Showing papers in "Logical Methods in Computer Science in 2007"
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TL;DR: The application field of coalgebras is extended, providing a new instance of the principle "process semantics via coinduction" of bisimilarity, namely coinductions in a Kleisli category.
Abstract: Trace semantics has been defined for various kinds of state-based systems, notably with different forms of branching such as non-determinism vs probability In this paper we claim to identify one underlying mathematical structure behind these "trace semantics," namely coinduction in a Kleisli category This claim is based on our technical result that, under a suitably order-enriched setting, a final coalgebra in a Kleisli category is given by an initial algebra in the category Sets Formerly the theory of coalgebras has been employed mostly in Sets where coinduction yields a finer process semantics of bisimilarity Therefore this paper extends the application field of coalgebras, providing a new instance of the principle "process semantics via coinduction"
226 citations
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TL;DR: In this article, a fixed-point algorithm for computing the set of states from which a player can win with probability 1 with a randomized observation-based strategy for a Buechi objective is presented.
Abstract: We study observation-based strategies for two-player turn-based games on
graphs with omega-regular objectives. An observation-based strategy relies on
imperfect information about the history of a play, namely, on the past sequence
of observations. Such games occur in the synthesis of a controller that does
not see the private state of the plant. Our main results are twofold. First, we
give a fixed-point algorithm for computing the set of states from which a
player can win with a deterministic observation-based strategy for any
omega-regular objective. The fixed point is computed in the lattice of
antichains of state sets. This algorithm has the advantages of being directed
by the objective and of avoiding an explicit subset construction on the game
graph. Second, we give an algorithm for computing the set of states from which
a player can win with probability 1 with a randomized observation-based
strategy for a Buechi objective. This set is of interest because in the absence
of perfect information, randomized strategies are more powerful than
deterministic ones. We show that our algorithms are optimal by proving matching
lower bounds.
131 citations
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TL;DR: In this paper, it was shown that the satisfiability problem for metric temporal logic over finite timed words is decidable, with non-primitive recursive complexity, and model checking the safety fragment of MTL is also decidable.
Abstract: Metric Temporal Logic (MTL) is a prominent specification formalism for
real-time systems. In this paper, we show that the satisfiability problem for
MTL over finite timed words is decidable, with non-primitive recursive
complexity. We also consider the model-checking problem for MTL: whether all
words accepted by a given Alur-Dill timed automaton satisfy a given MTL
formula. We show that this problem is decidable over finite words. Over
infinite words, we show that model checking the safety fragment of MTL--which
includes invariance and time-bounded response properties--is also decidable.
These results are quite surprising in that they contradict various claims to
the contrary that have appeared in the literature.
95 citations
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TL;DR: This paper investigates the expressive power of finite sets interpretations applied to infinite deterministic trees and finds that they can be used in the study of automatic and tree-automatic structures.
Abstract: We consider a new kind of interpretation over relational structures: finite sets interpretations. Those interpretations are defined by weak monadic second-order (WMSO) formulas with free set variables. They transform a given structure into a structure with a domain consisting of finite sets of elements of the orignal structure. The definition of these interpretations directly implies that they send structures with a decidable WMSO theory to structures with a decidable first-order theory. In this paper, we investigate the expressive power of such interpretations applied to infinite deterministic trees. The results can be used in the study of automatic and tree-automatic structures.
83 citations
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TL;DR: In this paper, a relational structure is defined as a core if all its endomorphisms are embeddings and the core is unique up to isomorphism, and it is shown that every hierarchical structure has a core.
Abstract: A relational structure is a core, if all its endomorphisms are embeddings.
This notion is important for computational complexity classification of
constraint satisfaction problems. It is a fundamental fact that every finite
structure has a core, i.e., has an endomorphism such that the structure induced
by its image is a core; moreover, the core is unique up to isomorphism. Weprove
that every \omega -categorical structure has a core. Moreover, every
\omega-categorical structure is homomorphically equivalent to a model-complete
core, which is unique up to isomorphism, and which is finite or \omega
-categorical. We discuss consequences for constraint satisfaction with \omega
-categorical templates.
