# Volume 15, Issue 3

2019

### 1. Guarded and Unguarded Iteration for Generalized Processes

Models of iterated computation, such as (completely) iterative monads, often depend on a notion of guardedness, which guarantees unique solvability of recursive equations and requires roughly that recursive calls happen only under certain guarding operations. On the other hand, many models of iteration do admit unguarded iteration. Solutions are then no longer unique, and in general not even determined as least or greatest fixpoints, being instead governed by quasi-equational axioms. Monads that support unguarded iteration in this sense are called (complete) Elgot monads. Here, we propose to equip (Kleisli categories of) monads with an abstract notion of guardedness and then require solvability of abstractly guarded recursive equations; examples of such abstractly guarded pre-iterative monads include both iterative monads and Elgot monads, the latter by deeming any recursive definition to be abstractly guarded. Our main result is then that Elgot monads are precisely the iteration-congruent retracts of abstractly guarded iterative monads, the latter being defined as admitting unique solutions of abstractly guarded recursive equations; in other words, models of unguarded iteration come about by quotienting models of guarded iteration.

### 2. Capturing Logarithmic Space and Polynomial Time on Chordal Claw-Free Graphs

We show that the class of chordal claw-free graphs admits LREC$_=$-definable canonization. LREC$_=$ is a logic that extends first-order logic with counting by an operator that allows it to formalize a limited form of recursion. This operator can be evaluated in logarithmic space. It follows that there exists a logarithmic-space canonization algorithm, and therefore a logarithmic-space isomorphism test, for the class of chordal claw-free graphs. As a further consequence, LREC$_=$ captures logarithmic space on this graph class. Since LREC$_=$ is contained in fixed-point logic with counting, we also obtain that fixed-point logic with counting captures polynomial time on the class of chordal claw-free graphs.

### 3. Subspace-Invariant AC$^0$ Formulas

We consider the action of a linear subspace $U$ of $\{0,1\}^n$ on the set of AC$^0$ formulas with inputs labeled by literals in the set $\{X_1,\overline X_1,\dots,X_n,\overline X_n\}$, where an element $u \in U$ acts on formulas by transposing the $i$th pair of literals for all $i \in [n]$ such that $u_i=1$. A formula is {\em $U$-invariant} if it is fixed by this action. For example, there is a well-known recursive construction of depth $d+1$ formulas of size $O(n{\cdot}2^{dn^{1/d}})$ computing the $n$-variable PARITY function; these formulas are easily seen to be $P$-invariant where $P$ is the subspace of even-weight elements of $\{0,1\}^n$. In this paper we establish a nearly matching $2^{d(n^{1/d}-1)}$ lower bound on the $P$-invariant depth $d+1$ formula size of PARITY. Quantitatively this improves the best known $\Omega(2^{\frac{1}{84}d(n^{1/d}-1)})$ lower bound for {\em unrestricted} depth $d+1$ formulas, while avoiding the use of the switching lemma. More generally, for any linear subspaces $U \subset V$, we show that if a Boolean function is $U$-invariant and non-constant over $V$, then its $U$-invariant depth $d+1$ formula size is at least $2^{d(m^{1/d}-1)}$ where $m$ is the minimum Hamming weight of a vector in $U^\bot \setminus V^\bot$.

### 4. Logical properties of random graphs from small addable classes

We establish zero-one laws and convergence laws for monadic second-order logic (MSO) (and, a fortiori, first-order logic) on a number of interesting graph classes. In particular, we show that MSO obeys a zero-one law on the class of connected planar graphs, the class of connected graphs of tree-width at most $k$ and the class of connected graphs excluding the $k$-clique as a minor. In each of these cases, dropping the connectivity requirement leads to a class where the zero-one law fails but a convergence law for MSO still holds.

### 5. Coherence for Frobenius pseudomonoids and the geometry of linear proofs

We prove coherence theorems for Frobenius pseudomonoids and snakeorators in monoidal bicategories. As a consequence we obtain a 3d notation for proofs in nonsymmetric multiplicative linear logic, with a geometrical notion of equivalence, and without the need for a global correctness criterion or thinning links. We argue that traditional proof nets are the 2d projections of these 3d diagrams.

