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Rank logic is dead, long live rank logic! [article]

Erich Grädel, Wied Pakusa
2015 arXiv   pre-print
Motivated by the search for a logic for polynomial time, we study rank logic (FPR) which extends fixed-point logic with counting (FPC) by operators that determine the rank of matrices over finite fields. While FPR can express most of the known queries that separate FPC from PTIME, nearly nothing was known about the limitations of its expressive power. In our first main result we show that the extensions of FPC by rank operators over different prime fields are incomparable. This solves an open
more » ... is solves an open question posed by Dawar and Holm and also implies that rank logic, in its original definition with a distinct rank operator for every field, fails to capture polynomial time. In particular we show that the variant of rank logic FPR* with an operator that uniformly expresses the matrix rank over finite fields is more expressive than FPR. One important step in our proof is to consider solvability logic FPS which is the analogous extension of FPC by quantifiers which express the solvability problem for linear equation systems over finite fields. Solvability logic can easily be embedded into rank logic, but it is open whether it is a strict fragment. In our second main result we give a partial answer to this question: in the absence of counting, rank operators are strictly more expressive than solvability quantifiers.
arXiv:1503.05423v1 fatcat:coz4qftg65go7dcttf6t3guwwa

Model-Theoretic Properties of ω-Automatic Structures

Faried Abu Zaid, Erich Grädel, Łukasz Kaiser, Wied Pakusa
2013 Theory of Computing Systems  
We investigate structural properties of ω-automatic presentations of infinite structures in order to sharpen our methods to determine whether a given structure is ω-automatic. We apply these methods to show that several classes of structures such as pairing functions and infinite integral domains do not have an ω-automatic model.
doi:10.1007/s00224-013-9508-6 fatcat:cmjmyiripfejxerkfltknqwf2a

Approximations of Isomorphism and Logics with Linear-Algebraic Operators [article]

Anuj Dawar, Erich Grädel, Wied Pakusa
2019 arXiv   pre-print
Invertible map equivalences are approximations of graph isomorphism that refine the well-known Weisfeiler-Leman method. They are parametrised by a number k and a set Q of primes. The intuition is that two graphs G and H which are equivalent with respect to k-Q-IM-equivalence cannot be distinguished by a refinement of k-tuples given by linear operators acting on vector spaces over fields of characteristic p, for any p in Q. These equivalences first appeared in the study of rank logic, but in
more » ... k logic, but in fact they can be used to delimit the expressive power of any extension of fixed-point logic with linear-algebraic operators. We define an infinitary logic with k variables and all linear-algebraic operators over finite vector spaces of characteristic p in Q and show that the k-Q-IM-equivalence is the natural notion of elementary equivalence for this logic. By means of a new and much deeper algebraic analysis of a generalized variant, for any prime p, of the CFI-structures due to Cai, F\"urer, and Immerman, we prove that, as long as Q is not the set of all primes, there is no k such that k-Q-IM-equivalence is the same as isomorphism. It follows that there are polynomial-time properties of graphs which are not definable in the infinitary logic with all Q-linear-algebraic operators and finitely many variables, which implies that no extension of fixed-point logic with linear-algebraic operators can capture PTIME, unless it includes such operators for all prime characteristics. Our analysis requires substantial algebraic machinery, including a homogeneity property of CFI-structures and Maschke's Theorem, an important result from the representation theory of finite groups.
arXiv:1902.06648v2 fatcat:dmyz6u4rqnhv3olcegxtmfe4oq

The Model-Theoretic Expressiveness of Propositional Proof Systems

Erich Grädel, Benedikt Pago, Wied Pakusa
2017 Leibniz International Proceedings in Informatics  
The formula Θ(G, H) consists of the following set of clauses: w∈Wi X[v → w] for each i < n, v ∈ V i (1) v∈Vi X[v → w] for each i < n, w ∈ W i (2) ¬(X[v 1 → w 1 ] ∧ X[v 2 → w 2 ]) if {(v 1 , w 1 ), (v 2  ... 
doi:10.18154/rwth-2020-09507 fatcat:s4ojkqjawbf4pf5wmi2xbnq6em

