Showing papers presented at "Computer Science Logic in 2003"

Games for synthesis of controllers with partial observation

Abstract: The synthesis of controllers for discrete event systems, as introduced by Ramadge and Wonham, amounts to computing winning strategies in parity games We show that in this framework it is possible to extend the specifications of the supervised systems as well as the constraints on the controllers by expressing them in the modal µ-calculusIn order to express unobservability constraints, we propose an extension of the modal µ-calculus in which one can specify whether an edge of a graph is a loop This extended µ-calculus still has the interesting properties of the classical one In particular it is equivalent to automata with loop testing The problems such as emptiness testing and elimination of alternation are solvable for such automataThe method proposed in this paper to solve a control problem consists in transforming this problem into a problem of satisfiability of a µ-calculus formula so that the set of models of this formula is exactly the set of controllers that solve the problem This transformation relies on a simple construction of the quotient of automata with loop testing by a deterministic transition system This is enough to deal with centralized control problems The solution of decentralized control problems uses a more involved construction of the quotient of two automataThis work extends the framework of Ramadge and Wonham in two directions We consider infinite behaviours and arbitrary regular specifications, while the standard framework deals only with specifications on the set of finite paths of processes We also allow dynamic changes of the sets of observable and controllable events

Simple Stochastic Parity Games

Abstract: Many verification, planning, and control problems can be modeled as games played on state-transition graphs by one or two players whose conflicting goals are to form a path in the graph. The focus here is on simple stochastic parity games, that is, two-player games with turn-based probabilistic transitions and ω-regular objectives formalized as parity (Rabin chain) winning conditions. An efficient translation from simple stochastic parity games to nonstochastic parity games is given. As many algorithms are known for solving the latter, the translation yields efficient algorithms for computing the states of a simple stochastic parity game from which a player can win with probability 1.

On Nash equilibria in stochastic games

Abstract: We study infinite stochastic games played by n-players on a finite graph with goals given by sets of infinite traces. The games are stochastic (each player simultaneously and independently chooses an action at each round, and the next state is determined by a probability distribution depending on the current state and the chosen actions), infinite (the game continues for an infinite number of rounds), nonzero sum (the players’ goals are not necessarily conflicting), and undiscounted. We show that if each player has a reachability objective, that is, if the goal for each player i is to visit some subset Ri of the states, then there exists an e-Nash equilibrium in memoryless strategies, for every e >0. However, exact Nash equilibria need not exist. We study the complexity of finding such Nash equilibria, and show that the payoff of some e-Nash equilibrium in memoryless strategies can be e-approximated in NP.

Finding approximate repetitions under Hamming distance

Abstract: The problem of computing periodicities with K possible mismatches is studied. Two main definitions are considered, and for both of them an O(nK logK + S) algorithm is proposed (n the word length and S the size of the output). This improves, in particular, the bound obtained by G. Landan and J. Schmidt in 1993 (Proceedings of the Fourth Annual Symposium on Combinatorial Pattern Matching, Lecture Notes in Computer Science, Vol. 684, Springer, Berlin, Padova, Italy, pp. 120-133). Finally, other possible definitions are briefly analyzed.

Quantified Constraints: Algorithms and Complexity

Abstract: The standard constraint satisfaction problem over an arbitrary finite domain can be expressed as follows: given a first-order sentence consisting of a conjunction of predicates, where all of the variables are existentially quantified, determine whether the sentence is true. This problem can be parameterized by the set of allowed constraint predicates. With each predicate, one can associate certain predicate-preserving operations, called polymorphisms, and the complexity of the parameterized problem is known to be determined by the polymorphisms of the allowed predicates. In this paper we consider a more general framework for constraint satisfaction problems which allows arbitrary quantifiers over constrained variables, rather than just existential quantifiers. We show that the complexity of such extended problems is determined by the surjective polymorphisms of the constraint predicates. We give examples to illustrate how this result can be used to identify tractable and intractable cases for the quantified constraint satisfaction problem over arbitrary finite domains.

