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11
The complexity of βreduction in low orders
, 2000
"... We study the complexity of βreduction for redexes of order 2,3 and 4. The obtained results are: evaluation of boolean expressions can be reduced to βreduction of order 2 and βreduction of order 2 is in O(nlog n), βreduction of order 3 is complete for PTIME, and βreduction of order 4 is complete ..."
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We study the complexity of βreduction for redexes of order 2,3 and 4. The obtained results are: evaluation of boolean expressions can be reduced to βreduction of order 2 and βreduction of order 2 is in O(nlog n), βreduction of order 3 is complete for PTIME, and βreduction of order 4 is complete for PSPACE. 1
Extracting Herbrand Disjunctions by Functional Interpretation
"... Abstract. Carrying out a suggestion by Kreisel, we adapt Gödel’s functional interpretation to ordinary firstorder predicate logic(PL) and thus devise an algorithm to extract Herbrand terms from PLproofs. The extraction is carried out in an extension of PL to higher types. The algorithm consists of ..."
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Abstract. Carrying out a suggestion by Kreisel, we adapt Gödel’s functional interpretation to ordinary firstorder predicate logic(PL) and thus devise an algorithm to extract Herbrand terms from PLproofs. The extraction is carried out in an extension of PL to higher types. The algorithm consists of two main steps: first we extract a functional realizer, next we compute the βnormalform of the realizer from which the Herbrand terms can be read off. Even though the extraction is carried out in the extended language, the terms are ordinary PLterms. In contrast to approaches to Herbrand’s theorem based on cut elimination or εelimination this extraction technique is, except for the normalization step, of low polynomial complexity, fully modular and furthermore allows an analysis of the structure of the Herbrand terms, in the spirit of Kreisel ([13]), already prior to the normalization step. It is expected that the implementation of functional interpretation in Schwichtenberg’s MINLOG system can be adapted to yield an efficient Herbrandterm extraction tool. 1.
The complexity of model checking higherorder fixpoint logic
 Logical Methods in Computer Science
"... Abstract. HigherOrder Fixpoint Logic (HFL) is a hybrid of the simply typed λcalculus and the modal µcalculus. This makes it a highly expressive temporal logic that is capable of expressing various interesting correctness properties of programs that are not expressible in the modal µcalculus. Thi ..."
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Abstract. HigherOrder Fixpoint Logic (HFL) is a hybrid of the simply typed λcalculus and the modal µcalculus. This makes it a highly expressive temporal logic that is capable of expressing various interesting correctness properties of programs that are not expressible in the modal µcalculus. This paper provides complexity results for its model checking problem. In particular we consider its fragments HFL k,m which are formed using types of bounded order k and arity m only. We establish kExpTimecompleteness for model checking each HFL k,m fragment. For the upper bound we use fixpoint elimination to obtain reachability games that are singlyexponential in the size of the formula and kfold exponential in the size of the underlying transition system. These games can be solved in deterministic linear time. As a simple consequence we obtain an ExpTime upper bound on the expression complexity of each HFL k,m. The lower bound is established by a reduction from the word problem for alternating (k − 1)fold exponential space bounded Turing Machines. Since there are fixed machines of that type whose word problems are kExpTimehard already we obtain, as a corollary, kExpTimecompleteness for the data complexity of HFL k,m already when m ≥ 4. This also yields a hierarchy result in expressive power. 1.
Bounding Skeletons, Locally Scoped Terms and Exact Bounds for Linear Head Reduction
"... Abstract. Bounding skeletons were recently introduced as a tool to study the length of interactions in Hyland/Ong game semantics. In this paper, we investigate the precise connection between them and execution of typed λterms. Our analysis sheds light on a new condition on λterms, called local sc ..."
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Abstract. Bounding skeletons were recently introduced as a tool to study the length of interactions in Hyland/Ong game semantics. In this paper, we investigate the precise connection between them and execution of typed λterms. Our analysis sheds light on a new condition on λterms, called local scope. We show that the reduction of locally scoped terms matches closely that of bounding skeletons. Exploiting this connection, we give upper bound to the length of linear head reduction for simplytyped locally scoped terms. General terms lose this connection to bounding skeletons. To compensate for that, we show that λlifting allows us to transform any λterm into a locally scoped one. We deduce from that an upper bound to the length of linear head reduction for arbitrary simplytyped λterms. In both cases, we prove the asymptotical optimality of the upper bounds by providing matching lower bounds. 1
THE COMPLEXITY OF MODEL CHECKING HIGHER ORDER FIXPOINT LOGIC
"... Abstract. Higher Order Fixpoint Logic (HFL) is a hybrid of the simply typed λcalculus and the modal µcalculus. This makes it a highly expressive temporal logic that is capable of expressing various interesting correctness properties of programs that are not expressible in the modal µcalculus. Thi ..."
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Abstract. Higher Order Fixpoint Logic (HFL) is a hybrid of the simply typed λcalculus and the modal µcalculus. This makes it a highly expressive temporal logic that is capable of expressing various interesting correctness properties of programs that are not expressible in the modal µcalculus. This paper provides complexity results for its model checking problem. In particular we consider its fragments HFL k,m which are formed using types of bounded order k and arity m only. We establish kExpTimecompleteness for model checking each HFL k,m fragment. For the upper bound we use fixpoint elimination to obtain reachability games that are singlyexponential in the size of the formula and kfold exponential in the size of the underlying transition system. These games can be solved in deterministic linear time. As a simple consequence we obtain an ExpTime upper bound on the expression complexity of each HFL k,m. The lower bound is established by a reduction from the word problem for alternating (k − 1)fold exponential space bounded Turing Machines. Since there are fixed machines of that type whose word problems are kExpTimehard already we obtain, as a corollary, kExpTimecompleteness for the data complexity of HFL k,m already when m ≥ 4. This also yields a hierarchy result in expressive power. 1.
Applications of Proof Interpretations
, 2006
"... In this thesis, the author describes the research carried out during his PhDstudies. The results presented in this thesis have previously been published in a ..."
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In this thesis, the author describes the research carried out during his PhDstudies. The results presented in this thesis have previously been published in a
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"... Extraction de programmes optimisés à partir de preuves (nonconstructives) par l’interprétation Dialectica (monotone) légère Pour l’obtention du titre de docteur au cadre du Laboratoire d’Informatique (LIX) de l ’ École Polytechnique et en cotutelle de thèse au cadre du Groupe de Logique Mathématiq ..."
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Extraction de programmes optimisés à partir de preuves (nonconstructives) par l’interprétation Dialectica (monotone) légère Pour l’obtention du titre de docteur au cadre du Laboratoire d’Informatique (LIX) de l ’ École Polytechnique et en cotutelle de thèse au cadre du Groupe de Logique Mathématique de l’Université de Munich (GKLI)
Argument Structure as an Interface Between Form and Meaning
"... Abstract. In this paper we argue for a specific form of semantic representations, which pair together a DRS with a list of statements that declare how the variables occurring in the DRS are to be handled under merge. This eliminates the need for an external accounting device. Additionally, it greatl ..."
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Abstract. In this paper we argue for a specific form of semantic representations, which pair together a DRS with a list of statements that declare how the variables occurring in the DRS are to be handled under merge. This eliminates the need for an external accounting device. Additionally, it greatly simplifies the amount of structure needed to account for the behaviour of contextually determined variables as well as argument identification in general. Factoring out structural from semantic information greatly simplifies the representations, and therefore it is expected to also simplifiy learning these structures. 1.