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11
Dynamic Logic
 Handbook of Philosophical Logic
, 1984
"... ed to be true under the valuation u iff there exists an a 2 N such that the formula x = y is true under the valuation u[x=a], where u[x=a] agrees with u everywhere except x, on which it takes the value a. This definition involves a metalogical operation that produces u[x=a] from u for all possibl ..."
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Cited by 817 (7 self)
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ed to be true under the valuation u iff there exists an a 2 N such that the formula x = y is true under the valuation u[x=a], where u[x=a] agrees with u everywhere except x, on which it takes the value a. This definition involves a metalogical operation that produces u[x=a] from u for all possible values a 2 N. This operation becomes explicit in DL in the form of the program x := ?, called a nondeterministic or wildcard assignment. This is a rather unconventional program, since it is not effective; however, it is quite useful as a descriptive tool. A more conventional way to obtain a square root of y, if it exists, would be the program x := 0 ; while x < y do x := x + 1: (1) In DL, such programs are firstclass objects on a par with formulas, complete with a collection of operators for forming compound programs inductively from a basis of primitive programs. To discuss the effect of the execution of a program on the truth of a formula ', DL uses a modal construct <>', which
HigherOrder Pushdown Trees Are Easy
, 2002
"... We show that the monadic secondorder theory of an infinite tree recognized by a higherorder pushdown automaton of any level is decidable. We also show that trees recognized by pushdown automata of level n coincide with trees generated by safe higherorder grammars of level n. Our decidability resu ..."
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Cited by 43 (2 self)
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We show that the monadic secondorder theory of an infinite tree recognized by a higherorder pushdown automaton of any level is decidable. We also show that trees recognized by pushdown automata of level n coincide with trees generated by safe higherorder grammars of level n. Our decidability result extends the result of Courcelle on algebraic (pushdown of level 1) trees and our own result on trees of level 2.
Regular sets of higherorder pushdown stacks
 In MFCS
, 2005
"... Abstract. It is a wellknown result that the set of reachable stack contents in a pushdown automaton is a regular set of words. We consider the more general case of higherorder pushdown automata and investigate, with a particular stress on effectiveness and complexity, the natural notion of regular ..."
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Cited by 15 (4 self)
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Abstract. It is a wellknown result that the set of reachable stack contents in a pushdown automaton is a regular set of words. We consider the more general case of higherorder pushdown automata and investigate, with a particular stress on effectiveness and complexity, the natural notion of regularity for higherorder stacks: a set of level k stacks is regular if it is obtained by a regular sequence of level k operations. We prove that any regular set of level k stacks admits a normalized representation and we use it to show that the regular sets of a given level form an effective Boolean algebra. In fact, this notion of regularity coincides with the notion of monadic second order definability over the canonical structure associated to level k stacks. Finally, we consider the link between regular sets of stacks and families of infinite graphs defined by higherorder pushdown systems.
Symbolic backwardsreachability analysis for higherorder pushdown systems
 IN FOSSACS
, 2007
"... Higherorder pushdown systems (PDSs) generalise pushdown systems through the use of higherorder stacks; that is, a nested “stack of stacks ” structure. These systems may be used to model higherorder programs and are closely related to the Caucal hierarchy of infinite graphs and safe higherorder ..."
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Cited by 9 (2 self)
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Higherorder pushdown systems (PDSs) generalise pushdown systems through the use of higherorder stacks; that is, a nested “stack of stacks ” structure. These systems may be used to model higherorder programs and are closely related to the Caucal hierarchy of infinite graphs and safe higherorder recursion schemes. We generalise higherorder PDSs to higherorder Alternating PDSs (APDSs) and consider the backwardsreachability problem over these systems. This builds on and extends previous work into pushdown systems and contextfree higherorder processes in a nontrivial manner. In particular, we show that the set of configurations from which a regular set of higherorder APDS configurations is reachable is regular and computable in nEXPTIME. In fact, the problem is nEXPTIMEcomplete. We show that this work has several applications in the verification of higherorder PDSs, such as lineartime modelchecking, alternationfree µcalculus modelchecking and the computation of winning regions of reachability games.
Symbolic Reachability Analysis of HigherOrder ContextFree Processes
 ContextFree Processes, FSTTCS’04, LNCS 3328
, 2004
"... Abstract. We consider the problem of symbolic reachability analysis of higherorder contextfree processes. These models are generalizations of the contextfree processes (also called BPA processes) where each process manipulates a data structure which can be seen as a nested stack of stacks. Our ma ..."