A gap property of deterministic tree languages

Abstract: We show that a tree language recognized by a deterministic parity automaton is either hard for the co-Buchi level and therefore cannot be recognized by a weak alternating automaton, or is on a very low level in the hierarchy of weak alternating automata. A topological counterpart of this property is that a deterministic tree language is either Π11 complete (and hence nonBorel), or it is on the level Π30 of the Borel hierarchy. We also give a new simple proof of the strictness of the hierarchy of weak alternating automata.

Deciding Monotonic Games.

Abstract: In an earlier work [ACJYK00] we presented a general framework for verification of infinite-state transition systems, where the transition relation is monotonic with respect to a well quasi-ordering on the set of states. In this paper, we investigate extending the framework from the context of transition systems to that of games. We show that monotonic games are in general undecidable. We identify a subclass of monotonic games, called downward closed games. We provide algorithms for analyzing downward closed games subject to winning conditions which are formulated as safety properties.

Resolution lower bounds for the weak functional pigeonhole principle

Abstract: We show that every resolution proof of the functional version FPHPnm of the pigeonhole principle (in which one pigeon may not split between several holes) must have size exp(Ω(n/(log m)2)). This implies an exp(Ω(n1/3)) bound when the number of pigeons m is arbitrary.

Atomic Cut Elimination for Classical Logic

Abstract: System SKS is a set of rules for classical propositional logic presented in the calculus of structures. Like sequent systems and unlike natural deduction systems, it has an explicit cut rule, which is admissible. In contrast to sequent systems, the cut rule can easily be reduced to atomic form. This allows for a very simple cut elimination procedure based on plugging in parts of a proof, like normalisation in natural deduction and unlike cut elimination in the sequent calculus. It should thus be a good common starting point for investigations into both proof search as computation and proof normalisation as computation.

Algebraic proof systems over formulas

Abstract: We introduce two algebraic propositional proof systems F-NS and F-PC. The main difference of our systems from (customary) Nullstellensatz and polynomial calculus is that the polynomials are represented as arbitrary formulas (rather than sums of monomials). Short proofs of Tseitin's tautologies in the constant-depth version of F-NS provide an exponential separation between this system and Polynomial Calculus.We prove that F-NS (and hence F-PC) polynomially simulates Frege systems, and that the constant-depth version of F-PC over finite field polynomially simulates constant-depth Frege systems with modular counting. We also present a short constant-depth F-PC (in fact, F-NS) proof of the propositional pigeon-hole principle. Finally, we introduce several extensions of our systems and pose numerous open questions.

Generating All Abductive Explanations for Queries on Propositional Horn Theories

Abstract: Abduction is a fundamental mode of reasoning, which has taken on increasing importance in Artificial Intelligence (AI) and related disciplines. Computing abductive explanations is an important problem, and there is a growing literature on this subject. We contribute to this endeavor by presenting new results on computing multiple resp. all of the possibly exponentially many explanations of an abductive query from a propositional Horn theory represented by a Horn CNF. Here the issues are whether a few explanations can be generated efficiently and, in case of all explanations, whether the computation is possible in polynomial total time (oroutput-polynomial time), i.e., in time polynomial in the combined size of the input and the output. We explore these issues for queries in CNF and important restrictions thereof. Among the results, we show that computing all explanations for a negative query literal from a Horn CNF is not feasible in polynomial total time unless P = NP, which settles an open issue. However, we show how to compute under restriction to acyclic Horn theories polynomially many explanations in input polynomial time and all explanations in polynomial total time, respectively. Complementing and extending previous results, this draws a detailed picture of the computational complexity of computing multiple explanations for queries on Horn theories.

Refined Complexity Analysis of Cut Elimination

Abstract: In [1,2] Zhang shows how the complexity of cut elimination depends on the nesting of quantifiers in cut formulas. By studying the role of contractions we can refine that analysis and show how the complexity depends on a combination of contractions and quantifier nesting. With the refined analysis the upper bound on cut elimination coincides with Statman’s lower bound. Every non-elementary growth example must display a combination of nesting of quantifiers and contractions similar to Statman’s lower bound example. The upper and lower bounds on cut elimination immediately translate into bounds on Herbrand’s theorem. Finally we discuss the role of quantifier alternations and show an elementary upper bound for the ∀ − ∧-case (resp. ∃ − ∨-case).

