Showing papers on "Undecidable problem published in 2015"

Undecidability of the spectral gap

Abstract: The spectral gap--the energy difference between the ground state and first excited state of a system--is central to quantum many-body physics. Many challenging open problems, such as the Haldane conjecture, the question of the existence of gapped topological spin liquid phases, and the Yang-Mills gap conjecture, concern spectral gaps. These and other problems are particular cases of the general spectral gap problem: given the Hamiltonian of a quantum many-body system, is it gapped or gapless? Here we prove that this is an undecidable problem. Specifically, we construct families of quantum spin systems on a two-dimensional lattice with translationally invariant, nearest-neighbour interactions, for which the spectral gap problem is undecidable. This result extends to undecidability of other low-energy properties, such as the existence of algebraically decaying ground-state correlations. The proof combines Hamiltonian complexity techniques with aperiodic tilings, to construct a Hamiltonian whose ground state encodes the evolution of a quantum phase-estimation algorithm followed by a universal Turing machine. The spectral gap depends on the outcome of the corresponding 'halting problem'. Our result implies that there exists no algorithm to determine whether an arbitrary model is gapped or gapless, and that there exist models for which the presence or absence of a spectral gap is independent of the axioms of mathematics.

Recursive Markov Decision Processes and Recursive Stochastic Games

Abstract: We introduce Recursive Markov Decision Processes (RMDPs) and Recursive Simple Stochastic Games (RSSGs), which are classes of (finitely presented) countable-state MDPs and zero-sum turn-based (perfect information) stochastic games. They extend standard finite-state MDPs and stochastic games with a recursion feature. We study the decidability and computational complexity of these games under termination objectives for the two players: one player's goal is to maximize the probability of termination at a given exit, while the other player's goal is to minimize this probability. In the quantitative termination problems, given an RMDP (or RSSG) and probability p, we wish to decide whether the value of such a termination game is at least p (or at most p); in the qualitative termination problem we wish to decide whether the value is 1. The important 1-exit subclasses of these models, 1-RMDPs and 1-RSSGs, correspond in a precise sense to controlled and game versions of classic stochastic models, including multitype Branching Processes and Stochastic Context-Free Grammars, where the objective of the players is to maximize or minimize the probability of termination (extinction). We provide a number of upper and lower bounds for qualitative and quantitative termination problems for RMDPs and RSSGs. We show both problems are undecidable for multi-exit RMDPs, but are decidable for 1-RMDPs and 1-RSSGs. Specifically, the quantitative termination problem is decidable in PSPACE for both 1-RMDPs and 1-RSSGs, and is at least as hard as the square root sum problem, a well-known open problem in numerical computation. We show that the qualitative termination problem for 1-RMDPs (i.e., a controlled version of branching processes) can be solved in polynomial time both for maximizing and minimizing 1-RMDPs. The qualitative problem for 1-RSSGs is in NP ∩ coNP, and is at least as hard as the quantitative termination problem for Condon's finite-state simple stochastic games, whose complexity remains a well known open problem. Finally, we show that even for 1-RMDPs, more general (qualitative and quantitative) model-checking problems with respect to linear-time temporal properties are undecidable even for a fixed property.

Qualitative analysis of POMDPs with temporal logic specifications for robotics applications

Abstract: We consider partially observable Markov decision processes (POMDPs), that are a standard framework for robotics applications to model uncertainties present in the real world, with temporal logic specifications. All temporal logic specifications in linear-time temporal logic (LTL) can be expressed as parity objectives. We study the qualitative analysis problem for POMDPs with parity objectives that asks whether there is a controller (policy) to ensure that the objective holds with probability 1 (almost-surely). While the qualitative analysis of POMDPs with parity objectives is undecidable, recent results show that when restricted to finite-memory policies the problem is EXPTIME-complete. While the problem is intractable in theory, we present a practical approach to solve the qualitative analysis problem. We designed several heuristics to deal with the exponential complexity, and have used our implementation on a number of well-known POMDP examples for robotics applications. Our results provide the first practical approach to solve the qualitative analysis of robot motion planning with LTL properties in the presence of uncertainty.

