Parameterized Linear Temporal Logics Meet Costs:
Still not Costlier than LTL††thanks: Supported by the project “TriCS” (ZI 1516/1-1) of the German Research Foundation (DFG). A preliminary version of this work appeared in the proceedings of GandALF 2015 [Zimmermann15gandalf].
We continue the investigation of parameterized extensions of Linear Temporal Logic (LTL) that retain the attractive algorithmic properties of LTL: a polynomial space model checking algorithm and a doubly-exponential time algorithm for solving games. Alur et al. and Kupferman et al. showed that this is the case for Parametric LTL (PLTL) and PROMPT-LTL respectively, which have temporal operators equipped with variables that bound their scope in time. Later, this was also shown to be true for Parametric LDL (PLDL), which extends PLTL to be able to express all -regular properties.
Here, we generalize PLTL to systems with costs, i.e., we do not bound the scope of operators in time, but bound the scope in terms of the cost accumulated during time. Again, we show that model checking and solving games for specifications in PLTL with costs is not harder than the corresponding problems for LTL. Finally, we discuss PLDL with costs and extensions to multiple cost functions.
Parameterized linear temporal logics address a serious shortcoming of Linear-temporal Logic (LTL) [Pnueli77]: LTL is not able to express timing constraints, e.g., while expresses that every request is eventually answered by a response , the waiting time between requests and responses might diverge. This is typically not the desired behavior, but cannot be ruled out by LTL.
To overcome this shortcoming, Alur et al. introduced parameterized LTL (PLTL) [AlurEtessamiLaTorrePeled01], which extends LTL with parameterized operators of the form and , where and are variables. The formula expresses that every request is answered within an arbitrary, but fixed number of steps . Here, is a variable valuation, a mapping of variables to natural numbers. Typically, one is interested in whether a PLTL formula is satisfied with respect to some variable valuation. For example, the model checking problem asks whether a given transition system satisfies a given PLTL specification with respect to some , i.e., whether every path satisfies with respect to . Similarly, solving infinite games amounts to determining whether there is an such that Player has a strategy such that every play that is consistent with the strategy satisfies the winning condition with respect to . Alur et al. showed that the PLTL model checking problem is PSpace-complete. Kupferman et al. later considered PROMPT–LTL [KupfermanPitermanVardi09], which can be seen as the fragment of PLTL without the parameterized always operator, and showed that PROMPT–LTL model checking is still PSpace-complete and that PROMPT–LTL realizability, an abstract notion of infinite game, is 2ExpTime-complete. While the results of Alur et al. relied on involved pumping arguments, the results of Kupferman et al. where all based on the so-called alternating-color technique, which basically allows to reduce PROMPT–LTL to LTL. Furthermore, the result on realizability was extended to infinite games on graphs [Zimmermann13], again using the alternating-color technique.
Another serious shortcoming of LTL (and its parameterized variants) is their expressiveness: LTL is equi-expressive to first-order logic with order [Kamp68] and thus not as expressive as -regular expressions. This shortcoming was addressed by a long line of temporal logics [GiacomoVardi13, LeuckerSanchez07, Vardi11, VardiWolper94, Wolper1983] with regular expressions, finite automata, or grammar operators to obtain the full expressivity of the -regular languages. One of these logics is Linear Dynamic Logic (LDL), which has temporal operators and , where is a regular expression. For example, the formula holds in a word , if every request at a position such that matches , there is a position such that holds at and matches . Intuitively, the diamond operator corresponds to the eventuality of LTL, but is guarded by a regular expression. Dually, the box operator is a guarded always. Although LDL is more expressive than LTL, its algorithmic properties are similar: model checking is PSpace-complete and solving games is 2ExpTime-complete [Vardi11].