64 citations
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TL;DR: In this paper, the authors describe algebraic and combinatorial characterisations of finite relational core structures admitting finitely many obstructions and show that it is decidable to determine whether a constraint satisfaction problem is first-order definable: they show the general problem to be NP-complete, and give a polynomial-time algorithm in the case of cores.
Abstract: We describe simple algebraic and combinatorial characterisations of finite
relational core structures admitting finitely many obstructions. As a
consequence, we show that it is decidable to determine whether a constraint
satisfaction problem is first-order definable: we show the general problem to
be NP-complete, and give a polynomial-time algorithm in the case of cores. A
slight modification of this algorithm provides, for first-order definable
CSP's, a simple poly-time algorithm to produce a solution when one exists. As
an application of our algebraic characterisation of first order CSP's, we
describe a large family of L-complete CSP's.
63 citations
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TL;DR: This work surveys the existing results relating the expressibility of regular languages in logical fragments of MSO[S] with algebraic properties of their minimal automata and shows that many of the best known results share the same underlying mechanics and rely on a very strong relation between logical substitutions and block-products of pseudovarieties of monoid.
Abstract: The study of finite automata and regular languages is a privileged meeting
point of algebra and logic. Since the work of Buchi, regular languages have
been classified according to their descriptive complexity, i.e. the type of
logical formalism required to define them. The algebraic point of view on
automata is an essential complement of this classification: by providing
alternative, algebraic characterizations for the classes, it often yields the
only opportunity for the design of algorithms that decide expressibility in
some logical fragment.
We survey the existing results relating the expressibility of regular
languages in logical fragments of MSO[S] with algebraic properties of their
minimal automata. In particular, we show that many of the best known results in
this area share the same underlying mechanics and rely on a very strong
relation between logical substitutions and block-products of pseudovarieties of
monoid. We also explain the impact of these connections on circuit complexity
theory.
56 citations
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TL;DR: In this article, the authors studied the existence of automatic presentations for various algebraic structures and proved that the complexity of the isomorphism problem for the class of all automatic structures is σ-1^1-complete.
Abstract: We study the existence of automatic presentations for various algebraic
structures. An automatic presentation of a structure is a description of the
universe of the structure by a regular set of words, and the interpretation of
the relations by synchronised automata. Our first topic concerns characterising
classes of automatic structures. We supply a characterisation of the automatic
Boolean algebras, and it is proven that the free Abelian group of infinite
rank, as well as certain Fraisse limits, do not have automatic presentations.
In particular, the countably infinite random graph and the random partial order
do not have automatic presentations. Furthermore, no infinite integral domain
is automatic. Our second topic is the isomorphism problem. We prove that the
complexity of the isomorphism problem for the class of all automatic structures
is \Sigma_1^1-complete.
53 citations
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TL;DR: For regular infinite lambda-trees it is decidable whether a given automaton has a run or not, and this decidability result holds for arbitrary recursion schemes of arbitrary level, without any syntactical restriction.
Abstract: Model checking properties are often described by means of finite automata.
Any particular such automaton divides the set of infinite trees into finitely
many classes, according to which state has an infinite run. Building the full
type hierarchy upon this interpretation of the base type gives a finite
semantics for simply-typed lambda-trees.
A calculus based on this semantics is proven sound and complete. In
particular, for regular infinite lambda-trees it is decidable whether a given
automaton has a run or not. As regular lambda-trees are precisely recursion
schemes, this decidability result holds for arbitrary recursion schemes of
arbitrary level, without any syntactical restriction.
53 citations
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TL;DR: In this article, a Markov chain is called decisive w.r.t. a set of target states F if it almost certainly eventually reaches either F or a state from which F can no longer be reached.