Language Emptiness of Continuous-Time Parametric Timed Automata

Abstract: Parametric timed automata extend the standard timed automata with the possibility to use parameters in the clock guards. In general, if the parameters are real-valued, the problem of language emptiness of such automata is undecidable even for various restricted subclasses. We thus focus on the case where parameters are assumed to be integer-valued, while the time still remains continuous. On the one hand, we show that the problem remains undecidable for parametric timed automata with three clocks and one parameter. On the other hand, for the case with arbitrary many clocks where only one of these clocks is compared with an arbitrary number of parameters, we show that the parametric language emptiness is decidable. The undecidability result tightens the bounds of a previous result which assumed six parameters, while the decidability result extends the existing approaches that deal with discrete-time semantics only. To the best of our knowledge, this is the first positive result in the case of continuous-time and unbounded integer parameters, except for the rather simple case of single-clock automata.

Theory in practice for system design and verification

Abstract: Methodology and tools for assisting developers in building high-confidence hardware and software at a reasonable cost has been one of the central themes in computer science since its inception. The formal methods research on this problem has focused on two complimentary goals: to provide mathematical abstractions to manage the complexity of the design and to develop analysis tools to check that the implementation works correctly as intended. Achieving these goals has proved to be extremely challenging for two reasons. First, the scale and complexity of systems being designed remains a moving target as computers have transformed from special-purpose and stand-alone number-crunching processors to networked devices interacting with the physical world. Second, once formalized, the computational problem of verifying that a system meets its specification is undecidable in the general case and has intractable complexity even in special cases.

Language Emptiness of Continuous-Time Parametric Timed Automata

Abstract: Parametric timed automata extend the standard timed automata with the possibility to use parameters in the clock guards. In general, if the parameters are real-valued, the problem of language emptiness of such automata is undecidable even for various restricted subclasses. We thus focus on the case where parameters are assumed to be integer-valued, while the time still remains continuous. On the one hand, we show that the problem remains undecidable for parametric timed automata with three clocks and one parameter. On the other hand, for the case with arbitrary many clocks where only one of these clocks is compared with (an arbitrary number of) parameters, we show that the parametric language emptiness is decidable. The undecidability result tightens the bounds of a previous result which assumed six parameters, while the decidability result extends the existing approaches that deal with discrete-time semantics only. To the best of our knowledge, this is the first positive result in the case of continuous-time and unbounded integer parameters, except for the rather simple case of single-clock automata.

Arbitrary Public Announcement Logic with Mental Programs

Abstract: We propose a variant of arbitrary public announcement logic which is decidable. In this variant, knowledge accessibility relations are defined by programs. Technically, programs are written in dynamic logic with propositional assignments. We prove that both the model checking problem and the satisfiability problem are decidable and AEXPpol-complete where AEXPpol is the class of decision problems decided by alternating Turing machines running in exponential time where the number of alternations is polynomial. Whereas arbitrary public announcement logic is undecidable, our framework is decidable and we provide a proof-of-concept to show its expressiveness: we use our framework to reason about epistemic properties and arbitrary announcements when agents are cameras located in the plane.

Extended symbolic finite automata and transducers

Abstract: Symbolic finite automata and transducers augment classic automata and transducers with symbolic alphabets represented as parametric theories. This extension enables to succinctly represent large and potentially infinite alphabets while preserving closure and decidability properties. Extended symbolic finite automata and transducers further extend these objects by allowing transitions to read consecutive input elements in a single step. In this paper we study the properties of these models. In contrast to the case of finite alphabets, we show how reading multiple symbols increases the expressiveness of the models, which causes some closure properties to stop holding and most decision problems to become undecidable. In particular we show how extended symbolic finite transducers are not closed under composition, and the equivalence problem is undecidable for both extended symbolic finite automata and transducers. We then introduce the subclass of Cartesian extended symbolic finite transducers in which guards are limited to conjunctions of unary predicates and we propose an equivalence algorithm for this subclass in the single-valued case. We also present a heuristic algorithm for composing extended symbolic finite transducers that works for many practical cases. Finally, we model real world programs with Cartesian extended symbolic finite transducers and use the proposed algorithms to prove their correctness.

