ATP and Presentation Service for Mizar Formalizations

This paper describes the Automated Reasoning for Mizar ( $\textsf{Miz}\mathbb{AR}$ ) service, which integrates several automated reasoning, artificial intelligence, and presentation tools with Mizar and its authoring environment. The service provides ATP assistance to Mizar authors in finding and explaining proofs, and offers generation of Mizar problems as challenges to ATP systems. The service is based on a sound translation from the Mizar language to that of first-order ATP systems, and relies on the recent progress in application of ATP systems in large theories containing tens of thousands of available facts. We present the main features of $\textsf{Miz}\mathbb{AR}$ services, followed by an account of initial experiments in finding proofs with the ATP assistance. Our initial experience indicates that the tool offers substantial help in exploring the Mizar library and in preparing new Mizar articles.     

The clock constraint specification language for building timed causality models

The uml Profile for Modeling and Analysis of Real-Time and Embedded (RTE) systems has recently been adopted by the OMG. Its Time Model extends the informal and simplistic Simple Time package proposed by Unified Modeling Language (UML2) and offers a broad range of capabilities required to model RTE systems including discrete/dense and chronometric/logical time. The Marte specification introduces a Time Structure inspired from several time models of the concurrency theory and proposes a new clock constraint specification language (ccsl) to specify, within the context of the uml, logical and chronometric time constraints. A semantic model in ccsl is attached to a (uml) model to give its timed causality semantics. In that sense, ccsl is comparable to the Ptolemy environment, in which directors give the semantics to models according to predefined models of computation and communication. This paper focuses on one historical model of computation of Ptolemy [Synchronous Data Flow (SDF)] and shows how to build SDF graphs by combining uml models and ccsl.     

Two-Way Automata Versus Logarithmic Space

We strengthen a previously known connection between the size complexity of two-way finite automata ( ) and the space complexity of Turing machines (tms). Specifically, we prove that

every s-state has a poly(s)-state that agrees with it on all inputs of length ≤s if and only if NL?L/poly, and

every s-state has a poly(s)-state that agrees with it on all inputs of length ≤2 ^s if and only if NLL?LL/polylog.

Here, and are the deterministic and nondeterministic , NL and L/poly are the standard classes of languages recognizable in logarithmic space by nondeterministic tms and by deterministic tms with access to polynomially long advice, and NLL and LL/polylog are the corresponding complexity classes for space O(loglogn) and advice length poly(logn). Our arguments strengthen and extend an old theorem by Berman and Lingas and can be used to obtain variants of the above statements for other modes of computation or other combinations of bounds for the input length, the space usage, and the length of advice.     

Complexity of Rational and Irrational Nash Equilibria

We introduce two new natural decision problems, denoted as ? RATIONAL NASH and ? IRRATIONAL NASH, pertinent to the rationality and irrationality, respectively, of Nash equilibria for (finite) strategic games. These problems ask, given a strategic game, whether or not it admits (i) a rational Nash equilibrium where all probabilities are rational numbers, and (ii) an irrational Nash equilibrium where at least one probability is irrational, respectively. We are interested here in the complexities of ? RATIONAL NASH and ? IRRATIONAL NASH. Towards this end, we study two other decision problems, denoted as NASH-EQUIVALENCE and NASH-REDUCTION, pertinent to some mutual properties of the sets of Nash equilibria of two given strategic games with the same number of players. The problem NASH-EQUIVALENCE asks whether or not the two sets of Nash equilibria coincide; we identify a restriction of its complementary problem that witnesses ? RATIONAL NASH. The problem NASH-REDUCTION asks whether or not there is a so called Nash reduction: a suitable map between corresponding strategy sets of players that yields a Nash equilibrium of the former game from a Nash equilibrium of the latter game; we identify a restriction of NASH-REDUCTION that witnesses ? IRRATIONAL NASH. As our main result, we provide two distinct reductions to simultaneously show that (i) NASH-EQUIVALENCE is co- $\mathcal{NP}$ -hard and ? RATIONAL NASH is $\mathcal{NP}$ -hard, and (ii) NASH-REDUCTION and ? IRRATIONAL NASH are both $\mathcal{NP}$ -hard, respectively. The reductions significantly extend techniques previously employed by Conitzer and Sandholm (Proceedings of the 18th Joint Conference on Artificial Intelligence, pp. 765–771, 2003; Games Econ. Behav. 63(2), 621–641, 2008).     