There are temporal logics whose expressiveness goes even beyond the -regular languages to capture properties of recursive programs, which are typically -contextfree. The visibly -contextfree languages [AlurM04] are an important class of languages located between the -regular ones and the -contextfree ones that enjoys desirable closure properties, which make it suitable to be employed in verification. Temporal logics that capture this class are visibly LTL [bozzelli14b], the fixed-point logic [bozzelli07], and visibly LDL (VLDL) [WeinertZimmermann15]. The logic visibly LTL enhances LTL with visibly rational expressions [bozzelli14a], and extends the linear-time -calculus [Vardi88] with non-local modalities. Finally, VLDL has the same temporal operators as LDL, but allows to use visibly pushdown automata instead of regular expressions as guards. For all these logics, model checking is ExpTime-complete, i.e., (under standard complexity theoretic assumptions) harder than the model checking problem for LTL. Furthermore, solving games with VLDL winning conditions is 3ExpTime-complete, again harder than solving LTL games. Thus, going beyond the -regular languages does increase the complexity of these problems at last.
All these logics tackle one shortcoming of LTL, but not both simultaneously. This was achieved for the first time by adding parameterized operators to LDL. The logic, called parameterized LDL (PLDL), has additional operators and with the expected semantics: the variables bound the scope of the operator. And even for this logic, which has parameters and is more expressive than LTL, model checking is still PSpace-complete and solving games is 2ExpTime-complete [FaymonvilleZimmermann14]. Again, these problems were solved by an application of the alternating-color technique. One has to overcome some technicalities, but the general proof technique is the same as for PROMPT–LTL.
The decision problems for the parameterized logics mentioned above are boundedness problems, e.g., one asks for an upper bound on the waiting times between requests and responses in case of the formula . Recently, more general boundedness problems in logics and automata received a lot of attention to obtain decidable quantitative extensions of monadic second-order logic and better synthesis algorithms. In general, boundedness problems are undecidable for automata with counters, but become decidable if the acceptance conditions can refer to boundedness properties of the counters, but the transition relation has no access to counter values. Recent advances include logics and automata with bounds [Bojanczyk04, BojanczykColcombet06], satisfiability algorithms for these logics [Bojanczyk11, Bojanczyk14, BojanczykTorunczyk12, Boom11], and regular cost-functions [Colcombet09]. However, these formalisms, while very expressive, are intractable and thus not suitable for verification and synthesis. Thus, less expressive formalisms were studied that appear more suitable for practical applications, e.g., finitary parity [ChatterjeeHenzingerHorn09], parity with costs [FijalkowZimmermann14], energy-parity [ChatterjeeDoyen10], mean-payoff-parity [ChatterjeeHenzingerJurdzinski05], consumption games [BCKN12], and the use of weighted automata for specifying quantitative properties [BCHJ09, CernyChatterjeeHenzingerRadhakrishnaSingh11]. In particular, the parity condition with cost is defined in graphs whose edges are weighted by natural numbers (interpreted as costs) and requires the existence of a bound such that almost every occurrence of an odd color is followed by an occurrence of a larger even color such that the cost between these positions is at most . Although strictly stronger than the classical parity condition, solving parity games with costs is as hard as solving parity games [FijalkowZimmermann14, MogaveroMS13].
We investigate parameterized temporal logics in a weighted setting similar to the one of parity conditions with costs: our graphs are equipped with cost-functions that label the edges with natural numbers and parameterized operators are now evaluated with respect to cost instead of time, i.e., the parameters bound the accumulated cost instead of the elapsed time. Thus, the formula requires that every request is answered with cost at most . We show the following results about PLTL with costs (cPLTL):
First, we refine the alternating-color technique to the cost-setting, which requires to tackle some technical problems induced by the fact that accumulated cost, unlike time, does not increase in every step, e.g., if an edge with cost zero is traversed. In particular, infinite paths with finite cost have to be taken care of appropriately.
Second, we show that Kupferman et al.’s proofs based on the alternating-color technique can be adapted to the cost-setting as well. For model checking, we again obtain PSpace-completeness while solving games is still 2ExpTime-complete.
Third, we consider PLDL with costs (cPLDL), which is defined as expected: the diamond and the box operator may be equipped with parameters bounding their scope. Again, the complexity does not increase: model checking is PSpace-complete while solving games is 2ExpTime-complete.
Fourth, we generalize both logics to a setting with multiple cost-functions. Now, the parameterized temporal operators have another parameter that determines the cost-function under which they are evaluated. Even these extensions do not increase complexity: model checking is again PSpace-complete while solving games is still 2ExpTime-complete.