A game approach to determinize timed automata

Abstract: Timed automata are frequently used to model real-time systems. Their determinization is a key issue for several validation problems. However, not all timed automata can be determinized, and determinizability itself is undecidable. In this paper, we propose a game-based algorithm which, given a timed automaton, tries to produce a language-equivalent deterministic timed automaton, otherwise a deterministic over-approximation. Our method generalizes two recent contributions: the determinization procedure of Baier et al. (Proceedings of the 36th international colloquium on automata, languages and programming (ICALP'09), 2009) and the approximation algorithm of Krichen and Tripakis (Form Methods Syst Des 34(3):238---304, 2009). Moreover, we extend it to apply to timed automata with invariants and $$\varepsilon $$?-transitions, and also consider other useful approximations: under-approximation, and combination of under- and over-approximations.

The Hunt for a Red Spider: Conjunctive Query Determinacy Is Undecidable

Abstract: We solve a well known, long-standing open problem in relational databases theory, showing that the conjunctive query determinacy problem (in its "unrestricted" version) is undecidable.

Chase Termination for Guarded Existential Rules

Abstract: The chase procedure is considered as one of the most fundamental algorithmic tools in database theory. It has been successfully applied to different database problems such as data exchange, and query answering and containment under constraints, to name a few. One of the central problems regarding the chase procedure is all-instance termination, that is, given a set of tuple-generating dependencies (TGDs) (a.k.a. existential rules), decide whether the chase under that set terminates, for every input database. It is well-known that this problem is undecidable, no matter which version of the chase we consider. The crucial question that comes up is whether existing restricted classes of TGDs, proposed in different contexts such as ontological query answering, make the above problem decidable. In this work, we focus our attention on the oblivious and the semi-oblivious versions of the chase procedure, and we give a positive answer for classes of TGDs that are based on the notion of guardedness. To the best of our knowledge, this is the first work that establishes positive results about the (semi-)oblivious chase termination problem. In particular, we first concentrate on the class of linear TGDs, and we syntactically characterize, via rich- and weak-acyclicity, its fragments that guarantee the termination of the oblivious and the semi-oblivious chase, respectively. Those syntactic characterizations, apart from being interesting in their own right, allow us to pinpoint the complexity of the problem, which is PSPACE-complete in general, and NL-complete if we focus on predicates of bounded arity, for both the oblivious and the semi-oblivious chase. We then proceed with the more general classes of guarded and weakly-guarded TGDs. Although we do not provide syntactic characterizations for its relevant fragments, as for linear TGDs, we show that the problem under consideration remains decidable. In fact, we show that it is 2EXPTIME-complete in general, and EXPTIME-complete if we focus on predicates of bounded arity, for both the oblivious and the semi-oblivious chase. Finally, we investigate the expressive power of the query languages obtained from our analysis, and we show that they are equally expressive with standard database query languages. Nevertheless, we have strong indications that they are more succinct.

Mortality of iterated piecewise affine functions over the integers: Decidability and complexity

Abstract: In the theory of discrete-time dynamical systems, one studies the limiting behaviour of processes defined by iterating a fixed function f over a given space. A much-studied case involves piecewise ane functions on R n . Blondel et al. (2001) studied the decidability of questions such as mortality for such functions with rational coecients. Mortality means that every trajectory includes a 0; if the iteration is seen as a loop while (x6 0) x := f(x), mortality means that the loop is guaranteed to terminate. Blondel et al. proved that the problems are undecidable when the dimension n of the state space is at least two. They assume that the variables range over the rationals; this is an essential assumption. From a program analysis (and discrete Computability) viewpoint, it would be more interesting to consider integer-valued variables. This paper establishes (un)decidability results for the integer setting. We show that also over integers, undecidability (moreover, 0 completeness) begins at two dimensions. We further investigate the eect of several restrictions on the iterated functions. Specifically, we consider bounding the size of the partition defining f, and restricting the coecients of the linear components. In the decidable cases, we give complexity results. The complexity is PTIME for ane functions, but for piecewise-ane ones it is PSPACE-complete. The undecidability proofs use some variants of the Collatz problem, which may be of independent interest. 1998 ACM Subject Classification F.1.0 Computation by abstract devices ‐ General

