Undecidability Results Research Articles

Certified algorithms for equilibrium states of local quantum Hamiltonians

Predicting observables in equilibrium states is a central yet notoriously hard question in quantum many-body systems. In the physically relevant thermodynamic limit, certain mathematical formulations of this task have even been shown to result in undecidable problems. Using a finite-size scaling of algorithms devised for finite systems often fails due to the lack of certified convergence bounds for this limit. In this work, we design certified algorithms for computing expectation values of observables in the equilibrium states of local quantum Hamiltonians, both at zero and positive temperature. Importantly, our algorithms output rigorous lower and upper bounds on these values. This allows us to show that expectation values of local observables can be approximated in finite time, contrasting related undecidability results. When the Hamiltonian is commuting on a 2-dimensional lattice, we prove fast convergence of the hierarchy at high temperature and as a result for a desired precision ε, local observables can be approximated by a convex optimization program of quasi-polynomial size in 1/ε.

On simulating Turing machines with matrix semigroups with integrality tests

We present a construction to simulate Turing machines with 3×3 matrices over rationals. The correctness of simulation is guaranteed by testing that the matrices have integral elements during the simulation. This construction implies an undecidability result for a special identity problem for semigroups of 3×3-matrices.

Bag Semantics Conjunctive Query Containment. Four Small Steps Towards Undecidability.

Query Containment Problem (QCP) is one of the most fundamental decision problems in database query processing and optimization. Complexity of QCP for conjunctive queries has been fully understood since 1970s. But, as Chaudhuri and Vardi noticed in their classical 1993 paper this understanding is based on the assumption that query answers are sets of tuples, and it does not transfer to the situation when multi-set (bag) semantics is considered. Now, 30 years later, decidability of QCP for bag semantics remains an open question, one of the most intriguing open questions in database theory. In this paper we show a series of undecidability results for some generalizations of this problem. We show, for example, that the problem whether, for given two boolean conjunctive queries φ s and φ b , and a linear function F, the inequality F(φ s (D)) =< φ b (D) holds for each database instance D, is undecidable.

Resynchronized Uniformization and Definability Problems for Rational Relations

Regular synchronization languages can be used to define rational relations of finite words, and to characterize subclasses of rational relations, like automatic or recognizable relations. We provide a systematic study of the decidability of uniformization and definability problems for subclasses of rational relations defined in terms of such synchronization languages. We rephrase known results in this setting and complete the picture by adding several new decidability and undecidability results.

Temporal logics with language parameters

We develop a generic framework to extend the logics LTL, CTL+ and CTL⁎ by automata-based connectives from formal language classes and analyse this framework with regard to regular languages, visibly pushdown languages, deterministic and non-deterministic context-free languages. More precisely, we consider how the use of different automata classes changes the expressive power of the logics and provide algorithms for the satisfiability and model checking problems induced by the use of different classes of automata. For the model checking problem, we treat not only finite Kripke transition systems, but also visibly pushdown systems and pushdown systems. We provide completeness or undecidability results in all cases and show that the extensions we consider can formulate properties not expressible in classical temporal logics or regular extensions thereof.

Decidability Boundaries for the Finite-Image Property of Weighted Finite Automata

A weighted finite automaton has the finite-image property if the image of the weighted language associated with it is finite. We show two undecidability results concerning the finite-image property of weighted finite automata over semirings, respectively strong bimonoids. Firstly, we give a computable idempotent commutative past-finite ordered semiring such that it is undecidable, for an arbitrary deterministic weighted finite automaton [Formula: see text] over that semiring, whether [Formula: see text] has the finite-image property. Secondly, we give a computable commutative past-finite monotonic ordered strong bimonoid such that it is undecidable, for an arbitrary weighted finite automaton [Formula: see text] over that strong bimonoid, whether [Formula: see text] has the finite-image property. This shows that recent decidability results for suitable weighted finite automata over past-finite monotonic strong bimonoids cannot be extended to natural classes of ordered semirings and ordered strong bimonoids without further assumptions.

An undecidability result for Separation Logic with theory reasoning

We show that the entailment problem is undecidable in Separation Logic with inductively defined spatial predicates, for rules satisfying the conditions given in [1], if theory reasoning is considered. The result holds for a wide class of theories, even those with a very low expressive power. For instance it applies to natural numbers with the successor function, or with the usual order relation.

