Undecidability of QLTL with two variables and one monadic predicate letter

We study the algorithmic properties of the quantified linear-time temporal logic QLTL in languages with restrictions on the number of individual variables as well as the number and arity of predicate letters. We prove that the satisfiability problem for QLTL in languages with two individual variables and one monadic predicate letter in Σ 11 -hard. Thus, QLTL is Π 11 -hard, and so not recursively enumerable, in such languages. The resultholds both for the increasing domain and the constant domain semantics and is obtained by reduction from a Σ 11 -hard N×N recurrent tiling problem. It follows from the proof for QLTL that similar results hold for the quantified branching-time temporal logic QCTL, and hence for the quantified alternating-time temporal logic QATL. The result presented in this paper strengthens a result by I. Hodkinson, F. Wolter, and M. Zakharyaschev, who have shown that the satisfiability problem for QLTL is Σ 11 -hard in languages with two individual variablesand an unlimited supply of monadic predicate letters.

Variants of derivation modes for which catalytic P systems with one catalyst are computationally complete

Catalytic P systems are among the first variants of membrane systems ever considered in this area. This variant of systems also features some prominent computational complexity questions, and in particular the problem of using only one catalyst: is one catalyst enough to allow for generating all recursively enumerable sets of multisets? Several additional ingredients have been shown to be sufficient for obtaining even computational completeness with only one catalyst. Last year we could show that the derivation mode $$max_{objects}$$ m a x objects , where we only take those multisets of rules which affect the maximal number of objects in the underlying configuration one catalyst is sufficient for obtaining computational completeness without any other ingredients. In this paper we follow this way of research and show that one catalyst is also sufficient for obtaining computational completeness when using specific variants of derivation modes based on non-extendable multisets of rules: we only take those non-extendable multisets whose application yields the maximal number of generated objects or else those non-extendable multisets whose application yields the maximal difference in the number of objects between the newly generated configuration and the current configuration. A similar computational completeness result can even be obtained when omitting the condition of non-extendability of the applied multisets when taking the maximal difference of objects or the maximal number of generated objects. Moreover, we reconsider simple P system with energy control—both symbol and rule energy-controlled P systems equipped with these new variants of derivation modes yield computational completeness.

MIP* = RE

Note from the Research Highlights Co-Chairs: A Research Highlights paper appearing in Communications is usually peer-reviewed prior to publication. The following paper is unusual in that it is still under review. However, the result has generated enormous excitement in the research community, and came strongly nominated by SIGACT, a nomination seconded by external reviewers. The complexity class NP characterizes the collection of computational problems that have efficiently verifiable solutions. With the goal of classifying computational problems that seem to lie beyond NP, starting in the 1980s complexity theorists have considered extensions of the notion of efficient verification that allow for the use of randomness (the class MA), interaction (the class IP), and the possibility to interact with multiple proofs, or provers (the class MIP). The study of these extensions led to the celebrated PCP theorem and its applications to hardness of approximation and the design of cryptographic protocols. In this work, we study a fourth modification to the notion of efficient verification that originates in the study of quantum entanglement. We prove the surprising result that every problem that is recursively enumerable, including the Halting problem, can be efficiently verified by a classical probabilistic polynomial-time verifier interacting with two all-powerful but noncommunicating provers sharing entanglement. The result resolves long-standing open problems in the foundations of quantum mechanics (Tsirelson's problem) and operator algebras (Connes' embedding problem).

Insertion-deletion systems with substitutions I

In this paper, we discuss the addition of substitutions as a further type of operations to (in particular, context-free) insertion-deletion systems, i.e., in addition to insertions and deletions we allow single letter replacements to occur. We investigate the effect of the addition of substitution rules on the context dependency of such systems, thereby also obtaining new characterizations of and even normal forms for context-sensitive (CS) and recursively enumerable (RE) languages and their phrase-structure grammars. More specifically, we prove that for each RE language, there is a system generating this language that only inserts and deletes strings of length two without considering the context of the insertion or deletion site, but which may change symbols (by a substitution operation) by checking a single symbol to the left of the substitution site. When we allow checking left and right single-letter context in substitutions, even context-free insertions and deletions of single letters suffice to reach computational completeness. When allowing context-free insertions only, checking left and right single-letter context in substitutions gives a new characterization of CS. This clearly shows the power of this new type of rules.