Assume-Guarantee Synthesis for Concurrent Reactive Programs with Partial Information

Abstract: Synthesis of program parts is particularly useful for concurrent systems. However, most approaches do not support common design tasks, like modifying a single process without having to re-synthesize or verify the whole system. Assume-guarantee synthesis AGS provides robustness against modifications of system parts, but thus far has been limited to the perfect information setting. This means that local variables cannot be hidden from other processes, which renders synthesis results cumbersome or even impossible to realize. We resolve this shortcoming by defining AGS under partial information. We analyze the complexity and decidability in different settings, showing that the problem has a high worst-case complexity and is undecidable in many interesting cases. Based on these observations, we present a pragmatic algorithm based on bounded synthesis, and demonstrate its practical applicability on several examples.

Translation-like Actions and Aperiodic Subshifts on Groups

Abstract: It is well known that if $G$ admits a f.g. subgroup $H$ with a weakly aperiodic SFT (resp. an undecidable domino problem), then $G$ itself has a weakly aperiodic SFT (resp. an undecidable domino problem). We prove that we can replace the property "$H$ is a subgroup of $G$" by "$H$ acts translation-like on $G$", provided $H$ is finitely presented. In particular: * If $G_1$ and $G_2$ are f.g. infinite groups, then $G_1 \times G_2$ has a weakly aperiodic SFT (and actually a undecidable domino problem). In particular the Grigorchuk group has an undecidable domino problem. * Every infinite f.g. $p$-group admits a weakly aperiodic SFT.

Lattice-Theoretic Progress Measures and Coalgebraic Model Checking (with Appendices).

Abstract: In the context of formal verification in general and model checking in particular, parity games serve as a mighty vehicle: many problems are encoded as parity games, which are then solved by the seminal algorithm by Jurdzinski. In this paper we identify the essence of this workflow to be the notion of progress measure, and formalize it in general, possibly infinitary, lattice-theoretic terms. Our view on progress measures is that they are to nested/alternating fixed points what invariants are to safety/greatest fixed points, and what ranking functions are to liveness/least fixed points. That is, progress measures are combination of the latter two notions (invariant and ranking function) that have been extensively studied in the context of (program) verification. We then apply our theory of progress measures to a general model-checking framework, where systems are categorically presented as coalgebras. The framework's theoretical robustness is witnessed by a smooth transfer from the branching-time setting to the linear-time one. Although the framework can be used to derive some decision procedures for finite settings, we also expect the proposed framework to form a basis for sound proof methods for some undecidable/infinitary problems.

Commutativity of Reducers

Abstract: In the Map-Reduce programming model for data parallel computation, a reducer computes an output from a list of input values associated with a key. The inputs however may not arrive at a reducer in a fixed order due to non-determinism in transmitting key-value pairs over the network. This gives rise to the reducer commutativity problem, that is, is the reducer computation independent of the order of its inputs? In this paper, we study the reducer commutativity problem formally. We introduce a syntactic subset of integer programs termed integer reducers to model real-world reducers. In spite of syntactic restrictions, we show that checking commutativity of integer reducers over unbounded lists of exact integers is undecidable. It remains undecidable even with input lists of a fixed length. The problem however becomes decidable for reducers over unbounded input lists of bounded integers. We propose an efficient reduction of commutativity checking to conventional assertion checking and report experimental results using various off-the-shelf program analyzers.