ANDES: efficient evaluation of NOT-twig queries in relational databases

Despite a large body of work on XPath query processing in relational environment, systematic study of queries containing not-predicates have received little attention in the literature. Particularly, several xml supports of industrial-strength commercial rdbms fail to efficiently evaluate such queries. In this paper, we present an efficient and novel strategy to evaluate not -twig queries in a tree-unaware relational environment. not -twig queries are XPath queries with ancestor–descendant and parent–child axis and contain one or more not-predicates. We propose a novel Dewey-based encoding scheme called Andes (ANcestor Dewey-based Encoding Scheme), which enables us to efficiently filter out elements satisfying a not-predicate by comparing their ancestor group identifiers. In this approach, a set of elements under the same common ancestor at a specific level in the xml tree is assigned same ancestor group identifier. Based on this scheme, we propose a novel sql translation algorithm for not-twig query evaluation. Experiments carried out confirm that our proposed approach built on top of an off-the-shelf commercial rdbms significantly outperforms state-of-the-art relational and native approaches. We also explore the query plans selected by a commercial relational optimizer to evaluate our translated queries in different input cardinality. Such exploration further validates the performance benefits of Andes.     

Amplifying circuit lower bounds against polynomial time, with applications

We give a self-reduction for the Circuit Evaluation problem (CircEval) and prove the following consequences.

1. Amplifying size–depth lower bounds. If CircEval has Boolean circuits of n ^k size and n ^1?δ depth for some k and δ, then for every ${\epsilon > 0}$ , there is a δ′ > 0 such that CircEval has circuits of ${n^{1 + \epsilon}}$ size and ${n^{1- \delta^{\prime}}}$ depth. Moreover, the resulting circuits require only ${\tilde{O}(n^{\epsilon})}$ bits of non-uniformity to construct. As a consequence, strong enough depth lower bounds for Circuit Evaluation imply a full separation of P and NC (even with a weak size lower bound).
2. Lower bounds for quantified Boolean formulas. Let c, d > 1 and e < 1 satisfy c < (1 ? e + d )/d. Either the problem of recognizing valid quantified Boolean formulas (QBF) is not solvable in TIME[n ^c ], or the Circuit Evaluation problem cannot be solved with circuits of n ^d size and n ^e depth. This implies unconditional polynomial-time uniform circuit lower bounds for solving QBF. We also prove that QBF does not have n ^c -time uniform NC circuits, for all c < 2.

     

Detecting Fixed Patterns in Chordal Graphs in Polynomial Time

The Contractibility problem takes as input two graphs G and H, and the task is to decide whether H can be obtained from G by a sequence of edge contractions. The Induced Minor and Induced Topological Minor problems are similar, but the first allows both edge contractions and vertex deletions, whereas the latter allows only vertex deletions and vertex dissolutions. All three problems are NP-complete, even for certain fixed graphs H. We show that these problems can be solved in polynomial time for every fixed H when the input graph G is chordal. Our results can be considered tight, since these problems are known to be W[1]-hard on chordal graphs when parameterized by the size of H. To solve Contractibility and Induced Minor, we define and use a generalization of the classic Disjoint Paths problem, where we require the vertices of each of the k paths to be chosen from a specified set. We prove that this variant is NP-complete even when k=2, but that it is polynomial-time solvable on chordal graphs for every fixed k. Our algorithm for Induced Topological Minor is based on another generalization of Disjoint Paths called Induced Disjoint Paths, where the vertices from different paths may no longer be adjacent. We show that this problem, which is known to be NP-complete when k=2, can be solved in polynomial time on chordal graphs even when k is part of the input. Our results fit into the general framework of graph containment problems, where the aim is to decide whether a graph can be modified into another graph by a sequence of specified graph operations. Allowing combinations of the four well-known operations edge deletion, edge contraction, vertex deletion, and vertex dissolution results in the following ten containment relations: (induced) minor, (induced) topological minor, (induced) subgraph, (induced) spanning subgraph, dissolution, and contraction. Our results, combined with existing results, settle the complexity of each of the ten corresponding containment problems on chordal graphs.     

