The model checking of higher-order recursion schemes has important applications in the verification of higher-order programs. Ong has previously shown that the modal mu-calculus model checking of trees generated by order-n recursion scheme is n-EXPTIME complete, but his algorithm and its correctness proof were rather complex. We give an alternative, type-based verification method: Given a modal mu-calculus formula, we can construct a type system in which a recursion scheme is typable if, and only if, the (possibly infinite, ranked) tree generated by the scheme satisfies the formula. The model checking problem is thus reduced to a type checking problem. Our type-based approach yields a simple verification algorithm, and its correctness proof (constructed without recourse to game semantics) is comparatively easy to understand. Furthermore, the algorithm is polynomial-time in the size of the recursion scheme, assuming that the formula and the largest order and arity of non-terminals of the recursion scheme are fixed.
{"title":"A Type System Equivalent to the Modal Mu-Calculus Model Checking of Higher-Order Recursion Schemes","authors":"N. Kobayashi, C. Ong","doi":"10.1109/LICS.2009.29","DOIUrl":"https://doi.org/10.1109/LICS.2009.29","url":null,"abstract":"The model checking of higher-order recursion schemes has important applications in the verification of higher-order programs. Ong has previously shown that the modal mu-calculus model checking of trees generated by order-n recursion scheme is n-EXPTIME complete, but his algorithm and its correctness proof were rather complex. We give an alternative, type-based verification method: Given a modal mu-calculus formula, we can construct a type system in which a recursion scheme is typable if, and only if, the (possibly infinite, ranked) tree generated by the scheme satisfies the formula. The model checking problem is thus reduced to a type checking problem. Our type-based approach yields a simple verification algorithm, and its correctness proof (constructed without recourse to game semantics) is comparatively easy to understand. Furthermore, the algorithm is polynomial-time in the size of the recursion scheme, assuming that the formula and the largest order and arity of non-terminals of the recursion scheme are fixed.","PeriodicalId":415902,"journal":{"name":"2009 24th Annual IEEE Symposium on Logic In Computer Science","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2009-08-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122426104","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The problem of Pairwise CFL-reachability is to decide whether two given program locations in different threads are simultaneously reachable in the presence of recursion in threads and scheduling constraints imposed by synchronization primitives. Pairwise CFL-reachability is the core problem underlying concurrent program analysis especially dataflow analysis. Unfortunately, it is undecidable even for the most commonly used synchronization primitive, i.e., mutex locks. Lock usage in concurrent programs can be characterized in terms of lock chains, where a sequence of mutex locks is said to be chained if the scopes of adjacent (nonnested) mutexes overlap. Although pairwise reachability is known to decidable for threads interacting via nested locks, i.e., chains of length one, these techniques don’t extend to programs with non-nested locks used in crucial applications like databases and device drivers. In this paper, we exploit the fact that lock usage patterns in real life programs do not produce unbounded lock chains. For such programs, we show that pairwise CFL-reachability becomes decidable. Our new results narrow the decidability gap for pairwise CFL-reachability by providing a more refined characterization for it in terms of boundedness of lock chains rather than the current state-of-the-art, i.e., nestedness of locks (chains of length one).
{"title":"Boundedness vs. Unboundedness of Lock Chains: Characterizing Decidability of Pairwise CFL-Reachability for Threads Communicating via Locks","authors":"Vineet Kahlon","doi":"10.1109/LICS.2009.45","DOIUrl":"https://doi.org/10.1109/LICS.2009.45","url":null,"abstract":"The problem of Pairwise CFL-reachability is to decide whether two given program locations in different threads are simultaneously reachable in the presence of recursion in threads and scheduling constraints imposed by synchronization primitives. Pairwise CFL-reachability is the core problem underlying concurrent program analysis especially dataflow analysis. Unfortunately, it is undecidable even for the most commonly used synchronization primitive, i.e., mutex locks. Lock usage in concurrent programs can be characterized in terms of lock chains, where a sequence of mutex locks is said to be chained if the scopes of adjacent (nonnested) mutexes overlap. Although pairwise reachability is known to decidable for threads interacting via nested locks, i.e., chains of length one, these techniques don’t extend to programs with non-nested locks used in crucial applications like databases and device drivers. In this paper, we exploit the fact that lock usage patterns in real life programs do not produce unbounded lock chains. For such programs, we show that pairwise CFL-reachability becomes decidable. Our new results narrow the decidability gap for pairwise CFL-reachability by providing a more refined characterization for it in terms of boundedness of lock chains rather than the current state-of-the-art, i.e., nestedness of locks (chains of length one).","PeriodicalId":415902,"journal":{"name":"2009 24th Annual IEEE Symposium on Logic In Computer Science","volume":"96 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2009-08-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122972644","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2009-08-11DOI: 10.1017/S095679681500009X
Thorsten Altenkirch, Neil Ghani, P. Hancock, Conor McBride, Peter Morris
We show that the syntactically rich notion of inductive families can be reduced to a core type theory with a fixed number of type constructors exploiting the novel notion of indexed containers. Indexed containers generalize simple containers, capturing strictly positive families instead of just strictly positive types, without having to extend the core type theory. Other applications of indexed containers include data type-generic programming and reasoning about polymorphic functions. The construction presented here has been formalized using the Agda system.