Fifth, we also study the optimization variant of the model checking and the game problem for these logics: here, one is interested in finding the optimal variable valuation for which a given transition system satisfies the specification. For example, for the request-response condition one is interested in minimizing the waiting times between requests and responses. For cPLTL and cPLDL, we show that the model checking optimization problem can be solved in polynomial space while the optimization problem for infinite games can be solved in triply-exponential time. These results are similar to the ones obtained for PLTL [AlurEtessamiLaTorrePeled01, Zimmermann13]. In particular, the exponential gap between the decision and the optimization variant of solving infinite games exists already for PLTL. Whether this gap can be closed is an open problem. A first step towards this direction was made by giving an approximation algorithm for this problem with doubly-exponential running time [TentrupWeinertZimmermann15].
The paper is structured as follows: in Section 2, we introduce cPLTL and discuss basic properties. Then, in Section 3, we extend the alternating-color technique to the setting with costs, which we apply in Section 4 to the model checking problem and in Section 5 to solve infinite games. In Section 6, we extend these results to cPLDL and to multiple cost-functions. Finally, in Section 8, we investigate model checking and game-solving as optimization problems.
2 Parametric LTL with Costs
Let be an infinite set of variables and let be a set of atomic propositions. The formulas of cPLTL are given by the grammar
where and . We use the derived operators and for some fixed , , and . Furthermore, we use and as shorthand for and , respectively. Additional derived operators are introduced on page 1.
The set of subformulas of a cPLTL formula is denoted by and we define the size of to be the cardinality of . Furthermore, we define
to be the set of variables parameterizing eventually operators in ,
to be the set of variables parameterizing always operators in . Furthermore, denotes the set of all variables appearing in .
cPLTL is evaluated on so-called cost-traces (traces for short) of the form
which encode the evolution of the system in terms of the atomic propositions that hold true in each time instance, and the cost of changing the system state. The cost of the trace is defined as , which might be infinite. A finite cost-trace is required to begin and end with an element of . The cost of a finite cost-trace is defined as .
Furthermore, we require the existence of a distinguished atomic proposition such that all cost-traces satisfy if and only if , i.e., indicates that the last step had non-zero cost. We use the proposition to reason about costs: for example, we are able to express whether a trace has cost zero and whether a trace has cost . In the following, we will ensure that all our systems only allow traces that satisfy this assumption.
Also, to evaluate formulas we need to instantiate the variables parameterizing the temporal operators. To this end, we define a variable valuation to be a mapping . Now, we can define the model relation between a cost-trace , a position of , a variable valuation , and a cPLTL formula as follows:
if and only if ,
if and only if ,
if and only if and ,
if and only if or ,
if and only if ,
if and only if there exists a such that and for every in the range ,
if and only if for every : either or there exists a in the range such that ,
if and only if there exists a with
such that , and
if and only if for every with
Note that we recover the semantics of PLTL as the special case where every is equal to one.
For the sake of brevity, we write instead of and say that is a model of with respect to . For variable-free formulas, we even drop the and write .
As usual for parameterized temporal logics, the use of variables has to be restricted: bounding eventually and always operators by the same variable leads to an undecidable satisfiability problem [AlurEtessamiLaTorrePeled01].
A cPLTL formula is well-formed, if .
In the following, we only consider well-formed formulas and omit the qualifier “well-formed”. Also, we will denote variables in by and variables in by , if the formula is clear from context.
We consider the following fragments of cPLTL. Let be a cPLTL formula:
is an LTL formula, if .
is a formula, if .
is a formula, if .
Every LTL, , and every formula is well-formed by definition.
The formula is satisfied with respect to , if every request (a position where holds) is followed by a response (a position where holds) such that the cost of the infix between the request and the response is at most .
The (max-) parity condition with costs [FijalkowZimmermann14] can be expressed111Note that the bound in the parity condition with costs may depend on the trace while one typically uses global bounds for cPLTL (see, e.g., Section 4 and Section 5). However, for games in finite arenas (and thus also for model checking) these two variants coincide [FijalkowZimmermann14]. in cPLTL via
where is the maximal color, which we assume w.l.o.g. to be even. However, the Streett condition with costs [FijalkowZimmermann14] cannot be expressed in cPLTL, as it is defined with respect to multiple cost functions, one for each Streett pair. We extend cPLTL to multiple cost functions in Section 7.