Definability of Cai-Fürer-Immerman Problems in Choiceless Polynomial Time

Wied Pakusa, Svenja Schalthöfer, Erkal Selman
2018 ACM Transactions on Computational Logic  
Choiceless Polynomial Time (CPT) is one of the most promising candidates in the search for a logic capturing Ptime. The question whether there is a logic that expresses exactly the polynomial-time computable properties of finite structures, which has been open for more than 30 years, is one of the most important and challenging problems in finite model theory. The strength of Choiceless Polynomial Time is its ability to perform isomorphism-invariant computations over structures, using
more » ... es, using hereditarily finite sets as data structures. But, as it preserves symmetries, it is choiceless in the sense that it cannot select an arbitrary element of a set-an operation which is crucial for many classical algorithms. CPT can define many interesting Ptime queries, including (the original version of) the Cai-Fürer-Immerman (CFI) query. The CFI query is particularly interesting because it separates fixed-point logic with counting from Ptime, and has since remained the main benchmark for the expressibility of logics within Ptime. The CFI construction associates with each connected graph a set of CFI-graphs that can be partitioned into exactly two isomorphism classes called odd and even CFI-graphs. The problem is to decide, given a CFI-graph, whether it is odd or even. In the original version, the underlying graphs are linearly ordered, and for this case, Dawar, Richerby and Rossman proved that the CFI query is CPT-definable. However, the CFI query over general graphs remains one of the few known examples for which CPT-definability is open. Our first contribution generalises the result by Dawar, Richerby and Rossman to the variant of the CFI query where the underlying graphs have colour classes of logarithmic size, instead of colour class size one. Secondly, we consider the CFI query over graph classes where the maximal degree is linear in the size of the graphs. For these classes, we establish CPT-definability using only sets of small, constant rank, which is known to be impossible for the general case. In our CFI-recognising procedures we strongly make use of the ability of CPT to create sets, rather than tuples only, and we further prove that, if CPT worked over tuples instead, no such procedure would be definable. We introduce a notion of "sequence-like objects" based on the structure of the graphs' symmetry groups, and we show that no CPT-program which only uses sequence-like objects can decide the CFI query over complete graphs, which have linear maximal degree. From a broader perspective, this generalises a result by Blass, Gurevich, and van den Bussche about the power of isomorphism-invariant machine models (for polynomial time) to a setting with counting. Lemma 19 (Action of Sym(V )). Let G = (V, E) be a complete graph and let S V = Aut(G) = Sym(V ). Then every π ∈ S V can be lifted to an automorphism ρ ∈ Γ = Aut(G T ) such that ρ(v * ) = ρ(w * ) if and only if π(v) = π(w) for all v, w ∈ V . Proof. Constructing ρ for a transposition (v, w) ∈ S V , one has to make sure that all edge gadgets {u, v} and {u, w}, and their neighbours in the vertex gadgets, are exchanged, and the parity of v * and w * is exchanged if necessary.
doi:10.1145/3154456 fatcat:b4nso4uqa5eqlfofh7lgczf2km

Defining Winning Strategies in Fixed-Point Logic

Felix Canavoi, Erich Gradel, Simon Lessenich, Wied Pakusa
2015 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science  
We study definability questions for positional winning strategies in infinite games on graphs. The quest for efficient algorithmic constructions of winning regions and winning strategies in infinite games, in particular parity games, is of importance in many branches of logic and computer science. A closely related, yet different, facet of this problem concerns the definability of winning regions and winning strategies in logical systems such as monadic second-order logic, least fixed-point
more » ... ast fixed-point logic LFP, the modal -calculus and some of its fragments. While a number of results concerning definability issues for winning regions have been established, so far almost nothing has been known concerning the definability of winning strategies. We make the notion of logical definability of positional winning strategies precise and study systematically the possibility of translations between definitions of winning regions and definitions of winning strategies. We present explicit LFP-definitions for winning strategies in games with relatively simple objectives, such as safety, reachability, eventual safety (Co-Büchi) and recurrent reachability (Büchi), and then prove, based on the Stage Comparison Theorem, that winning strategies for any class of parity games with a bounded number of priorities are LFP-definable. For parity games with an unbounded number of priorities, LFP-definitions of winning strategies are provably impossible on arbitrary (finite and infinite) game graphs. On finite game graphs however, this definability problem turns out to be equivalent to the fundamental open question about the algorithmic complexity of parity games. Indeed, based on a general argument about LFP-translations we prove that LFPdefinable winning strategies on the class of all finite parity games exist if, and only if, parity games can be solved in polynomial time, despite the fact that LFP is, in general, strictly weaker than polynomial time.
doi:10.1109/lics.2015.42 dblp:conf/lics/CanavoiGLP15 fatcat:3cndjmbxujgelmx63y7piyakqq