Uniform strategies, rational relations and jumping automata

Abstract: A general concept of uniform strategies has recently been proposed as a relevant notion in game theory for computer science, which subsumes various notions from the literature. It relies on properties involving sets of plays in two-player turn-based arenas equipped with arbitrary binary relations between plays; these properties are expressed in a language based on CTL * with a quantifier over related plays. There are two semantics for our quantifier, a strict one and a full one, that we study separately. Regarding the strict semantics, the existence of a uniform strategy is undecidable for rational binary relations, but introducing jumping tree automata and restricting attention to recognizable relations allows us to establish a 2-Exptime-complete complexity - and still capture a class of two-player imperfect-information games with epistemic temporal objectives. Regarding the full semantics, relying on information set automata we establish that the existence of a uniform strategy is decidable for rational relations and we provide a nonelementary synthesis procedure. We also exhibit an essentially optimal subclass of rational relations for which the problem becomes 2-Exptime-complete. Considering rich classes of relations makes the theory of uniform strategies powerful: it directly entails various results in logics of knowledge and time, some of them already known, and others new.

Undecidability in Binary Tag Systems and the Post Correspondence Problem for Five Pairs of Words

Abstract: Since Cocke and Minsky proved 2-tag systems universal, they have been extensively used to prove the universality of numerous computational models. Unfortunately, all known algorithms give universal 2-tag systems that have a large number of symbols. In this work, tag systems with only 2 symbols (the minimum possible) are proved universal via an intricate construction showing that they simulate cyclic tag systems. We immediately find applications of our result. We reduce the halting problem for binary tag systems to the Post correspondence problem for 5 pairs of words. This improves on 7 pairs, the previous bound for undecidability in this problem. Following our result, only the cases for 3 and 4 pairs of words remains open, as the problem is known to be decidable for 2 pairs. In a further application, we apply the reductions of Vesa Halava and others to show that the matrix mortality problem is undecidable for sets with six 3 x 3 matrices and for sets with two 18 x 18 matrices. The previous bounds for the undecidability in this problem was seven 3 x 3 matrices and two 21 x 21 matrices.

What's Decidable about Syntax-Guided Synthesis?

Abstract: Syntax-guided synthesis (SyGuS) is a recently proposed framework for program synthesis problems. The SyGuS problem is to find an expression or program generated by a given grammar that meets a correctness specification. Correctness specifications are given as formulas in suitable logical theories, typically amongst those studied in satisfiability modulo theories (SMT). In this work, we analyze the decidability of the SyGuS problem for different classes of grammars and correctness specifications. We prove that the SyGuS problem is undecidable for the theory of equality with uninterpreted functions (EUF).We identify a fragment of EUF, which we call regular-EUF, for which the SyGuS problem is decidable. We prove that this restricted problem is EXPTIME-complete and that the sets of solution expressions are precisely the regular tree languages. For theories that admit a unique, finite domain, we give a general algorithm to solve the SyGuS problem on tree grammars. Finite-domain theories include the bit-vector theory without concatenation. We prove SyGuS undecidable for a very simple bit-vector theory with concatenation, both for context-free grammars and for tree grammars. Finally, we give some additional results for linear arithmetic and bit-vector arithmetic along with a discussion of the implication of these results.

Robust Satisfiability of Systems of Equations

Abstract: We study the problem of robust satisfiability of systems of nonlinear equations, namely, whether for a given continuous function f:K → Rn on a finite simplicial complex K and α>0, it holds that each function g:K → Rn such that ║g−f║∞ ≤ α, has a root in K. Via a reduction to the extension problem of maps into a sphere, we particularly show that this problem is decidable in polynomial time for every fixed n, assuming dim K ≤ 2n−3. This is a substantial extension of previous computational applications of topological degree and related concepts in numerical and interval analysis. Via a reverse reduction, we prove that the problem is undecidable when dim K ≥ 2n−2, where the threshold comes from the stable range in homotopy theory. For the lucidity of our exposition, we focus on the setting when f is simplexwise linear. Such functions can approximate general continuous functions, and thus we get approximation schemes and undecidability of the robust satisfiability in other possible settings.