As for PLTL, one can also parameterize the until and the release operator and also consider bounds of the form “”. However, this does not increase expressiveness of the logic. Formally, we define
if and only if there exists a with
such that and for every in the range ,
if and only if for every with
: or there exists a in the range such that ,
if and only if there exists a with
such that , and
if and only if for every with satisfies .
if and only if there exists a with
such that and for every in the range , and
if and only if for every with
: or there exists a in the range such that .
Let denote equivalence of the formulas and , i.e., for every , every , and every , we have if and only if . Then, we have the following equivalences (which also restrict the use of variables as defined in Definition 1):
Note that we defined cPLTL formulas to be in negation normal form. Nevertheless, a negation can be pushed to the atomic propositions using the duality of the operators. Thus, we can define the negation of a cPLTL formula.
For every cPLTL formula there exists an efficiently constructible cPLTL formula s.t.
if and only if for every , every , and every ,
If is well-formed, then so is .
If is an LTL formula, then so is .
If is a formula, then is a formula and vice versa.
We construct by induction over the construction of using the dualities of the operators:
The latter four claims of Lemma 1 follow from the definition of while the first one can be shown by a straightforward induction over ’s construction. ∎
Another important property of parameterized logics is monotonicity: increasing (decreasing) the values for parameterized eventuality operators (parameterized always operators) preserves satisfaction.
Let be a cPLTL formula and let and be variable valuations satisfying for every and for every . If , then .
Especially, if we are interested in checking whether a formula is satisfied with respect to some , we can always recursively replace every subformula by , as this is equivalent to with respect to every variable valuation mapping to zero, which is the smallest possible value for . Note that we have to ignore the current truth value of , as it indicates the cost of the last transition, not the cost of the next one.
3 The Alternating-Color Technique for Costs
Fix a fresh atomic proposition . We say that a cost-trace
is a coloring of a cost trace
if and for every , i.e., and only differ in the truth values of the new proposition . A position is a changepoint of , if or if the truth value of in and differs. A block of is an infix of such that and are successive changepoints. If a coloring has only finitely many changepoints, then we refer to its suffix starting at the last changepoint as its tail, i.e., the coloring is the concatenation of a finite number of blocks and its tail.
Let . We say that is -bounded if every block and its tail (if it has one) has cost at most . Dually, we say that is -spaced, if every block has cost at least . Note that we do not have a requirement on the cost of the tail in this case.
Given a formula , let denote the LTL formula obtained from by recursively replacing every subformula by
Intuitively, the relativized formula requires to be satisfied within at most one changepoint. On bounded and spaced colorings, and are “equivalent”.
Lemma 3 (cp. Lemma 2.1 of [KupfermanPitermanVardi09])
Let be a cost-trace and let be a formula.
Let for some variable valuation . Then, for every -spaced coloring of , where .
Let for some -bounded coloring of . Then, , where for every .
Note that and its colorings coincide on their cost. Hence, when speaking about the cost of an infix or suffix, we do not have to specify whether we refer to or to a coloring of .
1.) Fix a -spaced coloring of , where . We show that implies by induction over the construction of .
The only non-trivial case is the one of a parameterized eventuality: thus, assume , i.e., there is a with and . By induction hypothesis, we have . As is -spaced, i.e., the cost of each block is at least , there is at most one changepoint between (and including) the positions and in . Hence, , if , and otherwise. Thus, .
2.) Dually, fix a -bounded coloring of and define the variable valuation with for every . We show that implies by induction over the construction of .
Again, the only non-trivial case is the one of a parameterized eventuality: thus, let . We assume (the other case is dual). Then, we have , i.e., is satisfied at some position such that there is at most one changepoint between (and including) the positions and in . As is -bounded, this implies that the cost of the infix is bounded by . Furthermore, applying the induction hypothesis yields . Hence, . ∎
4 Model Checking
A transition system consists of a finite directed graph , an initial state , a labeling function , and a cost function . We encode the weights in binary, although the algorithms we present in this section and their running times and space requirements are oblivious to the exact weights. Furthermore, we assume that every state has at least one successor to spare us from dealing with finite paths. Recall our requirement on cost-traces having a distinguished atomic property indicating the sign of the cost of the previous transition. Thus, we require to satisfy the following property: if , then for every edge leading to . Dually, if , then for every edge .