Built-in Coloring for Highly-Concurrent Doubly-Linked Lists

This paper presents a novel approach for highly-concurrent nonblocking implementations of doubly-linked lists, based on dynamically maintaining a coloring of the nodes in the list. In these implementations, operations on non-adjacent nodes in the linked-list proceed without interfering with each other. Roughly speaking, the operations are implemented by acquiring nodes in the operation’s data set, in the order of their colors, and then making the changes atomically. The length of waiting chains is restricted, thereby increasing concurrency, because the colors are taken from a small set. Operations carefully update the colors of the nodes they modify, so neighboring nodes in the list have different colors. A helping mechanism ensures progress in small neighborhoods of processes that keep taking steps. We use this approach in two new algorithms: CAS-Chromo uses an unary conditional primitive, cas, and allows insertions anywhere in the linked list and removals only at the ends, while DCAS-Chromo allows insertions and removals anywhere but uses a stronger primitive, dcas.     

OXPath: A language for scalable data extraction, automation, and crawling on the deep web

The evolution of the web has outpaced itself: A growing wealth of information and increasingly sophisticated interfaces necessitate automated processing, yet existing automation and data extraction technologies have been overwhelmed by this very growth. To address this trend, we identify four key requirements for web data extraction, automation, and (focused) web crawling: (1) interact with sophisticated web application interfaces, (2) precisely capture the relevant data to be extracted, (3) scale with the number of visited pages, and (4) readily embed into existing web technologies. We introduce OXPath as an extension of XPath for interacting with web applications and extracting data thus revealed—matching all the above requirements. OXPath’s page-at-a-time evaluation guarantees memory use independent of the number of visited pages, yet remains polynomial in time. We experimentally validate the theoretical complexity and demonstrate that OXPath’s resource consumption is dominated by page rendering in the underlying browser. With an extensive study of sublanguages and properties of OXPath, we pinpoint the effect of specific features on evaluation performance. Our experiments show that OXPath outperforms existing commercial and academic data extraction tools by a wide margin.     

A UTP semantics for Circus

Circus specifications define both data and behavioural aspects of systems using a combination of Z and CSP constructs. Previously, a denotational semantics has been given to Circus; however, a shallow embedding of Circus in Z, in which the mapping from Circus constructs to their semantic representation as a Z specification, with yet another language being used as a meta-language, was not useful for proving properties like the refinement laws that justify the distinguishing development technique associated with Circus. This work presents a final reference for the Circus denotational semantics based on Hoare and He’s Unifying Theories of Programming (UTP); as such, it allows the proof of meta-theorems about Circus including the refinement laws in which we are interested. Its correspondence with the CSP semantics is illustrated with some examples. We also discuss the library of lemmas and theorems used in the proofs of the refinement laws. Finally, we give an account of the mechanisation of the Circus semantics and of the mechanical proofs of the refinement laws.     

Prehistoric Graph in Modal Derivations and Self-Referentiality

By terms-allowed-in-formulas capacity, Artemov’s Logic of Proofs LP Artemov includes self-referential formulas of the form t:?(t) that play a crucial role in the realization of modal logic S4 in LP, and in the Brouwer–Heyting–Kolmogorov semantics of intuitionistic logic via LP. In an earlier work appeared in the Proceedings of CSR 2010 the author defined prehistoric loop in a sequent calculus of S4, and verified its necessity to self-referentiality in S4?LP realization. In this extended version we generalize results there to T and K4, two modal logics smaller than S4 that yet call for self-referentiality in their realizations into corresponding justification logics JT and J4.     