Abstract: We consider qualitative and quantitative verification problems for infinite- state Markov chains. We call a Markov chain decisive w.r.t. a given set of target states F if it almost certainly eventually reaches either F or a state from which F can no longer be reached. While all finite Markov chains are trivially decisive (for every set F), this also holds for many classes of infinite Markov chains. Infinite Markov chains which contain a finite attractor are decisive w.r.t. every set F. In particular, all Markov chains induced by probabilistic lossy channel systems (PLCS) con- tain a finite attractor and are thus decisive. Furthermore, all globally coarse Markov chains are decisive. The class of globally coarse Markov chains includes, e.g., those induced by probabilistic vector addition systems (PVASS) with upward-closed sets F, and all Markov chains induced by probabilistic noisy Turing machines (PNTM) (a generalization of the noisy Turing machines (NTM) of Asarin and Collins). We consider both safety and liveness problems for decisive Markov chains. Safety: What is the probability that a given set of states F is eventually reached. Liveness: What is the probability that a given set of states is reached infinitely often. There are three variants of these questions. (1) The qualitative problem, i.e., deciding if the probability is one (or zero); (2) the approximate quantitative problem, i.e., computing the probability up-to arbitrary precision; (3) the exact quantitative problem, i.e., computing probabilities exactly. 1. We express the qualitative problem in abstract terms for decisive Markov chains, and show an almost complete picture of its decidability for PLCS, PVASS and PNTM. 2. We also show that the path enumeration algorithm of Iyer and Narasimha terminates for decisive Markov chains and can thus be used to solve the approximate quantitative safety problem. A modified variant of this algorithm can be used to solve the approximate quantitative liveness problem. 3. Finally, we show that the exact probability of (repeatedly) reaching F cannot be effectively expressed (in a uniform way) in Tarski-algebra for either PLCS, PVASS or (P)NTM (unlike for probabilistic pushdown automata).
43 citations
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TL;DR: This work considers Dense-Timed Petri Nets, an extension of Petri nets in which each token is equipped with a real-valued clock and where the semantics is lazy (i.e., enabled transitions need not fir ...
Abstract: We consider Dense-Timed Petri Nets (TPN), an extension of Petri nets in which each token is equipped with a real-valued clock and where the semantics is lazy (i.e., enabled transitions need not fir ...
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TL;DR: The characterization theorem for extended abstract state machines with respect to general algorithms as axiomatized is proved, which proves that algorithms of this generality can complete a step without necessarily waiting for replies to all queries from that step.
Abstract: In earlier work, the Abstract State Machine Thesis -- that arbitrary
algorithms are behaviorally equivalent to abstract state machines -- was
established for several classes of algorithms, including ordinary, interactive,
small-step algorithms. This was accomplished on the basis of axiomatizations of
these classes of algorithms. In Part I (Interactive Small-Step Algorithms I:
Axiomatization), the axiomatization was extended to cover interactive
small-step algorithms that are not necessarily ordinary. This means that the
algorithms (1) can complete a step without necessarily waiting for replies to
all queries from that step and (2) can use not only the environment's replies
but also the order in which the replies were received. In order to prove the
thesis for algorithms of this generality, we extend here the definition of
abstract state machines to incorporate explicit attention to the relative
timing of replies and to the possible absence of replies. We prove the
characterization theorem for extended abstract state machines with respect to
general algorithms as axiomatized in Part I.
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TL;DR: In this article, it was shown that reachability-based secrecy actually implies equivalence-based security for digital signatures, symmetric and asymmetric encryption provided that the cryptographic primitives are probabilistic.
Abstract: Two styles of definitions are usually considered to express that a security
protocol preserves the confidentiality of a data s. Reachability-based secrecy
means that s should never be disclosed while equivalence-based secrecy states
that two executions of a protocol with distinct instances for s should be
indistinguishable to an attacker. Although the second formulation ensures a
higher level of security and is closer to cryptographic notions of secrecy,
decidability results and automatic tools have mainly focused on the first
definition so far.
This paper initiates a systematic investigation of the situations where
syntactic secrecy entails strong secrecy. We show that in the passive case,
reachability-based secrecy actually implies equivalence-based secrecy for
digital signatures, symmetric and asymmetric encryption provided that the
primitives are probabilistic. For active adversaries, we provide sufficient
(and rather tight) conditions on the protocol for this implication to hold.