Machine Characterizations of the Classes of the W-Hierarchy

Abstract: We give machine characterizations of the complexity classes of the W-hierarchy. Moreover, for every class of this hierarchy, we present a parameterized halting problem complete for this class.

On the Complexity of Existential Pebble Games

Abstract: Existential k-pebble games, k≥ 2, are combinatorial games played between two players, called the Spoiler and the Duplicator, on two structures. These games were originally introduced in order to analyze the expressive power of Datalog and related infinitary logics with finitely many variables. More recently, however, it was realized that existential k-pebble games have tight connections with certain consistency properties which play an important role in identifying tractable classes of constraint satisfaction problems and in designing heuristic algorithms for solving such problems. Specifically, it has been shown that strong k-consistency can be established for an instance of constraint satisfaction if and only if the Duplicator has a winnning strategy for the existential k-pebble game between two finite structures associated with the given instance of constraint satisfaction. In this paper, we pinpoint the computational complexity of determining the winner of the existential k-pebble game. The main result is that the following decision problem is EXPTIME-complete: given a positive integer k and two finite structures A and B, does the Duplicator win the existential k-pebble game on A and B? Thus, all algorithms for determining whether strong k-consistency can be established (when k is part of the input) are inherently exponential.

Constraint Satisfaction with Countable Homogeneous Templates

Abstract: For a fixed countable homogeneous relational structure Γ we study the computational problem whether a given finite structure of the same signature homomorphically maps to Γ. This problem is known as the constraint satisfaction problem CSP(Γ) for Γ and was intensively studied for finite Γ. We show that – as in the case of finite Γ – the computational complexity of CSP(Γ) for countable homogeneous Γ is determinded by the clone of polymorphisms of Γ. To this end we prove the following theorem which is of independent interest: The primitive positive definable relations over an ω-categorical structure Γ are precisely the relations that are invariant under the polymorphisms of Γ.

Extending the Dolev-Yao Intruder for Analyzing an Unbounded Number of Sessions

Abstract: We propose a protocol model which integrates two different ways of analyzing cryptographic protocols: i) analysis w.r.t. an unbounded number of sessions and bounded message size, and ii) analysis w.r.t. an a priori bounded number of sessions but with messages of unbounded size. We show that in this model secrecy is DEXPTIME-complete. This result is obtained by extending the Dolev-Yao intruder to simulate unbounded number of sessions.

A Logic for Probability in Quantum Systems

Abstract: Quantum computation deals with projective measurements and unitary transformations in finite dimensional Hilbert spaces. The paper presents a propositional logic designed to describe quantum computation at an operational level by supporting reasoning about the probabilities associated to such measurements: measurement probabilities, and transition probabilities (a quantum analogue of conditional probabilities). We present two axiomatizations, one for the logic as a whole and one for the fragment dealing just with measurement probabilities. These axiomatizations are proved to be sound and complete. The logic is also shown to be decidable, and we provide results characterizing its complexity in a number of cases.

Comparing the Succinctness of Monadic Query Languages over Finite Trees

Abstract: We study the succinctness of monadic second-order logic and a variety of monadic fixed point logics on trees. All these languages are known to have the same expressive power on trees, but some can express the same queries much more succinctly than others. For example, we show that, under some complexity theoretic assumption, monadic second-order logic is non-elementarily more succinct than monadic least fixed point logic, which in turn is non-elementarily more succinct than monadic datalog.

Towards a Proof System for Admissibility

Abstract: In [4] a basis for the admissible rules of intuitionistic propositional logic IPC was given. Here we strengthen this result by presenting a system ADM that has the following two properties. \(\phantom{~} A\vdash_{\sf ADM}B\) implies that A admissibly derives B. ADM is complete in the sense that for every formula A there exists a formula \(\phantom{~} A\vdash_{\sf ADM}\Lambda_A\) such that the admissibly derivable consequences of A are the (normal) consequences of \(\phantom{~} \Lambda_A\). This work is related to and partly relies upon research by Ghilardi on projective formulas [2, 3].