### 6. Controlling a population

We introduce a new setting where a population of agents, each modelled by a finite-state system, are controlled uniformly: the controller applies the same action to every agent. The framework is largely inspired by the control of a biological system, namely a population of yeasts, where the controller may only change the environment common to all cells. We study a synchronisation problem for such populations: no matter how individual agents react to the actions of the controller, the controller aims at driving all agents synchronously to a target state. The agents are naturally represented by a non-deterministic finite state automaton (NFA), the same for every agent, and the whole system is encoded as a 2-player game. The first player (Controller) chooses actions, and the second player (Agents) resolves non-determinism for each agent. The game with m agents is called the m -population game. This gives rise to a parameterized control problem (where control refers to 2 player games), namely the population control problem: can Controller control the m-population game for all m in N whatever Agents does?

### 7. A Forgotten Theory of Proofs ?

Looking at MacLane's thesis on proof theory in the light of combinatory logic

### 8. An efficient algorithm to decide periodicity of $b$-recognisable sets using LSDF convention

Let $b$ be an integer strictly greater than $1$. Each set of nonnegative integers is represented in base $b$ by a language over $\{0, 1, \dots, b - 1\}$. The set is said to be $b$-recognisable if it is represented by a regular language. It is known that ultimately periodic sets are $b$-recognisable, for every base $b$, and Cobham's theorem implies the converse: no other set is $b$-recognisable in every base $b$. We consider the following decision problem: let $S$ be a set of nonnegative integers that is $b$-recognisable, given as a finite automaton over $\{0,1, \dots, b - 1\}$, is $S$ periodic? Honkala showed in 1986 that this problem is decidable. Later on, Leroux used in 2005 the convention to write number representations with the least significant digit first (LSDF), and designed a quadratic algorithm to solve a more general problem. We use here LSDF convention as well and give a structural description of the minimal automata that accept periodic sets. Then, we show that it can be verified in linear time if a minimal automaton meets this description. In general, this yields a $O(b \log(n))$ procedure to decide whether an automaton with $n$ states accepts an ultimately periodic set of nonnegative integers.

### 9. Normalizing the Taylor expansion of non-deterministic {\lambda}-terms, via parallel reduction of resource vectors

It has been known since Ehrhard and Regnier's seminal work on the Taylor expansion of $\lambda$-terms that this operation commutes with normalization: the expansion of a $\lambda$-term is always normalizable and its normal form is the expansion of the Böhm tree of the term. We generalize this result to the non-uniform setting of the algebraic $\lambda$-calculus, i.e. $\lambda$-calculus extended with linear combinations of terms. This requires us to tackle two difficulties: foremost is the fact that Ehrhard and Regnier's techniques rely heavily on the uniform, deterministic nature of the ordinary $\lambda$-calculus, and thus cannot be adapted; second is the absence of any satisfactory generic extension of the notion of Böhm tree in presence of quantitative non-determinism, which is reflected by the fact that the Taylor expansion of an algebraic $\lambda$-term is not always normalizable. Our solution is to provide a fine grained study of the dynamics of $\beta$-reduction under Taylor expansion, by introducing a notion of reduction on resource vectors, i.e. infinite linear combinations of resource $\lambda$-terms. The latter form the multilinear fragment of the differential $\lambda$-calculus, and resource vectors are the target of the Taylor expansion of $\lambda$-terms. We show the reduction of resource vectors contains the image of any $\beta$-reduction step, from which we deduce that Taylor expansion and normalization commute on the nose. We moreover identify a class […]

### 10. Classical System of Martin-Lof's Inductive Definitions is not Equivalent to Cyclic Proofs

A cyclic proof system, called CLKID-omega, gives us another way of representing inductive definitions and efficient proof search. The 2005 paper by Brotherston showed that the provability of CLKID-omega includes the provability of LKID, first order classical logic with inductive definitions in Martin-Löf's style, and conjectured the equivalence. The equivalence has been left an open question since 2011. This paper shows that CLKID-omega and LKID are indeed not equivalent. This paper considers a statement called 2-Hydra in these two systems with the first-order language formed by 0, the successor, the natural number predicate, and a binary predicate symbol used to express 2-Hydra. This paper shows that the 2-Hydra statement is provable in CLKID-omega, but the statement is not provable in LKID, by constructing some Henkin model where the statement is false.