Computational properties of finite PNmatrices

Abstract Recent compositionality results in logic have highlighted the advantages of enlarging the traditional notion of logical matrix semantics, namely by incorporating non-determinism and partiality. Still, several important properties which are known to be computable for finite logical matrices have not been studied in the wider context of partial non-deterministic matrices (PNmatrices). In this paper, we study how incorporating non-determinism and/or partiality in logical matrices impacts on the computational properties of some natural problems regarding their induced logics and concretely their sets of theorems. We show that, while for some of these problems there is no relevant computational impact, there are problems whose computational complexity increases and still other problems that simply become undecidable. In particular, we show that the problem of checking whether the logics characterized by two finite PNmatrices have the same set of theorems is not decidable. This undecidability result explores the connection between PNmatrices and term-DAG-automata, where the universality problem is known to be undecidable. This link also motivates a final contribution, in the form of a pumping-like lemma, which can be used, in some cases, to show that a given logic cannot be characterized by a finite PNmatrix.

An undecidability result for the asymptotic theory of p-adic fields

Fix a prime p. We prove that the set of sentences true in all but finitely many finite extensions of Qp is undecidable in the language of valued fields with a cross-section. The proof goes via reduction to positive characteristic, ultimately adapting Pheidas' proof of the undecidability of Fp((t)) with a cross-section. This answers a variant of a question of Derakhshan-Macintyre.

Strong-separation Logic

Most automated verifiers for separation logic are based on the symbolic-heap fragment, which disallows both the magic-wand operator and the application of classical Boolean operators to spatial formulas. This is not surprising, as support for the magic wand quickly leads to undecidability, especially when combined with inductive predicates for reasoning about data structures. To circumvent these undecidability results, we propose assigning a more restrictive semantics to the separating conjunction. We argue that the resulting logic, strong-separation logic, can be used for symbolic execution and abductive reasoning just like “standard” separation logic, while remaining decidable even in the presence of both the magic wand and inductive predicates (we consider a list-segment predicate and a tree predicate)—a combination of features that leads to undecidability for the standard semantics.

Synthesizing ranking functions for loop programs via SVM

Termination of programs is probably the most famous undecidable problem in computer science. Despite this undecidability result, a lot of effort has been spent on improving algorithms that prove termination of loops, which is one of the fundamental aspects of software reliability analysis. These algorithms usually focus on finding an appropriate ranking function for the loop, which proves its termination. In this paper, we focus on handling the synthesis problem of nested ranking functions and multi-phase ranking functions for loop programs. We first reduce the problem of a nested ranking function synthesis to the existence problem of a hyperplane separating classes of data. This allows us to leverage Support-Vector Machines (SVM) techniques for the synthesis of nested ranking functions. SVM are supervised learning algorithms that are used to classify data; they work by finding a hyperplane separating data points parted into two classes. We show how to carefully define the data points so that the separating hyperplane gives rise to a nested ranking function for the loop. Then we use this algorithm for nested ranking functions synthesis as a subprocedure to devise a sound algorithm which incrementally synthesizes multi-phase ranking functions. Experimental results confirm the effectiveness of our SVM-based synthesis of nested and multi-phase ranking functions.

On some decidability questions concerning supports of rational series

A well-known construction of a weighted automaton over the integers, assigning zero to precisely the nonempty words from the equality set of a given Post's correspondence problem instance, is extended to the case of weights taken from an arbitrary integral domain that is not locally finite. Undecidability of problems for rational series such as universality or rationality of support thus generalises to such domains as well. In spite of being fairly simple, these findings imply that the support rationality problem can be undecidable over rings of positive characteristic, answering an open question of D. Kirsten and K. Quaas. In addition, a more general undecidability result for rational series over other than locally finite integral domains is proved, which subsumes, e.g., the undecidability of support rationality or context-freeness.

Existential decidability for addition and divisibility in holomorphy subrings of global fields

We investigate the problem of deciding whether a system of linear equations, together with divisibility conditions on the variables, has a solution over holomorphy subrings of global fields. We obtain decidability results when we allow poles at a cofinite set of places, and undecidability results when at a finite set of places.

Monadicity of Non-deterministic Logical Matrices is Undecidable

The notion of non-deterministic logical matrix (where connectives are interpreted as multi-functions) preserves many good properties of traditional semantics based on logical matrices (where connectives are interpreted as functions) whilst finitely characterizing a much wider class of logics, and has proven to be decisive in a myriad of recent compositional results in logic. Crucially, when a finite non-deterministic matrix satisfies monadicity (distinct truth-values can be separated by unary formulas) one can automatically produce an axiomatization of the induced logic. Furthermore, the resulting calculi are analytical and enable algorithmic proof-search and symbolic counter-model generation. For finite (deterministic) matrices it is well known that checking monadicity is decidable. We show that, in the presence of non-determinism, the property becomes undecidable. As a consequence, we conclude that there is no algorithm for computing the set of all multi-functions expressible in a given finite Nmatrix. The undecidability result is obtained by reduction from the halting problem for deterministic counter machines.