{"title":"Indexed Containers","authors":"Thorsten Altenkirch, Neil Ghani, P. Hancock, Conor McBride, Peter Morris","doi":"10.1017/S095679681500009X","DOIUrl":"https://doi.org/10.1017/S095679681500009X","url":null,"abstract":"We show that the syntactically rich notion of inductive families can be reduced to a core type theory with a fixed number of type constructors exploiting the novel notion of indexed containers. Indexed containers generalize simple containers, capturing strictly positive families instead of just strictly positive types, without having to extend the core type theory. Other applications of indexed containers include data type-generic programming and reasoning about polymorphic functions. The construction presented here has been formalized using the Agda system.","PeriodicalId":415902,"journal":{"name":"2009 24th Annual IEEE Symposium on Logic In Computer Science","volume":"74 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2009-08-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133986028","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2009-08-11DOI: 10.2168/LMCS-6(3:22)2010
Jérôme Leroux
The reachability problem for Vector Addition Systems (VASs) is a central problem of net theory. The general problem is known decidable by algorithms exclusively based on the classical Kosaraju-Lambert-Mayr-Sacerdote-Tenney decomposition. This decomposition is used in this paper to prove that the Parikh images of languages accepted by VASs are semi-pseudo-linear; a class that extends the semi-linear sets, a.k.a. the sets definable in the Presburger arithmetic. We provide an application of this result; we prove that a final configuration is not reachable from an initial one if and only if there exists a Presburger formula denoting a forward inductive invariant that contains the initial configuration but not the final one. Since we can decide if a Preburger formula denotes an inductive invariant, we deduce that there exist checkable certificates of non-reachability. In particular, there exists a simple algorithm for deciding the general VAS reachability problem based on two semi-algorithms. A first one that tries to prove the reachability by enumerating finite sequences of actions and a second one that tries to prove the non-reachability by enumerating Presburger formulas.
{"title":"The General Vector Addition System Reachability Problem by Presburger Inductive Invariants","authors":"Jérôme Leroux","doi":"10.2168/LMCS-6(3:22)2010","DOIUrl":"https://doi.org/10.2168/LMCS-6(3:22)2010","url":null,"abstract":"The reachability problem for Vector Addition Systems (VASs) is a central problem of net theory. The general problem is known decidable by algorithms exclusively based on the classical Kosaraju-Lambert-Mayr-Sacerdote-Tenney decomposition. This decomposition is used in this paper to prove that the Parikh images of languages accepted by VASs are semi-pseudo-linear; a class that extends the semi-linear sets, a.k.a. the sets definable in the Presburger arithmetic. We provide an application of this result; we prove that a final configuration is not reachable from an initial one if and only if there exists a Presburger formula denoting a forward inductive invariant that contains the initial configuration but not the final one. Since we can decide if a Preburger formula denotes an inductive invariant, we deduce that there exist checkable certificates of non-reachability. In particular, there exists a simple algorithm for deciding the general VAS reachability problem based on two semi-algorithms. A first one that tries to prove the reachability by enumerating finite sequences of actions and a second one that tries to prove the non-reachability by enumerating Presburger formulas.","PeriodicalId":415902,"journal":{"name":"2009 24th Annual IEEE Symposium on Logic In Computer Science","volume":"6 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2009-08-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130064022","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper presents a new lower bound for the discrete strategy improvement algorithm for solving parity games due to Voege and Jurdzinski. First, we informally show which structures are difficult to solve for the algorithm. Second, we outline a family of games on which the algorithm requires exponentially many strategy iterations, answering in the negative the long-standing question whether this algorithm runs in polynomial time. Additionally we note that the same family of games can be used to prove a similar result w.r.t. the strategy improvement variant by Schewe as well as the strategy iteration for solving discounted payoff games due to Puri.