### 11. Game characterizations and lower cones in the Weihrauch degrees

We introduce a parametrized version of the Wadge game for functions and show that each lower cone in the Weihrauch degrees is characterized by such a game. These parametrized Wadge games subsume the original Wadge game, the eraser and backtrack games as well as Semmes's tree games. In particular, we propose that the lower cones in the Weihrauch degrees are the answer to Andretta's question on which classes of functions admit game characterizations. We then discuss some applications of such parametrized Wadge games. Using machinery from Weihrauch reducibility theory, we introduce games characterizing every (transfinite) level of the Baire hierarchy via an iteration of a pruning derivative on countably branching trees.

### 12. Divergence and unique solution of equations

We study proof techniques for bisimilarity based on unique solution of equations. We draw inspiration from a result by Roscoe in the denotational setting of CSP and for failure semantics, essentially stating that an equation (or a system of equations) whose infinite unfolding never produces a divergence has the unique-solution property. We transport this result onto the operational setting of CCS and for bisimilarity. We then exploit the operational approach to: refine the theorem, distinguishing between different forms of divergence; derive an abstract formulation of the theorems, on generic LTSs; adapt the theorems to other equivalences such as trace equivalence, and to preorders such as trace inclusion. We compare the resulting techniques to enhancements of the bisimulation proof method (the `up-to techniques'). Finally, we study the theorems in name-passing calculi such as the asynchronous $\pi$-calculus, and use them to revisit the completeness part of the proof of full abstraction of Milner's encoding of the $\lambda$-calculus into the $\pi$-calculus for Lévy-Longo Trees.

### 13. On the commutativity of the powerspace constructions

We investigate powerspace constructions on topological spaces, with a particular focus on the category of quasi-Polish spaces. We show that the upper and lower powerspaces commute on all quasi-Polish spaces, and show more generally that this commutativity is equivalent to the topological property of consonance. We then investigate powerspace constructions on the open set lattices of quasi-Polish spaces, and provide a complete characterization of how the upper and lower powerspaces distribute over the open set lattice construction.

### 14. Companions, Causality and Codensity

In the context of abstract coinduction in complete lattices, the notion of compatible function makes it possible to introduce enhancements of the coinduction proof principle. The largest compatible function, called the companion, subsumes most enhancements and has been proved to enjoy many good properties. Here we move to universal coalgebra, where the corresponding notion is that of a final distributive law. We show that when it exists, the final distributive law is a monad, and that it coincides with the codensity monad of the final sequence of the given functor. On sets, we moreover characterise this codensity monad using a new abstract notion of causality. In particular, we recover the fact that on streams, the functions definable by a distributive law or GSOS specification are precisely the causal functions. Going back to enhancements of the coinductive proof principle, we finally obtain that any causal function gives rise to a valid up-to-context technique.

### 15. A categorical semantics for causal structure

We present a categorical construction for modelling causal structures within a general class of process theories that include the theory of classical probabilistic processes as well as quantum theory. Unlike prior constructions within categorical quantum mechanics, the objects of this theory encode fine-grained causal relationships between subsystems and give a new method for expressing and deriving consequences for a broad class of causal structures. We show that this framework enables one to define families of processes which are consistent with arbitrary acyclic causal orderings. In particular, one can define one-way signalling (a.k.a. semi-causal) processes, non-signalling processes, and quantum $n$-combs. Furthermore, our framework is general enough to accommodate recently-proposed generalisations of classical and quantum theory where processes only need to have a fixed causal ordering locally, but globally allow indefinite causal ordering. To illustrate this point, we show that certain processes of this kind, such as the quantum switch, the process matrices of Oreshkov, Costa, and Brukner, and a classical three-party example due to Baumeler, Feix, and Wolf are all instances of a certain family of processes we refer to as $\textrm{SOC}_n$ in the appropriate category of higher-order causal processes. After defining these families of causal structures within our framework, we give derivations of their operational behaviour using simple, diagrammatic axioms.