The Expressive Power of the N-Operator and the Decidability of Logic in Wittgenstein’s Tractatus

The present text discusses whether there is a tension between aphorisms 6.1-6.13 of the Tractatus and the Church-Turing theorem about the decidability of predicate logic. We attempt to establish the following points: (i) Aphorisms 6.1-6.13 are not consistent with the Church-Turing theorem. (ii) The logical symbolism of the Tractatus, built from the N-operator, can (and should) be interpreted as expressively complete with respect to first-order formulas. (iii) Wittgenstein’s reasons for believing that Logic is decidable were purely philosophical and the undecidability result shows that there are aspects of his criticisms of Frege and Russell that become unjustified in light of these results.

MOST SIMPLE EXTENSIONS OF ARE UNDECIDABLE

AbstractAll known structural extensions of the substructural logic $\textbf{FL}_{\textbf{e}}$ , the Full Lambek calculus with exchange/commutativity (corresponding to subvarieties of commutative residuated lattices axiomatized by $\{\vee , \cdot , 1\}$ -equations), have decidable theoremhood; in particular all the ones defined by knotted axioms enjoy strong decidability properties (such as the finite embeddability property). We provide infinitely many such extensions that have undecidable theoremhood, by encoding machines with undecidable halting problem. An even bigger class of extensions is shown to have undecidable deducibility problem (the corresponding varieties of residuated lattices have undecidable word problem); actually with very few exceptions, such as the knotted axioms and the other prespinal axioms, we prove that undecidability is ubiquitous. Known undecidability results for non-commutative extensions use an encoding that fails in the presence of commutativity, so and-branching counter machines are employed. Even these machines provide encodings that fail to capture proper extensions of commutativity, therefore we introduce a new variant that works on an exponential scale. The correctness of the encoding is established by employing the theory of residuated frames.

Decidability Questions for Insertion Systems and Related Models

Insertion systems or insertion grammars are a generative formalism in which words can only be generated by starting with some axioms and by iteratively inserting strings subject to certain contexts of a fixed maximal length. It is known that languages generated by such systems are always context sensitive and that the corresponding language classes are incomparable with the regular languages. On the other hand, it is possible to generate non-semilinear languages with systems having contexts of length two. Here, we study decidability questions for insertion systems. On the one hand, it can be seen that emptiness and universality are decidable. Moreover, the fixed membership problem is solvable in deterministic polynomial time. On the other hand, the usually studied decidability questions such as, for example, finiteness, inclusion, equivalence, regularity, inclusion in a regular language, and inclusion of a regular language turn out to be undecidable. Interestingly, the latter undecidability results can be carried over to other models which are basically able to handle the mechanism of inserting strings depending on contexts. In particular, new undecidability results are obtained for pure grammars, restarting automata, clearing restarting automata, and forgetting automata.

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.

Generalizations of Code Languages with Marginal Errors

The [Formula: see text]-prefix-free, [Formula: see text]-suffix-free and [Formula: see text]-infix-free languages generalize the prefix-free, suffix-free and infix-free languages by allowing marginal errors. For example, a string [Formula: see text] in a [Formula: see text]-prefix-free language [Formula: see text] can be a prefix of up to [Formula: see text] different strings in [Formula: see text]. We also define finitely prefix-free languages in which a string [Formula: see text] can be a prefix of finitely many strings. We present efficient algorithms that determine whether or not a given regular language is [Formula: see text]-prefix-free, [Formula: see text]-suffix-free or [Formula: see text]-infix-free, and analyze the time complexity of the algorithms. We establish undecidability results for deciding these properties for (linear) context-free languages.

Asynchronous session subtyping as communicating automata refinement

We study the relationship between session types and behavioural contracts, representing Communicating Finite State Machines (CFSMs), under the assumption that processes communicate asynchronously. Session types represent a syntax-based approach for the description of communication protocols, while behavioural contracts, formally expressing CFSMs, follow an operational approach. We show the existence of a fully abstract interpretation of session types into a fragment of contracts that maps session subtyping into binary compliance-preserving CFSMs/behavioural contract refinement. In this way, on the one hand, we enrich the theory of session types with an operational characterization and, on the other hand, we use recent undecidability results for asynchronous session subtyping to obtain an original undecidability result for asynchronous CFSMs/behavioural contract refinement.