{"title":"An Exponential Lower Bound for the Parity Game Strategy Improvement Algorithm as We Know it","authors":"Oliver Friedmann","doi":"10.1109/LICS.2009.27","DOIUrl":"https://doi.org/10.1109/LICS.2009.27","url":null,"abstract":"This paper presents a new lower bound for the discrete strategy improvement algorithm for solving parity games due to Voege and Jurdzinski. First, we informally show which structures are difficult to solve for the algorithm. Second, we outline a family of games on which the algorithm requires exponentially many strategy iterations, answering in the negative the long-standing question whether this algorithm runs in polynomial time. Additionally we note that the same family of games can be used to prove a similar result w.r.t. the strategy improvement variant by Schewe as well as the strategy iteration for solving discounted payoff games due to Puri.","PeriodicalId":415902,"journal":{"name":"2009 24th Annual IEEE Symposium on Logic In Computer Science","volume":"105 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2009-08-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123863763","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
One-counter processes are pushdown systems over a singleton stack alphabet (plus a stack-bottom symbol). We study the complexity of two closely related verification problems over one-counter processes: model checking with the temporal logic EF, where formulas are given as directed acyclic graphs, and weak bisimilarity checking against finite systems. We show that both problems are $P^NP$-complete. This is achieved by establishing a close correspondence with the membership problem for a natural fragment of Presburger Arithmetic, which we show to be$P^NP$-complete. This fragment is also a suitable representation for the global versions of the problems. We also show that there already exists a fixed EF formula(resp. a fixed finite system) such that model checking (resp. weak bisimulation) over one-counter processes is hard for $P^{NP[log]}$. However, the complexity drops to $P$ if the one-counter process is fixed.
{"title":"On the Computational Complexity of Verifying One-Counter Processes","authors":"Stefan Göller, Richard Mayr, A. Lin","doi":"10.1109/LICS.2009.37","DOIUrl":"https://doi.org/10.1109/LICS.2009.37","url":null,"abstract":"One-counter processes are pushdown systems over a singleton stack alphabet (plus a stack-bottom symbol). We study the complexity of two closely related verification problems over one-counter processes: model checking with the temporal logic EF, where formulas are given as directed acyclic graphs, and weak bisimilarity checking against finite systems. We show that both problems are $P^NP$-complete. This is achieved by establishing a close correspondence with the membership problem for a natural fragment of Presburger Arithmetic, which we show to be$P^NP$-complete. This fragment is also a suitable representation for the global versions of the problems. We also show that there already exists a fixed EF formula(resp. a fixed finite system) such that model checking (resp. weak bisimulation) over one-counter processes is hard for $P^{NP[log]}$. However, the complexity drops to $P$ if the one-counter process is fixed.","PeriodicalId":415902,"journal":{"name":"2009 24th Annual IEEE Symposium on Logic In Computer Science","volume":"602 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2009-08-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123206618","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The discussion in the computer-science literature of the relative merits of linear- versus branching-time frameworks goes back to the early 1980s. One of the beliefs dominating this discussion has been that the linear-time framework is not expressive enough semantically, making linear-time logics lacking in expressiveness. In this work we examine the branching-linear issue from the perspective of process equivalence, which is one of the most fundamental concepts in concurrency theory, as defining a notion of equivalence essentially amounts to defining semantics for processes. We accept three principles that have been recently proposed for concurrent-process equivalence. The first principle takes contextual equivalence as the primary notion of equivalence. The second principle requires the description of a process to specify all relevant behavioral aspects of the process. The third principle requires observable process behavior to be reflected in its input/output behavior. It has been recently shown that under these principles trace semantics for nondeterministic transducers is fully abstract. Here we consider two extensions of the earlier model: probabilistic transducers and asynchronous transducers. We show that in both cases trace semantics is fully abstract.
{"title":"Trace Semantics is Fully Abstract","authors":"Sumit Nain, Moshe Y. Vardi","doi":"10.1109/LICS.2009.12","DOIUrl":"https://doi.org/10.1109/LICS.2009.12","url":null,"abstract":"The discussion in the computer-science literature of the relative merits of linear- versus branching-time frameworks goes back to the early 1980s. One of the beliefs dominating this discussion has been that the linear-time framework is not expressive enough semantically, making linear-time logics lacking in expressiveness. In this work we examine the branching-linear issue from the perspective of process equivalence, which is one of the most fundamental concepts in concurrency theory, as defining a notion of equivalence essentially amounts to defining semantics for processes. We accept three principles that have been recently proposed for concurrent-process equivalence. The first principle takes contextual equivalence as the primary notion of equivalence. The second principle requires the description of a process to specify all relevant behavioral aspects of the process. The third principle requires observable process behavior to be reflected in its input/output behavior. It has been recently shown that under these principles trace semantics for nondeterministic transducers is fully abstract. Here we consider two extensions of the earlier model: probabilistic transducers and asynchronous transducers. We show that in both cases trace semantics is fully abstract.","PeriodicalId":415902,"journal":{"name":"2009 24th Annual IEEE Symposium on Logic In Computer Science","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2009-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122278630","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Graded modal logic is the formal language obtained from ordinary modal logic by endowing its modal operators with cardinality constraints. Under the familiar possible-worlds semantics, these augmented modal operators receive interpretations such as "It is true at no fewer than 15 accessible worlds that...", or "It is true at no more than 2 accessible worlds that...". We investigate the complexity of satisfiability for this language over some familiar classes of frames. This problem is more challenging than its ordinary modal logic counterpart---especially in the case of transitive frames, where graded modal logic lacks the tree-model property. We obtain tight complexity bounds for the problem of determining the satisfiability of a given graded modal logic formula over the classes of frames characterized by any combination of reflexivity, seriality, symmetry, transitivity and the Euclidean property.