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TL;DR: Higher-Order Fixpoint Logic (HFL) as discussed by the authors is a hybrid of the simply typed λ-calculus and the modal μcalculus, which makes it a highly expressive temporal logic that is capable of expressing various interes.
Abstract: Higher-Order Fixpoint Logic (HFL) is a hybrid of the simply typed λ-calculus and the modal μ-calculus. This makes it a highly expressive temporal logic that is capable of expressing various interes ...
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TL;DR: In this paper, the Abstract State Machine Thesis was extended to cover interactive small-step algorithms that are not necessarily ordinary, and the proof of the axiomatization was established.
Abstract: In earlier work, the Abstract State Machine Thesis -- that arbitrary
algorithms are behaviorally equivalent to abstract state machines -- was
established for several classes of algorithms, including ordinary, interactive,
small-step algorithms. This was accomplished on the basis of axiomatizations of
these classes of algorithms. Here we extend the axiomatization and, in a
companion paper, the proof, to cover interactive small-step algorithms that are
not necessarily ordinary. This means that the algorithms (1) can complete a
step without necessarily waiting for replies to all queries from that step and
(2) can use not only the environment's replies but also the order in which the
replies were received.
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TL;DR: In this article, the model-checking and parameter synthesis problems of TCTL over discrete-timed automata where parameters are allowed both in the model and in the property (temporal formula) were studied.
Abstract: In this paper, we study the model-checking and parameter synthesis problems
of the logic TCTL over discrete-timed automata where parameters are allowed
both in the model (timed automaton) and in the property (temporal formula). Our
results are as follows. On the negative side, we show that the model-checking
problem of TCTL extended with parameters is undecidable over discrete-timed
automata with only one parametric clock. The undecidability result needs
equality in the logic. On the positive side, we show that the model-checking
and the parameter synthesis problems become decidable for a fragment of the
logic where equality is not allowed. Our method is based on automata theoretic
principles and an extension of our method to express durations of runs in timed
automata using Presburger arithmetic.
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TL;DR: In this article, a new approach for performing predicate abstraction based on symbolic decision procedures is presented, which takes a set of predicates in the theory and symbolically executes a decision procedure on all the subsets over the set of predicate predicates.
Abstract: We present a new approach for performing predicate abstraction based on
symbolic decision procedures. Intuitively, a symbolic decision procedure for a
theory takes a set of predicates in the theory and symbolically executes a
decision procedure on all the subsets over the set of predicates. The result of
the symbolic decision procedure is a shared expression (represented by a
directed acyclic graph) that implicitly represents the answer to a predicate
abstraction query.
We present symbolic decision procedures for the logic of Equality and
Uninterpreted Functions (EUF) and Difference logic (DIFF) and show that these
procedures run in pseudo-polynomial (rather than exponential) time. We then
provide a method to construct symbolic decision procedures for simple mixed
theories (including the two theories mentioned above) using an extension of the
Nelson-Oppen combination method. We present preliminary evaluation of our
Procedure on predicate abstraction benchmarks from device driver verification
in SLAM.
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TL;DR: The special case of k = 1 for trees, shows that single-head deterministic tree-walking automata with nested pebbles are characterized by first-order logic with unary determin- istic transitive closure.
Abstract: String languages recognizable in (deterministic) log-space are characterized either by two-way (deterministic) multi-head automata, or following Immerman, by first- order logic with (deterministic) transitive closure. Here we elaborate this result, and match the number of heads to the arity of the transitive closure. More precisely, first-order logic with k-ary deterministic transitive closure has the same power as deterministic automata walking on their input with k heads, additionally using a finite set of nested pebbles. This result is valid for strings, ordered trees, and in general for families of graphs having a fixed automaton that can be used to traverse the nodes of each of the graphs in the family. Other examples of such families are grids, toruses, and rectangular mazes. For nondeterministic automata, the logic is restricted to positive occurrences of transitive closure. The special case of k = 1 for trees, shows that single-head deterministic tree-walking automata with nested pebbles are characterized by first-order logic with unary determin- istic transitive closure. This refines our earlier result that placed these automata between first-order and monadic second-order logic on trees.