A path through is a sequence with and for every . Its cost-trace is defined as
which satisfies our assumption on the proposition .
The transition system satisfies a cPLTL formula with respect to a variable valuation , if the trace of every path through satisfies with respect to . The cPLTL model checking problem asks, given a transition system and a cPLTL formula , whether satisfies with respect to some .
The cPLTL model checking problem is PSpace-complete.
The proof we give below is a generalization of the one for PROMPT–LTL by Kupferman et al. [KupfermanPitermanVardi09]. We begin by showing PSpace-membership. First note that we can restrict ourselves to formulas: given a cPLTL formula , let denote the formula obtained by recursively replacing every subformula by . Due to Lemma 11 and the discussion below it, every transition system satisfies with respect to some if and only if satisfies with respect to some .
Next, we show how to apply the alternating-color: recall that the classical algorithm for LTL model checking searches for a fair path, i.e., one that visits infinitely many accepting states, in the product of with a Büchi automaton recognizing the models of the negated specification. If such a path exists, then does not satisfy the specification, as the fair path contains a path through and an accepting run of the automaton on its trace, i.e., the trace does not satisfy the specification. If there is no such fair path, then satisfies the specification.
For cPLTL we have to find such a path for every in order to show that does not satisfy the specification with respect to any . To this end, one relativizes the specification as described in Section 3 and builds an automaton for the negation of the relativized formula in conjunction with a formula that ensures that every ultimately periodic model is both -bounded and -spaced for some appropriate and . Then, we search for a pumpable fair path in the product of the system and the Büchi automaton recognizing the models of the negated specification, which is non-deterministically labeled by . Applying Lemma 3 and pumping a fair path through the product appropriately yields a counterexample for every . Thus, model checking is reduced to finding a pumpable fair path. Let us stress again that this algorithm is similar to the one for PROMPT–LTL, we just have to pay attention to some intricacies stemming from the fact that we want to bound the cost, not the waiting time: there might be paths with finite cost, which have to be dealt with appropriately.
Recall that is the distinguished atomic proposition used to relativize cPLTL formulas. A colored Büchi graph with costs consists of a finite direct graph , an initial vertex , a labeling function , a cost-function , and a set of accepting vertices. A path is pumpable, if each of its blocks induced by contains a vertex repetition such that the cycle formed by the repetition has non-zero cost222Note that our definition is more involved than the one of Kupferman et al., since we require a cycle with non-zero cost instead of any circle.. Note that we do not have a requirement on the cost of the tail, if the path has one. The path is fair, if it visits infinitely often. The pumpable non-emptiness problem asks, given a colored Büchi graph with costs, whether it has an initial pumpable fair path.
If a colored Büchi graph with costs has an initial pumpable fair path, then also one of the form with , where is the number of vertices of the graph.
Let be an arbitrary initial pumpable fair path. First, assume it has only finitely many changepoints. If there are two blocks that start with the same vertex, then we can remove all blocks in between and obtain another initial pumpable fair path. Thus, we can assume that has at most blocks. Furthermore, the length of each block can be bounded by by removing cycles while retaining the state repetition with non-zero cost and at least one accepting vertex (provided the block has one). Now, consider the tail: by removing infixes one can find a cycle of length at most containing an accepting vertex and a path of length at most leading from the last changepoint to a vertex on the cycle. Hence, we define to be the prefix containing all blocks and the path leading to the cycle and define to be the cycle. Then, we have and is an initial pumpable fair path.
On the other hand, if contains infinitely many changepoints, then we can remove blocks and shorten other blocks as described above until we have constructed a prefix such that has the desired properties. In this case, we can assume that the first position of is a changepoint by “rotating” appropriately and appending a suitable prefix of it to . ∎
Let be a transition system and let be a formula. Furthermore, consider the LTL formula
which is satisfied by a cost-trace, if the trace has infinitely many changepoints if and only if333Here, we use our assumption on indicating the sign of the costs. it has cost . Now, let be a nondeterministic Büchi automaton recognizing the models of the LTL formula , which we can pick such that its number of states is bounded exponentially in . Now, define the colored Büchi graph with costs
if and only if and ,
[cp. Lemma 4.2 of [KupfermanPitermanVardi09]] does not satisfy with respect to any if and only if has an initial pumpable fair path.