Shorthand Universal Cycles for Permutations

The set of permutations of ??n??={1,??,n} in one-line notation is ??(n). The shorthand encoding of a ₁?a _n ????(n) is a ₁?a _n?1. A shorthand universal cycle for permutations (SP-cycle) is a circular string of length n! whose substrings of length n?1 are the shorthand encodings of ??(n). When an SP-cycle is decoded, the order of ??(n) is a Gray code in which successive permutations differ by the prefix-rotation ?? _i =(1 2 ? i) for i??{n?1,n}. Thus, SP-cycles can be represented by n! bits. We investigate SP-cycles with maximum and minimum ??weight?? (number of ?? _n?1s in the Gray code). An SP-cycle n a n b?n z is ??periodic?? if its ??sub-permutations?? a,b,??,z equal ??(n?1). We prove that periodic min-weight SP-cycles correspond to spanning trees of the (n?1)-permutohedron. We provide two constructions: B(n) and C(n). In B(n) the spanning trees use ??half-hunts?? from bell-ringing, and in C(n) the sub-permutations use cool-lex order by Williams (SODA, 987?C996, 2009). Algorithmic results are: (1)?memoryless decoding of B(n) and C(n), (2)?O((n?1)!)-time generation of B(n) and C(n) using sub-permutations, (3)?loopless generation of B(n)??s binary representation n bits at a time, and (4)?O(n+??(n))-time ranking of B(n)??s permutations where ??(n) is the cost of computing a permutation??s inversion vector. Results (1)?C(4) improve on those for the previous SP-cycle construction D(n) by Ruskey and Williams (ACM Trans. Algorithms 6(3):Art.?45, 2010), which we characterize here using ??recycling??.     

Visibly rational expressions

Regular expressions (RE) are an algebraic formalism for expressing regular languages, widely used in string search and as a specification language in verification. In this paper, we introduce and investigate visibly rational expressions (VRE), an extension of RE for the class of visibly pushdown languages (VPL). We show that VRE capture precisely the class of VPL. Moreover, we identify an equally expressive fragment of VRE which admits a quadratic time compositional translation into the automata acceptors of VPL. We also prove that, for this fragment, universality, inclusion and language equivalence are EXPTIME-complete. Finally, we provide an extension of VRE for VPL over infinite words.     

Reasoning with Qualitative Positional Information for Domestic Domains in the Situation Calculus

In this paper, we present a thorough integration of qualitative representations and reasoning for positional information for domestic service robotics domains into our high-level robot control. In domestic settings for service robots like in the RoboCup@Home competitions, complex tasks such as “get the cup from the kitchen and bring it to the living room” or “find me this and that object in the apartment” have to be accomplished. At these competitions the robots may only be instructed by natural language. As humans use qualitative concepts such as “near” or “far”, the robot needs to cope with them, too. For our domestic robot, we use the robot programming and plan language Readylog, our variant of Golog. In previous work we extended the action language Golog, which was developed for the high-level control of agents and robots, with fuzzy set-based qualitative concepts. We now extend our framework to positional fuzzy fluents with an associated positional context called frames. With that and our underlying reasoning mechanism we can transform qualitative positional information from one context to another to account for changes in context such as the point of view or the scale. We demonstrate how qualitative positional fluents based on a fuzzy set semantics can be deployed in domestic domains and showcase how reasoning with these qualitative notions can seamlessly be applied to a fetch-and-carry task in a RoboCup@Home scenario.     

On the power of unambiguity in log-space

We report progress on the NL versus UL problem.

We show that counting the number of s-t paths in graphs where the number of s-v paths for any v is bounded by a polynomial can be done in FUL: the unambiguous log-space function class. Several new upper bounds follow from this including ${{{ReachFewL} \subseteq {UL}}}$ and ${{{LFew} \subseteq {UL}^{FewL}}}$

We investigate the complexity of min-uniqueness—a central notion in studying the NL versus UL problem. In this regard we revisit the class OptL[log n] and introduce UOptL[log n], an unambiguous version of OptL[log n]. We investigate the relation between UOptL[log n] and other existing complexity classes.

We consider the unambiguous hierarchies over UL and UOptL[log n]. We show that the hierarchy over UOptL[log n] collapses. This implies that ${{{ULH} \subseteq {L}^{{promiseUL}}}}$ thus collapsing the UL hierarchy.

We show that the reachability problem over graphs embedded on 3 pages is complete for NL. This contrasts with the reachability problem over graphs embedded on 2 pages, which is log-space equivalent to the reachability problem in planar graphs and hence is in UL.