### 16. Quantitative Automata under Probabilistic Semantics

Automata with monitor counters, where the transitions do not depend on counter values, and nested weighted automata are two expressive automata-theoretic frameworks for quantitative properties. For a well-studied and wide class of quantitative functions, we establish that automata with monitor counters and nested weighted automata are equivalent. We study for the first time such quantitative automata under probabilistic semantics. We show that several problems that are undecidable for the classical questions of emptiness and universality become decidable under the probabilistic semantics. We present a complete picture of decidability for such automata, and even an almost-complete picture of computational complexity, for the probabilistic questions we consider.

### 17. On the Succinctness of Atoms of Dependency

Propositional team logic is the propositional analog to first-order team logic. Non-classical atoms of dependence, independence, inclusion, exclusion and anonymity can be expressed in it, but for all atoms except dependence only exponential translations are known. In this paper, we systematically compare their succinctness in the existential fragment, where the splitting disjunction only occurs positively, and in full propositional team logic with unrestricted negation. By introducing a variant of the Ehrenfeucht-Fra\"{i}ssé game called formula size game into team logic, we obtain exponential lower bounds in the existential fragment for all atoms. In the full fragment, we present polynomial upper bounds also for all atoms.

### 18. Bounded Quantifier Instantiation for Checking Inductive Invariants

We consider the problem of checking whether a proposed invariant $\varphi$ expressed in first-order logic with quantifier alternation is inductive, i.e. preserved by a piece of code. While the problem is undecidable, modern SMT solvers can sometimes solve it automatically. However, they employ powerful quantifier instantiation methods that may diverge, especially when $\varphi$ is not preserved. A notable difficulty arises due to counterexamples of infinite size. This paper studies Bounded-Horizon instantiation, a natural method for guaranteeing the termination of SMT solvers. The method bounds the depth of terms used in the quantifier instantiation process. We show that this method is surprisingly powerful for checking quantified invariants in uninterpreted domains. Furthermore, by producing partial models it can help the user diagnose the case when $\varphi$ is not inductive, especially when the underlying reason is the existence of infinite counterexamples. Our main technical result is that Bounded-Horizon is at least as powerful as instrumentation, which is a manual method to guarantee convergence of the solver by modifying the program so that it admits a purely universal invariant. We show that with a bound of 1 we can simulate a natural class of instrumentations, without the need to modify the code and in a fully automatic way. We also report on a prototype implementation on top of Z3, which we used to verify several examples by Bounded-Horizon of bound 1.

### 19. Abstract Completion, Formalized

Completion is one of the most studied techniques in term rewriting and fundamental to automated reasoning with equalities. In this paper we present new correctness proofs of abstract completion, both for finite and infinite runs. For the special case of ground completion we present a new proof based on random descent. We moreover extend the results to ordered completion, an important extension of completion that aims to produce ground-complete presentations of the initial equations. We present new proofs concerning the completeness of ordered completion for two settings. Moreover, we revisit and extend results of Métivier concerning canonicity of rewrite systems. All proofs presented in the paper have been formalized in Isabelle/HOL.

