdown languages, which, so far, constituted the ends of two pillars of decidable program logics. Here we show that this logic is not only more expressive than either of its two fragments, but in fact even more expressive than their union. Hence, the decidability border amongst program logics has been properly pushed up. We complete the picture by providing results separating all the PDL-based and modal fixpoint logics with regular, visibly pushdown and arbitrary context-free constructions.
Separation logics are a family of extensions of Hoare logic for reasoning about programs that mutate memory. These logics are abstract because they are independent of any particular concrete memory model. Their assertion languages, called propositional abstract separation logics, extend the logic of (Boolean) Bunched Implications (BBI) in various ways. We develop a modular proof theory for various propositional abstract separation logics using cut-free labelled sequent calculi. We first extend the cut-fee labelled sequent calculus for BBI of Hou et al to handle Calcagno et als original logic of separation algebras by adding sound rules for partial-determinism and cancellativity, while preserving cut-elimination. We prove the completeness of our calculus via a sound intermediate calculus that enables us to construct counter-models from the failure to find a proof. We then capture other propositional abstract separation logics by adding sound rules for indivisible unit and disjointness, while maintaining completeness. We present a theorem prover based on our labelled calculus for these propositional abstract separation logics.
Given two structures $G$ and $H$ distinguishable in $fo k$ (first-order logic with $k$ variables), let $A^k(G,H)$ denote the minimum alternation depth of a $fo k$ formula distinguishing $G$ from $H$. Let $A^k(n)$ be the maximum value of $A^k(G,H)$ over $n$-element structures. We prove the strictness of the quantifier alternation hierarchy of $fo 2$ in a strong quantitative form, namely $A^2(n)ge n/8-2$, which is tight up to a constant factor. For each $kge2$, it holds that $A^k(n)>log_{k+1}n-2$ even over colored trees, which is also tight up to a constant factor if $kge3$. For $kge 3$ the last lower bound holds also over uncolored trees, while the alternation hierarchy of $fo 2$ collapses even over all uncolored graphs. We also show examples of colored graphs $G$ and $H$ on $n$ vertices that can be distinguished in $fo 2$ much more succinctly if the alternation number is increased just by one: while in $Sigma_{i}$ it is possible to distinguish $G$ from $H$ with bounded quantifier depth, in $Pi_{i}$ this requires quantifier depth $Omega(n^2)$. The quadratic lower bound is best possible here because, if $G$ and $H$ can be distinguished in $fo k$ with $i$ quantifier alternations, this can be done with quantifier depth $n^{2k-2}$.
We study counting propositional logic as an extension of propositional logic with counting quantifiers. We prove that the complexity of the underlying decision problem perfectly matches the appropriate level of Wagners counting hierarchy, but also that the resulting logic admits a satisfactory proof-theoretical treatment. From the latter, a type system for a probabilistic lambda-calculus is derived in the spirit of the Curry-Howard correspondence, showing the potential of counting propositional logic as a useful tool in several fields of theoretical computer science.
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