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TL;DR: In this paper, the authors prove a conjecture by A. Pnueli and strengthen it showing a sequence of counting modalities none of which is expressible in the temporal logic generated by the previous modalities, over the real line, or over the positive real line.
Abstract: We prove a conjecture by A. Pnueli and strengthen it showing a sequence of
"counting modalities" none of which is expressible in the temporal logic
generated by the previous modalities, over the real line, or over the positive
reals. Moreover, there is no finite temporal logic that can express all of them
over the real line, so that no finite metric temporal logic is expressively
complete.
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TL;DR: It is proved that constraint languages consisting of relations that are invariant under a short sequence of Jonsson terms are tractable by showing that such languages have bounded width.
Abstract: Constraint languages that arise from finite algebras have recently been the
object of study, especially in connection with the Dichotomy Conjecture of
Feder and Vardi. An important class of algebras are those that generate
congruence distributive varieties and included among this class are lattices,
and more generally, those algebras that have near-unanimity term operations. An
algebra will generate a congruence distributive variety if and only if it has a
sequence of ternary term operations, called Jonsson terms, that satisfy certain
equations.
We prove that constraint languages consisting of relations that are invariant
under a short sequence of Jonsson terms are tractable by showing that such
languages have bounded relational width.
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TL;DR: An abstraction-based model checking method which relies on re- finement of an under-approximation of the feasible behaviors of the system under analysis to decide termination or to refine the abstraction by generating new abstraction predicates.
Abstract: We propose an abstraction-based model checking method which relies on re- finement of an under-approximation of the feasible behaviors of the system under analysis. The method preserves errors to safety properties, since all analyzed behaviors are feasible by definition. The method does not require an abstract transition relation to be gener- ated, but instead executes the concrete transitions while storing abstract versions of the concrete states, as specified by a set of abstraction predicates. For each explored transition the method checks, with the help of a theorem prover, whether there is any loss of precision introduced by abstraction. The results of these checks are used to decide termination or to refine the abstraction by generating new abstraction predicates. If the (possibly infi- nite) concrete system under analysis has a finite bisimulation quotient, then the method is guaranteed to eventually explore an equivalent finite bisimilar structure. We illustrate the application of the approach for checking concurrent programs.
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TL;DR: In this paper, a call-by-value version of PCF is investigated under a complexity-theoretically motivated type system, and two semantics are constructed for ATR.
Abstract: This paper investigates what is essentially a call-by-value version of PCF
under a complexity-theoretically motivated type system. The programming
formalism, ATR, has its first-order programs characterize the polynomial-time
computable functions, and its second-order programs characterize the type-2
basic feasible functionals of Mehlhorn and of Cook and Urquhart. (The ATR-types
are confined to levels 0, 1, and 2.) The type system comes in two parts, one
that primarily restricts the sizes of values of expressions and a second that
primarily restricts the time required to evaluate expressions. The
size-restricted part is motivated by Bellantoni and Cook's and Leivant's
implicit characterizations of polynomial-time. The time-restricting part is an
affine version of Barber and Plotkin's DILL. Two semantics are constructed for
ATR. The first is a pruning of the naive denotational semantics for ATR. This
pruning removes certain functions that cause otherwise feasible forms of
recursion to go wrong. The second semantics is a model for ATR's time
complexity relative to a certain abstract machine. This model provides a
setting for complexity recurrences arising from ATR recursions, the solutions
of which yield second-order polynomial time bounds. The time-complexity
semantics is also shown to be sound relative to the costs of interpretation on
the abstract machine.
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TL;DR: In this paper, an interpolant-based method for strengthening predicate abstraction transition relations is presented, which guarantees convergence given an adequate set of predicates, without requiring an exact image computation, and empirically shows that the method converges more rapidly than an earlier method based on counterexample analysis.