### 20. Parity Games with Weights

Quantitative extensions of parity games have recently attracted significant interest. These extensions include parity games with energy and payoff conditions as well as finitary parity games and their generalization to parity games with costs. Finitary parity games enjoy a special status among these extensions, as they offer a native combination of the qualitative and quantitative aspects in infinite games: The quantitative aspect of finitary parity games is a quality measure for the qualitative aspect, as it measures the limit superior of the time it takes to answer an odd color by a larger even one. Finitary parity games have been extended to parity games with costs, where each transition is labeled with a nonnegative weight that reflects the costs incurred by taking it. We lift this restriction and consider parity games with costs with arbitrary integer weights. We show that solving such games is in NP $\cap$ coNP, the signature complexity for games of this type. We also show that the protagonist has finite-state winning strategies, and provide tight pseudo-polynomial bounds for the memory he needs to win the game. Naturally, the antagonist may need infinite memory to win. Moreover, we present tight bounds on the quality of winning strategies for the protagonist. Furthermore, we investigate the problem of determining, for a given threshold $b$, whether the protagonist has a strategy of quality at most $b$ and show this problem to be EXPTIME-complete. The protagonist […]

### 21. Query learning of derived $\omega$-tree languages in polynomial time

We present the first polynomial time algorithm to learn nontrivial classes of languages of infinite trees. Specifically, our algorithm uses membership and equivalence queries to learn classes of $\omega$-tree languages derived from weak regular $\omega$-word languages in polynomial time. The method is a general polynomial time reduction of learning a class of derived $\omega$-tree languages to learning the underlying class of $\omega$-word languages, for any class of $\omega$-word languages recognized by a deterministic Büchi acceptor. Our reduction, combined with the polynomial time learning algorithm of Maler and Pnueli [1995] for the class of weak regular $\omega$-word languages yields the main result. We also show that subset queries that return counterexamples can be implemented in polynomial time using subset queries that return no counterexamples for deterministic or non-deterministic finite word acceptors, and deterministic or non-deterministic Büchi $\omega$-word acceptors. A previous claim of an algorithm to learn regular $\omega$-trees due to Jayasrirani, Begam and Thomas [2008] is unfortunately incorrect, as shown in Angluin [2016].

### 22. Alternating, private alternating, and quantum alternating realtime automata

We present new results on realtime alternating, private alternating, and quantum alternating automaton models. Firstly, we show that the emptiness problem for alternating one-counter automata on unary alphabets is undecidable. Then, we present two equivalent definitions of realtime private alternating finite automata (PAFAs). We show that the emptiness problem is undecidable for PAFAs. Furthermore, PAFAs can recognize some nonregular unary languages, including the unary squares language, which seems to be difficult even for some classical counter automata with two-way input. Regarding quantum finite automata (QFAs), we show that the emptiness problem is undecidable both for universal QFAs on general alphabets, and for alternating QFAs with two alternations on unary alphabets. On the other hand, the same problem is decidable for nondeterministic QFAs on general alphabets. We also show that the unary squares language is recognized by alternating QFAs with two alternations.

### 23. The Data Complexity of Ontology-Mediated Queries with Closed Predicates

In the context of ontology-mediated querying with description logics (DLs), we study the data complexity of queries in which selected predicates can be closed (OMQCs). We provide a non-uniform analysis, aiming at a classification of the complexity into tractable and non-tractable for ontologies in the lightweight DLs DL-Lite and EL, and the expressive DL ALCHI. At the level of ontologies, we prove a dichotomy between FO-rewritable and coNP-complete for DL-Lite and between PTime and coNP-complete for EL. The meta problem of deciding tractability is proved to be in PTime. At the level of OMQCs, we show that there is no dichotomy (unless NP equals PTime) if both concept and role names can be closed. If only concept names can be closed, we tightly link the complexity of query evaluation to the complexity of surjective CSPs. We also identify a class of OMQCs based on ontologies formulated in DL-Lite that are guaranteed to be tractable and even FO-rewritable.