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Cited by 8 (1 self)
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Abstract. We consider the problem of symbolic reachability analysis of higherorder contextfree processes. These models are generalizations of the contextfree processes (also called BPA processes) where each process manipulates a data structure which can be seen as a nested stack of stacks. Our main result is that, for any higherorder contextfree process, the set of all predecessors of a given regular set of configurations is regular and effectively constructible. This result generalizes the analogous result which is known for level 1 contextfree processes. We show that this result holds also in the case of backward reachability analysis under a regular constraint on configurations. As a corollary, we obtain a symbolic model checking algorithm for the temporal logic E(U,X) with regular atomic predicates, i.e., the fragment of CTL restricted to the EU and EX modalities. 1
Iterated pushdown automata and sequences of rational numbers
, 2006
"... We introduce a link between automata of level k and treestructures. Thismethod leads to new decidability results about integer sequences. We also reduce some equality problems for sequences of rational numbers to the equivalence problem for deterministic automata of level k. ..."
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Cited by 3 (0 self)
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We introduce a link between automata of level k and treestructures. Thismethod leads to new decidability results about integer sequences. We also reduce some equality problems for sequences of rational numbers to the equivalence problem for deterministic automata of level k.
On Global Model Checking Trees Generated by HigherOrder Recursion Schemes
, 2009
"... Higherorder recursion schemes are systems of rewrite rules on typed nonterminal symbols, which can be used to define infinite trees. The Global Modal MuCalculus Model Checking Problem takes as input such a recursion scheme together with a modal µcalculus sentence and asks for a finite representa ..."
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Higherorder recursion schemes are systems of rewrite rules on typed nonterminal symbols, which can be used to define infinite trees. The Global Modal MuCalculus Model Checking Problem takes as input such a recursion scheme together with a modal µcalculus sentence and asks for a finite representation of the set of nodes in the tree generated by the scheme at which the sentence holds. Using a method that appeals to game semantics, we show that for an ordern recursion scheme, one can effectively construct a nondeterministic ordern collapsible pushdown automaton representing this set. The level of the automaton is strict in the sense that in general no nondeterministic order(n −1) automaton could do likewise (assuming the requisite hierarchy theorem). The question of determinisation is left open. As a corollary we can also construct an ordern collapsible pushdown automaton representing the constructible winning region of an ordern collapsible pushdown parity game.
Global ModelChecking of HigherOrder Pushdown Systems
, 2008
"... Pushdown systems equip a finite state system with an unbounded stack memory, and are thus infinite state. By recording the call history on the stack, these systems provide a natural model for recursive procedure calls. Modelchecking for pushdown systems has been wellstudied. The most successful im ..."
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Cited by 1 (1 self)
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Pushdown systems equip a finite state system with an unbounded stack memory, and are thus infinite state. By recording the call history on the stack, these systems provide a natural model for recursive procedure calls. Modelchecking for pushdown systems has been wellstudied. The most successful implementation of these techniques is the tool Moped, which has become an important component of verification suites such as Terminator, SLAM and Blast. Higherorder pushdown systems allow a more complex memory structure: a higherorder stack is a stack of lowerorder stacks. These systems form a robust hierarchy closely related to the Caucal hierarchy and higherorder recursion schemes. This latter connection demonstrates their importance as models for programs with higherorder functions. We study the global modelchecking problem for higherorder pushdown systems. In particular, we show how to compute the winning regions of twoplayer games with reachability, Büchi and parity conditions. Our approach extends the saturation methods of Bouajjani, Esparza and Maler for order1 pushdown systems, and Bouajjani and Meyer for higherorder pushdown systems with a single control state. These techniques begin with an automaton recognising (higherorder) stacks,
On Parikh Images of HigherOrder Pushdown Automata
"... We introduce the notion of semipolynomial sets, generalizing the notion of semilinear sets, and show that each semipolynomial set is the Parikh image of level 2 pushdown automata, which represent a special class of higherorder pushdown automata. ..."
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We introduce the notion of semipolynomial sets, generalizing the notion of semilinear sets, and show that each semipolynomial set is the Parikh image of level 2 pushdown automata, which represent a special class of higherorder pushdown automata.
Author manuscript, published in "ICALP'05, Portugal (2005)" Unsafe Grammars and Panic Automata
, 2008
"... Abstract. We show that the problem of checking if an infinite tree generated by a higherorder grammar of level 2 (hyperalgebraic) satisfies a given µcalculus formula (or, equivalently, if it is accepted by an alternating parity automaton) is decidable, actually 2Exptimecomplete. Consequently, th ..."
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Abstract. We show that the problem of checking if an infinite tree generated by a higherorder grammar of level 2 (hyperalgebraic) satisfies a given µcalculus formula (or, equivalently, if it is accepted by an alternating parity automaton) is decidable, actually 2Exptimecomplete. Consequently, the monadic secondorder theory of any hyperalgebraic tree is decidable, so that the safety restriction can be removed from our previous decidability result. The last result has been independently obtained by Aehlig, de Miranda and Ong. Our proof goes via a characterization of possibly unsafe secondorder grammars by a new variant of higherorder pushdown automata, which we call panic automata. In addition to the standard pop 1 and pop 2 operations, these automata have an option of a destructive move called panic. The modelchecking problem is then reduced to the problem of deciding the winner in a parity game over a suitable 2nd order pushdown system.