Abstract: In predicate abstraction, exact image computation is problematic, requiring
in the worst case an exponential number of calls to a decision procedure. For
this reason, software model checkers typically use a weak approximation of the
image. This can result in a failure to prove a property, even given an adequate
set of predicates. We present an interpolant-based method for strengthening the
abstract transition relation in case of such failures. This approach guarantees
convergence given an adequate set of predicates, without requiring an exact
image computation. We show empirically that the method converges more rapidly
than an earlier method based on counterexample analysis.
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TL;DR: In this paper, the problem of typing lambda-terms in second-order dual light affine logic (DLAL) is addressed, and a procedure which, starting with a term typed in system F, determines whether it is typable in DLAL and outputs a concrete typing if there exists any.
Abstract: In a previous work Baillot and Terui introduced Dual light affine logic
(DLAL) as a variant of Light linear logic suitable for guaranteeing complexity
properties on lambda calculus terms: all typable terms can be evaluated in
polynomial time by beta reduction and all Ptime functions can be represented.
In the present work we address the problem of typing lambda-terms in
second-order DLAL. For that we give a procedure which, starting with a term
typed in system F, determines whether it is typable in DLAL and outputs a
concrete typing if there exists any. We show that our procedure can be run in
time polynomial in the size of the original Church typed system F term.
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TL;DR: For a two-variable formula ψ(X,Y) of Monadic Logic of Order (MLO), the authors showed that the Church synthesis problem for P is computable if and only if the monadic theory of fixme is decidable.
Abstract: For a two-variable formula ψ(X,Y) of Monadic Logic of Order (MLO) the
Church Synthesis Problem concerns the existence and construction of an operator
Y=F(X) such that ψ(X,F(X)) is universally valid over Nat.
B\"{u}chi and Landweber proved that the Church synthesis problem is
decidable; moreover, they showed that if there is an operator F that solves the
Church Synthesis Problem, then it can also be solved by an operator defined by
a finite state automaton or equivalently by an MLO formula. We investigate a
parameterized version of the Church synthesis problem. In this version ψ
might contain as a parameter a unary predicate P. We show that the Church
synthesis problem for P is computable if and only if the monadic theory of
is decidable. We prove that the B\"{u}chi-Landweber theorem can be
extended only to ultimately periodic parameters. However, the MLO-definability
part of the B\"{u}chi-Landweber theorem holds for the parameterized version of
the Church synthesis problem.
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TL;DR: In this paper, it was shown that Martin-Lof dependent type theory with a program for Spector double negation shift, similar to bar recursion, has the strong normalisation property.
Abstract: In 1961, Spector presented an extension of Godel's system T by a new schema of definition called bar recursion. With this new schema, he was able to give an interpretation of Analysis, extending Godel's Dialectica interpretation of Arithmetic, and completing preliminary results of Kreisel. Tait proved a normalisation theorem for Spector's bar recursion, by embedding it in a system with infinite terms. In a paper by Berardi, Bezem and Coquand, an alternative form of bar recursion was introduced. This allowed to give an interpretation of Analysis by modified realisability, instead of Dialectica interpretation. It presented also a normalisation proof for this new schema, but this proof, which used Tait's method of introducing infinite terms, was quite complex. It was simplified significantly by U. Berger, who used instead a modification of Plotkin's computational adequacy theorem, and could prove strong normalisation. In a way, the idea is to replace infinite terms by elements of a domain interpretation. This domain has the property that a term is strongly normalisable if its semantics is not ⊥. The main contribution of this paper is to show that, using ideas from intersection types and Martin-Lof's domain interpretation of type theory, one can in turn simplify further U. Berger's argument. Contrary to him, we build a domain model for an untyped programming language. Compared to other works, there is no need of an extra hypothesis to deduce strong normalisation from the domain interpretation. A noteworthy feature of this domain model is that it is in a natural way a complete lattice, and in particular it has a top element which can be seen as the interpretation of a top-level exception in programming language. We think that this model can be the basis of modular proofs of strong normalisation for various type systems. As a main application, we show that Martin-Lof dependent type theory extended with a program for Spector double negation shift, similar to bar recursion, has the strong normalisation property.
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TL;DR: In this paper, the authors study the decidability of FO(R), a first-order logic extended by reachability predicates, for finite transition systems, and show that it is in general undecidable.