On relativisation and complexity gap for resolution-based proof systems

Abstract: We study the proof complexity of Taut, the class of Second-Order Existential (SO∃) logical sentences which fail in all finite models. The Complexity-Gap theorem for Tree-like Resolution says that the shortest Tree-like Resolution refutation of any such sentence Φ is either fully exponential, \(2^{\Omega \left(n\right)}\), or polynomial, \(n^{O\left(1\right)}\), where n is the size of the finite model. Moreover, there is a very simple model-theoretics criteria which separates the two cases: the exponential lower bound holds if and only if Φ holds in some infinite model.

Tree-width and the monadic quantifier hierarchy

Abstract: It is well known that on classes of graphs of bounded tree-width, every monadic second-order property is decidable in polynomial time. The converse is not true without further assumptions. It follows from the work of Robertson and Seymour, that if a class of graphs K has unbounded tree-width and is closed under minors, then K contains all planar graphs. But on planar graphs, three-colorability is NP-complete. Hence, if P ≠ NP and on K every existential monadic second-order property is in P, then K has bounded tree-width. In other words, for K closed under minors, K is of bounded tree-width iff all monadic second-order properties are decidable in P.In this note we prove that in order to characterize classes of graphs of bounded tree-width where the monadic quantifier hierarchy collapses, closure under minors can be replaced by closure under topological minors. Closure under minors of K implies that K is in P, whereas we also note that there is a class of graphs K closed under topological minors which is not even r.e.We also show, that closure under induced subgraphs or even under subgraphs alone does not suffice to show that the collapse of the monadic quantifier hierarchy on K implies that K is of bounded tree-width or clique-width.Other characterizations of classes of bounded tree-width in terms of collapses of the monadic quantifier hierarchy to levels above the existential are discussed.

Modular semantics and logics of classes

Abstract: The semantics of class-based languages can be defined in terms of objects only [1,7,8] if classes are viewed as objects with a constructor method. One obtains a store in which method closures are held together with field values. Such a store is also called “higher-order” and does not come for free [13]. It is much harder to prove properties of such stores and as a consequence (soundness of) programming logics can become rather contrived (see [2]).

Calculi of Meta-variables

Abstract: The notion of meta-variable plays a fundamental role when we define formal systems such as logical and computational calculi. Yet it has been usually understood only informally as is seen in most textbooks of logic. Based on our observations of the usages of meta-variables in textbooks, we propose two formal systems that have the notion of meta-variable.

Computational Aspects of Σ-Definability over the Real Numbers without the Equality Test

Abstract: In this paper we study the expressive power and algorithmic properties of the language of Σ-formulas intended to represent computability over the real numbers In order to adequately represent computability, we extend the reals by the structure of hereditarily finite sets In this setting it is crucial to consider the real numbers without equality since the equality test is undecidable over the reals We prove Engeler’s Lemma for Σ-definability over the reals without the equality test which relates Σ-definability with definability in the constructive infinitary language \(L_{\omega_1\omega}\) Thus, a relation over the real numbers is Σ-definable if and only if it is definable by a disjunction of a recursively enumerable set of quantifier free formulas This result reveals computational aspects of Σ-definability and also gives topological characterisation of Σ-definable relations over the reals without the equality test

Validity of CTL Queries Revisited

Abstract: We systematically investigate temporal logic queries in model checking, adding to the seminal paper by William Chan at CAV 2000. Chan’s temporal logic queries are CTL specifications where one unspecified subformula is to be filled in by the model checker in such a way that the specification becomes true. Chan defined a fragment of CTL queries called \(\mbox{CTL}^{v}\) which guarantees the existence of a unique strongest solution. The starting point of our paper is a counterexample to this claim. We then show how the research agenda of Chan can be realized by modifying his fragment appropriately. To this aim, we investigate the criteria required by Chan, and define two new fragments \(\mbox{CTL}^{v}_{new}\) and \(\mbox{CTL}^{d}\) where the first is the one originally intended; the latter fragment also provides unique strongest solutions where possible but admits also cases where the set of solutions is empty.