Let not satisfy with respect to any variable valuation. Fix , where is the largest cost in , and define the valuation by for every . As does not satisfy with respect to , there is a path through with . Thus, due to Lemma 3.2, every -bounded coloring of does not satisfy .
Now, let be a -bounded and -spaced coloring of which starts with not holding true. Such a coloring can always be constructed, as is the largest cost appearing in . Note that satisfies by construction. Thus, we have , i.e., there is an accepting run of on . Consider the path
where , which is fair by construction. We claim that it is pumpable: consider a block, which is -spaced. Thus, it contains at least many edges with non-zero cost, enough to enforce a vertex repetition with non-zero cost in between. To this end, one takes the sets of vertices visited between the -th and the -th edge with non-zero cost (including the -th edge). This yields non-empty sets of vertices of that coincide in their third component, as we are within one block. However, there are only many such vertices, which yields the desired repetition.
Now, consider the converse implication and let be an arbitrary variable valuation. We show that does not satisfy with respect to . Due to Lemma 11, it is sufficient to show that does not satisfy with respect to the valuation mapping every variable to .
Fix an initial pumpable fair path of . It has a vertex repetition in every block such that the induced cycle has non-zero cost. We pump each such cycle times to obtain the path
By construction, is a path through and
is a coloring of its trace . Also, is an accepting run of on , i.e., . Lastly, is -spaced by construction, as the cost-function of is induced by the one of .
Now, we are ready to prove Theorem 4.1.
PSpace-hardness holds already for LTL model checking [SistlaClarke85], which is a special case of cPLTL model checking. Membership is witnessed by the following algorithm: check whether the colored Büchi graph has an initial pumpable fair path, which is correct due to Lemma 5. But as the graph is of exponential size, it has to be constructed and tested for non-emptiness on-the-fly.
Due to Lemma 4, it suffices to check for the existence of an ultimately periodic path such that , i.e., is exponential in the size of and quadratic in the size of . To this end, one guesses a vertex (the first vertex of ) and checks the following reachability properties:
Is reachable from via a path where each block contains a cycle with non-zero cost?
Is reachable from via a non-empty path that visits an accepting vertex and which either has no changepoint or where each block contains a cycle with non-zero cost? In this case, we also require that and the last vertex on the path from to guessed in item 1.) differ on their third component in order to make a changepoint. This spares us from having a block that spans and .
All these reachability problems can be solved in non-deterministic polynomial space, as a successor of a vertex of can be guessed and verified in polymonial time and the length of the paths to be guessed is bounded by , which can be represented with polynomially many bits. ∎
Furthermore, by applying both directions of the proof of Lemma 5, we obtain an exponential upper bound on the values of a satisfying variable valuation, if one exists. This is asymptotically tight, as one can already show exponential lower bounds for PROMPT–LTL [KupfermanPitermanVardi09].
Fix a transition system and a cPLTL-formula such that satisfies with respect to some . Then, satisfies with respect to a valuation that is bounded exponentially in the size of and linearly in the number of states of and in the maximal cost in .
Dually, one can show the existence of an exponential variable valuation that witnesses whether a given specification is satisfied with respect to every variable valuation. The following lemma states the contrapositive of this statement, which we prove using pumping arguments that are similar to the ones for the analogous results for and [FaymonvilleZimmermann15].
Fix a transition system and a -formula such that does not satisfy with respect to every . Then, does not satisfy with respect to a valuation that is bounded exponentially in the size of and linearly in the number of states of and in the maximal cost in .
Let be a Büchi automaton recognizing the models of , which is of exponential size in . Define , where is the largest cost in , and let be the variable valuation mapping every variable to . We consider the contrapositive and show: if there is an such that does not satisfy with respect to , then does not satisfy with respect to .
Thus, assume there is an and a path such that . Due to upwards-monotonicity we can assume w.l.o.g. that maps all variables to the same value, call it .
Let be a -bounded and -spaced -coloring of that starts with not holding true in the first position, which can always be constructed as is the largest cost. Applying Lemma 3.1 shows that satisfies . Furthermore, it satisfies by construction. Fix some accepting run of on and consider an arbitrary block of : if the run does not visit an accepting state during the block, we can remove (if necessary) infixes of the block where the run reaches the same state before and after the infix and where the state of at the beginning and the end of the infix are the same, until the block has length at most and thus cost at most .