On the generative capacity of matrix insertion-deletion systems of small sum-norm

A matrix insertion-deletion system (or matrix ins-del system) is described by a set of insertion-deletion rules presented in matrix form, which demands all rules of a matrix to be applied in the given order. These systems were introduced to model very simplistic fragments of sequential programs based on insertion and deletion as elementary operations as can be found in biocomputing. We are investigating such systems with limited resources as formalized in descriptional complexity. A traditional descriptional complexity measure of such a matrix ins-del system is its size $$s=(k;n,i',i'';m,j',j'')$$ s = ( k ; n , i ′ , i ′ ′ ; m , j ′ , j ′ ′ ) , where the parameters from left to right represent the maximal matrix length, maximal insertion string length, maximal length of left contexts in insertion rules, maximal length of right contexts in insertion rules; the last three are deletion counterparts of the previous three parameters. We call the sum $$n+i'+i''+m+j'+j''$$ n + i ′ + i ′ ′ + m + j ′ + j ′ ′ the sum-norm of s. We show that matrix ins-del systems of sum-norm 4 and sizes (3; 1, 0, 0; 1, 2, 0), (3; 1, 0, 0; 1, 0, 2), (2; 1, 2, 0; 1, 0, 0), (2; 1, 0, 2; 1, 0, 0), and (2; 1, 1, 1; 1, 0, 0) describe the recursively enumerable languages. Moreover, matrix ins-del systems of sizes (3; 1, 1, 0; 1, 0, 0), (3; 1, 0, 1; 1, 0, 0), (2; 2, 1, 0; 1, 0, 0) and (2; 2, 0, 1; 1, 0, 0) can describe at least the regular closure of the linear languages. In fact, we show that if a matrix ins-del system of size s can describe the class of linear languages $$\mathrm {LIN}$$ LIN , then without any additional resources, matrix ins-del systems of size s also describe the regular closure of $$\mathrm {LIN}$$ LIN . Finally, we prove that matrix ins-del systems of sizes (2; 1, 1, 0; 1, 1, 0) and (2; 1, 0, 1; 1, 0, 1) can describe at least the regular languages.

Improved Descriptional Complexity Results for Simple Semi-Conditional Grammars

A simple semi-conditional (SSC) grammar is a form of regulated rewriting system where the derivations are controlled either by a permitting string alone or by a forbidden string alone and this condition is specified in the rule. The maximum length i (j, resp.) of the permitting (forbidden, resp.) strings serves as a measure of descriptional complexity known as the degree of such grammars. In addition to the degree, the numbers of nonterminals and of conditional rules are also counted into the descriptional complexity measures of these grammars. We improve on some previously obtained results on the computational completeness of SSC grammars by minimizing the number of nonterminals and / or the number of conditional rules for a given degree (i, j). More specifically we prove, using a refined analysis of a normal form for type-0 grammars due to Geffert, that every recursively enumerable language is generated by an SSC grammar of (i) degree (2, 1) with eight conditional rules and nine nonterminals, (ii) degree (3, 1) with seven conditional rules and seven nonterminals (iii) degree (4, 1) with six conditional rules and seven nonterminals and (iv) degree (4, 1) with eight conditional rules and six nonterminals.