     

On Rewriting Rules in Mizar

This paper presents some tentative experiments in using a special case of rewriting rules in Mizar (Mizar homepage: http://www.mizar.org/): rewriting a term as its subterm. A similar technique, but based on another Mizar mechanism called functor identification (Korni?owicz 2009) was used by Caminati, in his paper on basic first-order model theory in Mizar (Caminati, J Form Reason 3(1):49–77, 2010, Form Math 19(3):157–169, 2011). However for this purpose he was obligated to introduce some artificial functors. The mechanism presented in the present paper looks promising and fits the Mizar paradigm.     

AutoRELAX: automatically RELAXing a goal model to address uncertainty

Dynamically adaptive systems (DAS) must cope with system and environmental conditions that may not have been fully understood or anticipated during development. RELAX is a fuzzy logic-based specification language for identifying and assessing sources of environmental uncertainty, thereby making DAS requirements more tolerant of unanticipated conditions. This paper presents AutoRELAX, an approach that automatically generates RELAXed goal models to address environmental uncertainty. Specifically, AutoRELAX identifies goals to RELAX, which RELAX operators to apply, and the shape of the fuzzy logic function that establishes the goal satisfaction criteria. AutoRELAX generates different solutions by making tradeoffs between minimizing the number of RELAXed goals and maximizing delivered functionality by reducing the number of adaptations triggered by minor and adverse environmental conditions. In a recent extension, AutoRELAX uses a stepwise adaptation of weights to balance these two competing concerns and thereby further improve the utility of AutoRELAX. We apply it to two industry-based applications involving network management and a robotic controller, respectively.     

Small Space Analogues of Valiant’s Classes and the Limitations of Skew Formulas

In the uniform circuit model of computation, the width of a boolean circuit exactly characterizes the “space” complexity of the computed function. Looking for a similar relationship in Valiant’s algebraic model of computation, we propose width of an arithmetic circuit as a possible measure of space. In the uniform setting, we show that our definition coincides with that of VPSPACE at polynomial width. We introduce the class VL as an algebraic variant of deterministic log-space L; VL is a subclass of VP. Further, to define algebraic variants of non-deterministic space-bounded classes, we introduce the notion of “read-once” certificates for arithmetic circuits. We show that polynomial-size algebraic branching programs (an algebraic analog of NL) can be expressed as read-once exponential sums over polynomials in ${{\sf VL}, {\it i.e.}\quad{\sf VBP} \in \Sigma^R \cdot {\sf VL}}$ . Thus, read-once exponential sums can be viewed as a reasonable way of capturing space-bounded non-determinism. We also show that Σ ^R ·VBP = VBP, i.e. VBPs are stable under read-once exponential sums. Though the best upper bound we have for Σ ^R ·VL itself is VNP, we can obtain better upper bounds for width-bounded multiplicatively disjoint (md-) circuits. Without the width restriction, md- arithmetic circuits are known to capture all of VP. We show that read-once exponential sums over md- constant-width arithmetic circuits are within VP and that read-once exponential sums over md- polylog-width arithmetic circuits are within VQP. We also show that exponential sums of a skew formula cannot represent the determinant polynomial.     

Highly-Efficient Wait-Free Synchronization

We study a simple technique, originally presented by Herlihy (ACM Trans. Program. Lang. Syst. 15(5):745–770, 1993), for executing concurrently, in a wait-free manner, blocks of code that have been programmed for sequential execution and require significant synchronization in order to be performed in parallel. We first present an implementation of this technique, called Sim, which employs a collect object. We describe a simple implementation of a collect object from a single shared object that supports atomic Add (or XOR) in addition to read; this implementation has step complexity O(1). By plugging in to Sim this implementation, Sim exhibits constant step complexity as well. This allows us to derive lower bounds on the step complexity of implementations of several shared objects, like Add, XOR, collect, and snapshot objects, from LL/SC objects. We then present a practical version of Sim, called PSim, which is implemented in a real shared-memory machine. From a theoretical perspective, PSim has worse step complexity than Sim, its theoretical analog; in practice though, we experimentally show that PSim is highly-efficient: it outperforms several state-of-the-art lock-based and lock-free synchronization techniques, and this given that it is wait-free, i.e. that it satisfies a stronger progress condition than all the algorithms that it outperforms. We have used PSim to get highly-efficient wait-free implementations of stacks and queues.