Abstract: Formal verification using the model checking paradigm has to deal with two
aspects: The system models are structured, often as products of components, and
the specification logic has to be expressive enough to allow the formalization
of reachability properties. The present paper is a study on what can be
achieved for infinite transition systems under these premises. As models we
consider products of infinite transition systems with different synchronization
constraints. We introduce finitely synchronized transition systems, i.e.
product systems which contain only finitely many (parameterized) synchronized
transitions, and show that the decidability of FO(R), first-order logic
extended by reachability predicates, of the product system can be reduced to
the decidability of FO(R) of the components. This result is optimal in the
following sense: (1) If we allow semifinite synchronization, i.e. just in one
component infinitely many transitions are synchronized, the FO(R)-theory of the
product system is in general undecidable. (2) We cannot extend the expressive
power of the logic under consideration. Already a weak extension of first-order
logic with transitive closure, where we restrict the transitive closure
operators to arity one and nesting depth two, is undecidable for an
asynchronous (and hence finitely synchronized) product, namely for the infinite
grid.
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TL;DR: In this paper, the authors identify a form of problem structure concerned with the symmetrical or asymmetrical nature of the cost of achieving the individual planning goals, and quantify this sort of structure with a simple numeric parameter called AsymRatio, ranging between 0 and 1.
Abstract: In Verification and in (optimal) AI Planning, a successful method is to
formulate the application as boolean satisfiability (SAT), and solve it with
state-of-the-art DPLL-based procedures. There is a lack of understanding of why
this works so well. Focussing on the Planning context, we identify a form of
problem structure concerned with the symmetrical or asymmetrical nature of the
cost of achieving the individual planning goals. We quantify this sort of
structure with a simple numeric parameter called AsymRatio, ranging between 0
and 1. We run experiments in 10 benchmark domains from the International
Planning Competitions since 2000; we show that AsymRatio is a good indicator of
SAT solver performance in 8 of these domains. We then examine carefully crafted
synthetic planning domains that allow control of the amount of structure, and
that are clean enough for a rigorous analysis of the combinatorial search
space. The domains are parameterized by size, and by the amount of structure.
The CNFs we examine are unsatisfiable, encoding one planning step less than the
length of the optimal plan. We prove upper and lower bounds on the size of the
best possible DPLL refutations, under different settings of the amount of
structure, as a function of size. We also identify the best possible sets of
branching variables (backdoors). With minimum AsymRatio, we prove exponential
lower bounds, and identify minimal backdoors of size linear in the number of
variables. With maximum AsymRatio, we identify logarithmic DPLL refutations
(and backdoors), showing a doubly exponential gap between the two structural
extreme cases. The reasons for this behavior -- the proof arguments --
illuminate the prototypical patterns of structure causing the empirical
behavior observed in the competition benchmarks.
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TL;DR: In this article, an explicit coinduction principle for recursively defined stochastic processes is given, which applies to any closed property, not just equality, and works even when solutions are not unique.
Abstract: We give an explicit coinduction principle for recursively-defined stochastic
processes. The principle applies to any closed property, not just equality, and
works even when solutions are not unique. The rule encapsulates low-level
analytic arguments, allowing reasoning about such processes at a higher
algebraic level. We illustrate the use of the rule in deriving properties of a
simple coin-flip process.
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TL;DR: In this article, the authors present probabilistic extensions of interval temporal logic (ITL) and duration calculus (DC) with infinite intervals and complete Hilbert-style proof sys- tems for them.
Abstract: The paper presents probabilistic extensions of interval temporal logic (ITL) and duration calculus (DC) with infinite intervals and complete Hilbert-style proof sys- tems for them. The completeness results are a strong completeness theorem for the system of probabilistic ITL with respect to an abstract semantics and a relative completeness the- orem for the system of probabilistic DC with respect to real-time semantics. The proposed systems subsume probabilistic real-time DC as known from the literature. A correspon- dence between the proposed systems and a system of probabilistic interval temporal logic with finite intervals and expanding modalities is established too.