On the other hand, assume the run visits at least one accepting state during the block. Fix one such position. Then, we can remove infixes as above between the beginning of the block and the position before the accepting state is visited and between the position after the accepting state is reached and before the end of the block. What remains is a block whose length is at most , at it has most many positions before the designated position, this position itself, and at most many after the designated position. Hence, the block has cost at most .
5 Infinite Games
An arena consists of a finite directed graph , a partition of , an initial vertex , a labeling , and a cost function . Again, we encode the weights in binary, although the algorithms we present here and their running times and space requirements are oblivious to the exact weights. Also, we again assume that every vertex has at least one successor to avoid dealing with finite paths. Also, we again ensure our requirement on the proposition to indicate the sign of the costs in a cost-trace: if , then we require for every edge leading to . Dually, if , then for every edge .
A play is a path through starting in and its cost-trace is defined as
A strategy for Player is a mapping with for every and . A play is consistent with if for every with .
A cPLTL game consists of an arena and a winning condition , which is a cPLTL formula. A strategy for Player is winning with respect to some variable valuation , if the trace of every play that is consistent with satisfies the winning condition with respect to .
We are interested in determining whether Player has a winning strategy for a given cPLTL game, and in determining a winning strategy for her if this is the case, which we refer to as solving the game.
Determining whether Player has a winning strategy in a given cPLTL game is 2ExpTime-complete. Furthermore, a winning strategy (if one exists) can be computed in doubly-exponential time.
Our proof technique is a generalization of the one for infinite games with PLTL winning conditions [Zimmermann13], which in turn extended Kupferman et al.’s solution for the PROMPT–LTL realizability problem [KupfermanPitermanVardi09]. First, we note that it is again sufficient to consider formulas, as we are interested in the existence of a variable valuation (see the discussion below Lemma 11). Next, we apply the alternating-color technique: to this end, we modify the arena to allow Player to produce colorings of plays of the original arena and use the relativized winning condition, i.e., we reduce the problem to a game with LTL winning condition. The winner (and a winning strategy) of such a game can be computed in doubly-exponential time [PnueliRosner89, PnueliRosner89a].
To allow for the coloring, we double the vertices of the arena, additionally label one copy with and the other not, and split every move into two: first, the player whose turn it is picks an outgoing edge, then Player decides in which copy she wants to visit the target, thereby picking the truth value of .
Formally, given an arena , the extended arena consists of
for every and and
A path through the new arena has the form for some path through , where and . Also, we have . Note that we use the costs in only to argue the correctness of our construction, not to define the winning condition for the game in .
Also, note that the additional choice vertices of the form have to be ignored when it comes to evaluating the winning condition on the trace of a play. Thus, we consider games with LTL winning conditions under so-called blinking semantics: Player wins a play under blinking semantics, if satisfies the winning condition ; otherwise, Player wins. Winning strategies under blinking semantics are defined as expected. Determining whether Player has a winning strategy for a given game with LTL winning condition under blinking semantics is 2ExpTime-complete, which can be shown by a slight variation of the proof for LTL games under classical semantics [PnueliRosner89, PnueliRosner89a]. Furthermore, if Player has a winning strategy for such a game, then also a finite-state one of at most doubly-exponential size in .
Such a strategy for an arena is given by a memory structure with a finite set of memory states, an initial memory state , and an update function , and by a next-move function satisfying for every and every . The function is defined via and . Then, the strategy implemented by and is defined by . The size of is (slightly abusively) defined as .
Given a game with winning condition , define as above and let , where . Recall that is satisfied by a cost-trace, if the trace has infinitely many changepoints if and only if it has cost .
[cp. Lemma 3.1 of [KupfermanPitermanVardi09]] Player has a winning strategy for with respect to some if and only if she has a winning strategy for under blinking semantics.
Let be a winning strategy for Player in with respect to some fixed and define . We define a strategy for as follows:
if , which implies . Thus, at a non-choice vertex, Player mimics the behavior of . At choice vertices, she alternates between the two copies of the arena every time the cost has exceeded : let