{"title":"A Note on the Complexity of the Satisfiability Problem for Graded Modal Logics","authors":"Yevgeny Kazakov, Ian Pratt-Hartmann","doi":"10.1109/LICS.2009.17","DOIUrl":"https://doi.org/10.1109/LICS.2009.17","url":null,"abstract":"Graded modal logic is the formal language obtained from ordinary modal logic by endowing its modal operators with cardinality constraints. Under the familiar possible-worlds semantics, these augmented modal operators receive interpretations such as \"It is true at no fewer than 15 accessible worlds that...\", or \"It is true at no more than 2 accessible worlds that...\". We investigate the complexity of satisfiability for this language over some familiar classes of frames. This problem is more challenging than its ordinary modal logic counterpart---especially in the case of transitive frames, where graded modal logic lacks the tree-model property. We obtain tight complexity bounds for the problem of determining the satisfiability of a given graded modal logic formula over the classes of frames characterized by any combination of reflexivity, seriality, symmetry, transitivity and the Euclidean property.","PeriodicalId":415902,"journal":{"name":"2009 24th Annual IEEE Symposium on Logic In Computer Science","volume":"6 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2009-05-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114193322","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2009-05-13DOI: 10.2168/LMCS-6(3:10)2010
K. Chatterjee, L. Doyen, T. Henzinger
Weighted automata are nondeterministic automata with numerical weights on transitions. They can define quantitative languages L that assign to each word w a real number L(w). In the case of infinite words, the value of a run is naturally computed as the maximum, limsup, liminf, limit average, or discounted sum of the transition weights. We study expressiveness and closure questions about these quantitative languages. We first show that the set of words with value greater than a threshold can be non-omega-regular for deterministic limit-average and discounted-sum automata, while this set is always omega-regular when the threshold is isolated (i.e., some neighborhood around the threshold contains no word). In the latter case, we prove that the omega-regular language is robust against small perturbations of the transition weights. We next consider automata with transition weights 0 or 1 and show that they are as expressive as general weighted automata in the limit-average case, but not in the discounted-sum case. Third, for quantitative languages L1 and L2, we consider the operations max(L1, L2), min(L1, L2), and 1-L1, which generalize the boolean operations on languages, as well as the sum L1 + L2. We establish the closure properties of all classes of quantitative languages with respect to these four operations.
{"title":"Expressiveness and Closure Properties for Quantitative Languages","authors":"K. Chatterjee, L. Doyen, T. Henzinger","doi":"10.2168/LMCS-6(3:10)2010","DOIUrl":"https://doi.org/10.2168/LMCS-6(3:10)2010","url":null,"abstract":"Weighted automata are nondeterministic automata with numerical weights on transitions. They can define quantitative languages L that assign to each word w a real number L(w). In the case of infinite words, the value of a run is naturally computed as the maximum, limsup, liminf, limit average, or discounted sum of the transition weights. We study expressiveness and closure questions about these quantitative languages. We first show that the set of words with value greater than a threshold can be non-omega-regular for deterministic limit-average and discounted-sum automata, while this set is always omega-regular when the threshold is isolated (i.e., some neighborhood around the threshold contains no word). In the latter case, we prove that the omega-regular language is robust against small perturbations of the transition weights. We next consider automata with transition weights 0 or 1 and show that they are as expressive as general weighted automata in the limit-average case, but not in the discounted-sum case. Third, for quantitative languages L1 and L2, we consider the operations max(L1, L2), min(L1, L2), and 1-L1, which generalize the boolean operations on languages, as well as the sum L1 + L2. We establish the closure properties of all classes of quantitative languages with respect to these four operations.","PeriodicalId":415902,"journal":{"name":"2009 24th Annual IEEE Symposium on Logic In Computer Science","volume":"3 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2009-05-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134603323","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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