Goal-Directed Calculi for Gödel-Dummett Logics

Abstract: In this work we present goal-directed calculi for the Godel-Dummett logic LC and its finite-valued counterparts, LC n (n ≥ 2). We introduce a terminating hypersequent calculus for the implicational fragment of LC with local rules and a single identity axiom. We also give a labelled goal-directed calculus with invertible rules and show that it is co-NP. Finally we derive labelled goal-directed calculi for LC n .

The surprising power of restricted programs and Gödel's functionals

Abstract: Consider the following imperative programming language. The programs operate on registers storing natural numbers, the input \(\vec{x}\) is stored in certain registers, and a number b, called the base, is fixed to \(\max(\vec{x},1)+1\) before the execution starts. The single primitive instruction \(\verb/X+/\) increases the number stored in the register \(\verb/X/\) by 1 modulo b. There are two control structures: the loop \(\verb+while X{P}+\) executing the program \(\verb+P+\) repeatedly as long as the content of the register \(\verb/X/\) is different from 0; the composition \(\verb+P+\texttt{;} \verb+Q+\) executing first the program \(\verb/P/\), then the program \(\verb/Q/\). This is the whole language. The language is natural, extremely simple, yet powerful. We will prove that it captures \(\mbox{\sc linspace}\), i.e. the numerical relations decidable by such programs are exactly those decidable by Turing machines working in linear space. Variations of the language capturing other important deterministic complexity classes, like e.g. \(\mbox{\sc logspace}\), \(\mbox{\sc p}\) and \(\mbox{\sc pspace}\) are possible (see Kristiansen and Voda [5]).

On Relativisation and Complexity Gap.

Abstract: We study the proof complexity of Taut, the class of Second-Order Existential (SO∃) logical sentences which fail in all finite models. The Complexity-Gap theorem for Tree-like Resolution says that the shortest Tree-like Resolution refutation of any such sentence Φ is either fully exponential, \(2^{\Omega \left(n\right)}\), or polynomial, \(n^{O\left(1\right)}\), where n is the size of the finite model. Moreover, there is a very simple model-theoretics criteria which separates the two cases: the exponential lower bound holds if and only if Φ holds in some infinite model.

Bistability: An Extensional Characterization of Sequentiality

Abstract: We give a simple order-theoretic construction of a cartesian closed category of sequential functions. It is based on biordered sets analogous to Berry’s bidomains, except that the stable order is replaced with a new notion, the bistable order, and instead of preserving stably bounded greatest lower bounds, functions are required to preserve bistably bounded least upper bounds and greatest lower bounds. We show that bistable cpos and bistable and continuous functions form a CCC, yielding models of functional languages such as the simply-typed λ-calculus and SPCF. We show that these models are strongly sequential and use this fact to prove universality and full abstraction results.

Logical Relations for Dynamic Name Creation

Abstract: Pitts and Stark’s nu-calculus is a typed lambda-calculus which forms a basis for the study of interaction between higher-order functions and dynamically created names. A similar approach has received renewed attention recently through Sumii and Pierce’s cryptographic lambda-calculus, which deals with security protocols. Logical relations are a powerful tool to prove properties of such a calculus, notably observational equivalence. While Pitts and Stark construct a logical relation for the nu-calculus, it rests heavily on operational aspects of the calculus and is hard to be extended. We propose an alternative Kripke logical relation for the nu-calculus, which is derived naturally from the categorical model of the nu-calculus and the general notion of Kripke logical relation. This is also related to the Kripke logical relation for the name creation monad by Goubault-Larrecq et al. (CSL’2002), which the authors claimed had similarities with Pitts and Stark’s logical relation. We show that their Kripke logical relation for names is strictly weaker than Pitts and Stark’s. We also show that our Kripke logical relation, which extends the definition of Goubault-Larrecq et al., is equivalent to Pitts and Stark’s up to first-order types; our definition rests on purely semantic constituents, and dispenses with the detours through operational semantics that Pitts and Stark use.