The triviality problem for profinite completions

Abstract: We prove that there is no algorithm that can determine whether or not a finitely presented group has a non-trivial finite quotient; indeed, this property remains undecidable among the fundamental groups of compact, non-positively curved square complexes. We deduce that many other properties of groups are undecidable. For hyperbolic groups, there cannot exist algorithms to determine largeness, the existence of a linear representation with infinite image (over any infinite field), or the rank of the profinite completion.

On Σ11‐complete equivalence relations on the generalized Baire space

Abstract: Working with uncountable structures of fixed cardinality, we investigate the complexity of certain equivalence relations and show that if , then many of them are -complete, in particular the isomorphism relation of dense linear orders. Then we show that it is undecidable in whether or not the isomorphism relation of a certain well behaved theory (stable, NDOP, NOTOP) is -complete (it is, if , but can be forced not to be).

Robust Multidimensional Mean-Payoff Games are Undecidable

Abstract: Mean-payoff games play a central role in quantitative synthesis and verification In a single-dimensional game a weight is assigned to every transition and the objective of the protagonist is to assure a non-negative limit-average weight In the multidimensional setting, a weight vector is assigned to every transition and the objective of the protagonist is to satisfy a boolean condition over the limit-average weight of each dimension, eg, LimAvg(x 1) ≤ 0 ∨ LimAvg(x 2) ≥ 0 ∧ LimAvg(x 3) ≥ 0 We recently proved that when one of the players is restricted to finite-memory strategies then the decidability of determining the winner is inter-reducible with Hilbert’s Tenth problem over rationals (a fundamental long-standing open problem) In this work we consider arbitrary (infinite-memory) strategies for both players and show that the problem is undecidable

Deciding the value 1 problem for probabilistic leaktight automata

Abstract: The value 1 problem is a decision problem for probabilistic automata over finite words: given a probabilistic automaton, are there words accepted with probability arbitrarily close to 1? This problem was proved undecidable recently; to overcome this, several classes of probabilistic automata of different nature were proposed, for which the value 1 problem has been shown decidable. In this paper, we introduce yet another class of probabilistic automata, called leaktight automata, which strictly subsumes all classes of probabilistic automata whose value 1 problem is known to be decidable. We prove that for leaktight automata, the value 1 problem is decidable (in fact, PSPACE-complete) by constructing a saturation algorithm based on the computation of a monoid abstracting the behaviours of the automaton. We rely on algebraic techniques developed by Simon to prove that this abstraction is complete. Furthermore, we adapt this saturation algorithm to decide whether an automaton is leaktight. Finally, we show a reduction allowing to extend our decidability results from finite words to infinite ones, implying that the value 1 problem for probabilistic leaktight parity automata is decidable.

Two-variable Logic with Counting and a Linear Order

Abstract: We study the finite satisfiability problem for the two-variable fragment of the first-order logic extended with counting quantifiers (C2) and interpreted over linearly ordered structures. We show that the problem is undecidable in the case of two linear orders (in presence of two other binary symbols). In the case of one linear order it is NEXPTIME-complete, even in presence of the successor relation. Surprisingly, the complexity of the problem explodes when we add one binary symbol more: C2 with one linear order and its successor, in presence of other binary predicate symbols, is decidable, but it is as expressive (and as complex) as Vector Addition Systems.

Hierarchical Information Patterns and Distributed Strategy Synthesis

Abstract: Infinite games with imperfect information are deemed to be undecidable unless the information flow is severely restricted. One fundamental decidable case occurs when there is a total ordering among players, such that each player has access to all the information that the following ones receive.

On the Value Problem in Weighted Timed Games.

Abstract: A weighted timed game is a timed game with extra quantitative information representing e.g. energy consumption. Optimizing the weight for reaching a target is a natural question, which has already been investigated for ten years. Existence of optimal strategies is known to be undecidable in general, and only very restricted classes of games have been identified for which optimal weight and almost-optimal strategies can be computed. In this paper, we show that the value problem is undecidable in weighted timed games. We then introduce a large subclass of weighted timed games (for which the undecidability proof above applies), and provide an algorithm to compute arbitrary approximations of the value in such games. To the best of our knowledge, this is the first approximation scheme for an undecidable class of weighted timed games.