When catalytic P systems with one catalyst can be computationally complete

Catalytic P systems are among the first variants of membrane systems ever considered in this area. This variant of systems also features some prominent computational complexity questions, and in particular the problem of using only one catalyst in the whole system: is one catalyst enough to allow for generating all recursively enumerable sets of multisets? Several additional ingredients have been shown to be sufficient for obtaining computational completeness even with only one catalyst. In this paper, we show that one catalyst is sufficient for obtaining computational completeness if either catalytic rules have weak priority over non-catalytic rules or else instead of the standard maximally parallel derivation mode, we use the derivation mode maxobjects, i.e., we only take those multisets of rules which affect the maximal number of objects in the underlying configuration.

Inconsistent Models for Relevant Arithmetics

This paper develops in certain directions the work of Meyer in [3], [4], [5] and [6] (see also Routley [10] and Asenjo [11]). In those works, Peano’s axioms for arithmetic were formulated with a logical base of the relevant logic R, and it was proved finitistically that the resulting arithmetic, called R♯, was absolutely consistent. It was pointed out that such a result escapes incau- tious formulations of Goedel’s second incompleteness theorem, and provides a basis for a revived Hilbert programme. The absolute consistency result used as a model arithmetic modulo two. Modulo arithmetics are not or- dinarily thought of as an extension of Peano arithmetic, since some of the propositions of the latter, such as that zero is the successor of no number, fail in the former. Consequently a logical base which, unlike classical logic, tolerates contradictory theories was used for the model. The logical base for the model was the three-valued logic RM3 (see e.g. [1] or [8]), which has the advantage that while it is an extension of R, it is finite valued and so easier to handle. The resulting model-theoretic structure (called in this paper RM32) is interesting in its own right in that the set of sentences true therein consti- tutes a negation inconsistent but absolutely consistent arithmetic which is an extension of R♯. In fact, in the light of the result of [6], it is an extension of Peano arithmetic with a base of a classical logic, P♯. A generalisation of the structure is to modulo arithmetics with the same logical base RM3, but with varying moduli (called RM3i here). We first study the properties of these arithmetics in this paper. The study is then generalised by vary- ing the logical base, to give the arithmetics RMni, of logical base RMn and modulus i. Not all of these exist, however, as arithmetical properties and logical properties interact, as we will show. The arithmetics RMni give rise, on intersection, to an inconsistent arithmetic RMω which is not of modulo i for any i. We also study its properties, and, among other results, we show by finitistic means that the more natural relevant arithmetics R♯ and R♯♯ are incomplete (whether or not consistent and recursively enumerable). In the rest of the paper we apply these techniques to several topics, particularly relevant quantum arithmetic in which we are able to show (unlike classical quantum arithmetic) that the law of distribution remains unprovable. Aside from its intrinsic interest, we regard the present exercise as a demonstration that inconsistent theories and models are of mathematical worth and interest.

Undecidability of the Logic of Partial Quasiary Predicates

We obtain an effective embedding of the classical predicate logic into the logic of partial quasiary predicates. The embedding has the property that an image of a non-theorem of the classical logic is refutable in a model of the logic of partial quasiary predicates that has the same cardinality as the classical countermodel of the non-theorem. Therefore, we also obtain an embedding of the classical predicate logic of finite models into the logic of partial quasiary predicates over finite structures. As a consequence, we prove that the logic of partial quasiary predicates is undecidable—more precisely, $\varSigma ^0_1$-complete—over arbitrary structures and not recursively enumerable—more precisely, $\varPi ^0_1$-complete—over finite structures.

Action Logic is Undecidable

Action logic is the algebraic logic (inequational theory) of residuated Kleene lattices. One of the operations of this logic is the Kleene star, which is axiomatized by an induction scheme. For a stronger system that uses an -rule instead (infinitary action logic), Buszkowski and Palka (2007) proved -completeness (thus, undecidability). Decidability of action logic itself was an open question, raised by Kozen in 1994. In this article, we show that it is undecidable, more precisely, -complete. We also prove the same undecidability results for all recursively enumerable logics between action logic and infinitary action logic, for fragments of these logics with only one of the two lattice (additive) connectives, and for action logic extended with the law of distributivity.