### 24. Approximability in the GPAC

Most of the physical processes arising in nature are modeled by either ordinary or partial differential equations. From the point of view of analog computability, the existence of an effective way to obtain solutions of these systems is essential. A pioneering model of analog computation is the General Purpose Analog Computer (GPAC), introduced by Shannon as a model of the Differential Analyzer and improved by Pour-El, Lipshitz and Rubel, Costa and Graça and others. Its power is known to be characterized by the class of differentially algebraic functions, which includes the solutions of initial value problems for ordinary differential equations. We address one of the limitations of this model, concerning the notion of approximability, a desirable property in computation over continuous spaces that is however absent in the GPAC. In particular, the Shannon GPAC cannot be used to generate non-differentially algebraic functions which can be approximately computed in other models of computation. We extend the class of data types using networks with channels which carry information on a general complete metric space $X$; for example $X=C(R,R)$, the class of continuous functions of one real (spatial) variable. We consider the original modules in Shannon's construction (constants, adders, multipliers, integrators) and we add \emph{(continuous or discrete) limit} modules which have one input and one output. We then define an L-GPAC to be a network built with $X$-stream channels and […]

### 25. Interaction Graphs: Exponentials

This paper is the fourth of a series exposing a systematic combinatorial approach to Girard's Geometry of Interaction (GoI) program. The GoI program aims at obtaining particular realisability models for linear logic that accounts for the dynamics of cut-elimination. This fourth paper tackles the complex issue of defining exponential connectives in this framework. For that purpose, we use the notion of \emph{graphings}, a generalisation of graphs which was defined in earlier work. We explain how to define a GoI for Elementary Linear Logic (ELL) with second-order quantification, a sub-system of linear logic that captures the class of elementary time computable functions.

### 26. A diagrammatic calculus of fermionic quantum circuits

We introduce the fermionic ZW calculus, a string-diagrammatic language for fermionic quantum computing (FQC). After defining a fermionic circuit model, we present the basic components of the calculus, together with their interpretation, and show how the main physical gates of interest in FQC can be represented in our language. We then list our axioms, and derive some additional equations. We prove that the axioms provide a complete equational axiomatisation of the monoidal category whose objects are systems of finitely many local fermionic modes (LFMs), with maps that preserve or reverse the parity of states, and the tensor product as monoidal product. We achieve this through a procedure that rewrites any diagram in a normal form. As an example, we show how the statistics of a fermionic Mach-Zehnder interferometer can be calculated in the diagrammatic language. We conclude by giving a diagrammatic treatment of the dual-rail encoding, a standard method in optical quantum computing used to perform universal quantum computation.

### 27. Regular tree languages in low levels of the Wadge Hierarchy

In this article we provide effective characterisations of regular languages of infinite trees that belong to the low levels of the Wadge hierarchy. More precisely we prove decidability for each of the finite levels of the hierarchy; for the class of the Boolean combinations of open sets $BC(\Sigma_1^0)$ (i.e. the union of the first $\omega$ levels); and for the Borel class $\Delta_2^0$ (i.e. for the union of the first $\omega_1$ levels).

### 28. Higher-dimensional automata modeling shared-variable systems

The purpose of this paper is to provide a construction to model shared-variable systems using higher-dimensional automata which is compositional in the sense that the parallel composition of completely independent systems is modeled by the standard tensor product of HDAs and nondeterministic choice is represented by the coproduct.

### 29. Definability and Interpolation within Decidable Fixpoint Logics

We look at characterizing which formulas are expressible in rich decidable logics such as guarded fixpoint logic, unary negation fixpoint logic, and guarded negation fixpoint logic. We consider semantic characterizations of definability, as well as effective characterizations. Our algorithms revolve around a finer analysis of the tree-model property and a refinement of the method of moving back and forth between relational logics and logics over trees.

