Consider any subformula and FLC environment for every free in The heart of the construction is decode that shows how to decode in HFL the transition system representing an FLC formula..
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with the HFL formula flc-diag expressing a property of the form prescribed by
Theorem 2 The properties of the terms decode and init defined in Table 4 are
given by the following theorem:
Theorem 3 Let be a closed well-named FLC formula over the action set A.
1.
2.
Consider any subformula and FLC environment
for every free in
The heart of the construction is decode that shows how to decode (in HFL) the
transition system representing an FLC formula Its definition given in Table 4
is easiest understood on the basis of its property given in Theorem 3, with
the variable read as standing for the function representing an
environment and the variable in each of the cases read as standing
for the singleton set On an argument the formula is decoded in
cases according to its outermost form which in turn is inferred based on which
of the propositional constants holds in (standing for For all
constructs other than variables and fixed points, their corresponding cases can be
understood by close analogy with the HFL-translation of these constructs given
in Table 3 together with the understanding that and yield singleton
sets including the corresponding subformulas of and that for constant literals
term yields the set of states in which the literal holds If is a variable,
we evaluate the environment on the set (as given by the property
of which is yielded by the term If is a fixed point formula, we
correspondingly bind (using or v) a new variable and decode the subformula
of (given by but in an environment that is obtained by modifying the
current environment to map (given by to (the case-term used for
the environment argument to in the fixed point cases yields this updated
environment) This ensures that when decoding the subformulas of any use
of a variable corresponding to this recursive definition will be decoded as
The decoding of the fixed-point cases explains the presence of the environment
argument in defining decode Finally, it is worth noting that: (1) decode is a
recursive definition of a higher-order function, and (2) because decode is defined
by case-analysis, it is not monotonic in the argument (standing for the formula
being decoded) These features of HFL are therefore crucial to its definition
As an easy corollary of Theorem 3, we get the relevant properties of the terms
flc-sem and flc-diag
Corollary 3 For any closed well-named FLC formula over the action set A,
we have that
Trang 2A Higher Order Modal Fixed Point Logic 527
1.
Combined with Theorem 2 this gives us that the HFL formula flc-diag is a
characteristic formula for a property of finite transition systems that is
inex-pressible in FLC, and thus HFL is strictly more expressive than FLC even over
finite transition systems
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Trang 4Author Index
Abdulla, Parosh Aziz 35
Amadio, Roberto M 68
Andrews, Tony 1
Baldan, Paolo 83
Baudru, Nicolas 99
Berger, Martin 115
131 Bollig, Benedikt 146
Borgström, Johannes 161
Bozga, Liana 177
Brázdil, Tomáš 193
Briais, Sébastien 161
Brookes, Stephen 16
Bruns, Glenn 209
Bugliesi, Michele 225
Caires, Luís 240
Cîrstea, Corina 258
Clarke, Edmund 276
Colazzo, Dario 225
Corradini, Andrea 83
Crafa, Silvia 225
Dal Zilio, Silvano 68
Danos, Vincent 292
Ene, Cristian 177
Gay, Simon 497
Groote, Jan Friso 308
Hirschkoff, Daniel 325
Jagadeesan, Radha 209
Jeffrey, Alan 209
Jonsson, Bengt 35
König, Barbara 83
340 355 Krivine, Jean 292
193, 371 Lakhnech, Yassine 177
Laroussinie, F 387
Leroux, Jérôme 402
Leucker, Martin 146
Lozes, Étienne 240
Ma, Qin 417 Maranget, Luc 417 Markey, Nicolas 387, 432 Melliès, Paul-André 448 Mokrushin, Leonid 340 Morin, Rémi 99 Nestmann, Uwe 161 Nilsson, Marcus 35 O’Hearn, Peter W 49 Pattinson, Dirk 258 Qadeer, Shaz 1 Rajamani, Sriram K 1 Raskin, Jean-François 432 Ravara, António 497
355 Rehof, Jakob 1 Riely, James 209 Saksena, Mayank 35 Schnoebelen, Philippe 371, 387
193 355 Sutre, Grégoire 402 Tabuada, Paulo 466 Talupur, Muralidhar 276 Thiagarajan, P.S 340 Touili, Tayssir 276 Varacca, Daniele 481 Vasconcelos, Vasco 497 Veith, Helmut 276 Viswanathan, Mahesh 512 Viswanathan, Ramesh 512 Völzer, Hagen 481 Walukiewicz, Igor 131 Willemse, Tim 308 Winskel, Glynn 481 Xie, Yichen 1
Yi, Wang 340
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