### 30. On noncommutative extensions of linear logic

Pomset logic introduced by Retoré is an extension of linear logic with a self-dual noncommutative connective. The logic is defined by means of proof-nets, rather than a sequent calculus. Later a deep inference system BV was developed with an eye to capturing Pomset logic, but equivalence of system has not been proven up to now. As for a sequent calculus formulation, it has not been known for either of these logics, and there are convincing arguments that such a sequent calculus in the usual sense simply does not exist for them. In an on-going work on semantics we discovered a system similar to Pomset logic, where a noncommutative connective is no longer self-dual. Pomset logic appears as a degeneration, when the class of models is restricted. Motivated by these semantic considerations, we define in the current work a semicommutative multiplicative linear logic}, which is multiplicative linear logic extended with two nonisomorphic noncommutative connectives (not to be confused with very different Abrusci-Ruet noncommutative logic). We develop a syntax of proof-nets and show how this logic degenerates to Pomset logic. However, a more interesting problem than just finding yet another noncommutative logic is to find a sequent calculus for this logic. We introduce decorated sequents, which are sequents equipped with an extra structure of a binary relation of reachability on formulas. We define a decorated sequent calculus for semicommutative logic and prove that it is cut-free, […]

### 31. The parameterized space complexity of model-checking bounded variable first-order logic

The parameterized model-checking problem for a class of first-order sentences (queries) asks to decide whether a given sentence from the class holds true in a given relational structure (database); the parameter is the length of the sentence. We study the parameterized space complexity of the model-checking problem for queries with a bounded number of variables. For each bound on the quantifier alternation rank the problem becomes complete for the corresponding level of what we call the tree hierarchy, a hierarchy of parameterized complexity classes defined via space bounded alternating machines between parameterized logarithmic space and fixed-parameter tractable time. We observe that a parameterized logarithmic space model-checker for existential bounded variable queries would allow to improve Savitch's classical simulation of nondeterministic logarithmic space in deterministic space $O(\log^2n)$. Further, we define a highly space efficient model-checker for queries with a bounded number of variables and bounded quantifier alternation rank. We study its optimality under the assumption that Savitch's Theorem is optimal.

### 32. Nash Equilibrium and Bisimulation Invariance

Game theory provides a well-established framework for the analysis of concurrent and multi-agent systems. The basic idea is that concurrent processes (agents) can be understood as corresponding to players in a game; plays represent the possible computation runs of the system; and strategies define the behaviour of agents. Typically, strategies are modelled as functions from sequences of system states to player actions. Analysing a system in such a setting involves computing the set of (Nash) equilibria in the concurrent game. However, we show that, with respect to the above model of strategies (arguably, the "standard" model in the computer science literature), bisimilarity does not preserve the existence of Nash equilibria. Thus, two concurrent games which are behaviourally equivalent from a semantic perspective, and which from a logical perspective satisfy the same temporal logic formulae, may nevertheless have fundamentally different properties (solutions) from a game theoretic perspective. Our aim in this paper is to explore the issues raised by this discovery. After illustrating the issue by way of a motivating example, we present three models of strategies with respect to which the existence of Nash equilibria is preserved under bisimilarity. We use some of these models of strategies to provide new semantic foundations for logics for strategic reasoning, and investigate restricted scenarios where bisimilarity can be shown to preserve the existence of Nash […]

### 33. The Complexity of Flat Freeze LTL

We consider the model-checking problem for freeze LTL on one-counter automata (OCA). Freeze LTL extends LTL with the freeze quantifier, which allows one to store different counter values of a run in registers so that they can be compared with one another. As the model-checking problem is undecidable in general, we focus on the flat fragment of freeze LTL, in which the usage of the freeze quantifier is restricted. In a previous work, Lechner et al. showed that model checking for flat freeze LTL on OCA with binary encoding of counter updates is decidable and in 2NEXPTIME. In this paper, we prove that the problem is, in fact, NEXPTIME-complete no matter whether counter updates are encoded in unary or binary. Like Lechner et al., we rely on a reduction to the reachability problem in OCA with parameterized tests (OCA(P)). The new aspect is that we simulate OCA(P) by alternating two-way automata over words. This implies an exponential upper bound on the parameter values that we exploit towards an NP algorithm for reachability in OCA(P) with unary updates. We obtain our main result as a corollary. As another application, relying on a reduction by Bundala and Ouaknine, one obtains an alternative proof of the known fact that reachability in closed parametric timed automata with one parametric clock is in NEXPTIME.