Characterising Choiceless Polynomial Time with First-Order Interpretations

Erich Gradel, Wied Pakusa, Svenja Schalthofer, Lukasz Kaiser
2015 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science  
Choiceless Polynomial Time (CPT) is one of the candidates in the quest for a logic for polynomial time. It is a strict extension of fixed-point logic with counting, but to date the question is open whether it expresses all polynomial-time properties of finite structures. We present here alternative characterisations of Choiceless Polynomial Time (with and without counting) based on iterated first-order interpretations. The fundamental mechanism of Choiceless Polynomial Time is the manipulation
more » ... s the manipulation of hereditarily finite sets over the input structure by means of set-theoretic operations and comprehension terms. While this is very convenient and powerful for the design of abstract computations on structures, it makes the analysis of the expressive power of CPT rather difficult. We aim to reduce this functional framework operating on higher-order objects to an approach that evaluates formulae on less complex objects. We propose a more model-theoretic formalism, called polynomial-time interpretation logic (PIL), that replaces the machinery of hereditarily finite sets and comprehension terms by traditional first-order interpretations, and handles counting by Härtig quantifiers. In our framework, computations on finite structures are captured by iterations of interpretations, and a run is a sequence of states, each of which is a finite structure of a fixed vocabulary. Our main result is that PIL has precisely the same expressive power as Choiceless Polynomial Time. We also analyse the structure of PIL and show that many of the logical formalisms or database languages that have been proposed in the quest for a logic for polynomial time reappear as fragments of PIL, obtained by restricting interpretations in a natural way (e.g. by omitting congruences or using only onedimensional interpretations).
doi:10.1109/lics.2015.68 dblp:conf/lics/GradelPSK15 fatcat:o6zudj655jdxzk54icw22ulauu

Definability of linear equation systems over groups and rings

Anuj Dawar, Erich Grädel, Bjarki Holm, Eryk Kopczynski, Wied Pakusa, Marc Herbstritt
2012 Annual Conference for Computer Science Logic  
doi:10.4230/lipics.csl.2012.213 dblp:conf/csl/DawarGHKP12 fatcat:trwv7mjlurbcndckxjlrlodrhm

A Finite-Model-Theoretic View on Propositional Proof Complexity [article]

Erich Grädel and Martin Grohe and Benedikt Pago and Wied Pakusa
2018 arXiv   pre-print
PAKUSA [ c] defines a linear order on { c} × Γ( c), as claimed.  ... 
arXiv:1802.09377v2 fatcat:uliyrq6kgjh3fnskxcuq6kq7py

Definability of linear equation systems over groups and rings

Anuj Dawar, Eryk Kopczynski, Bjarki Holm, Erich Grädel, Wied Pakusa, Arnaud Durand
2013 Logical Methods in Computer Science  
PAKUSA variable v − j for each v j , and the equation v j Lemma 4 . 4 ( 44 Non-solvability over chain rings).  ...  Pakusa [27] observed that the same approach works for the definability of the determinant and characteristic polynomial of matrices over prime rings Z p n .  ... 
doi:10.2168/lmcs-9(4:12)2013 fatcat:miw5iydrhjd5plmirn2yz6zv54

Editor: Stephan Kreutzer

Erich Grädel, Wied Pakusa
Leibniz International Proceedings in Informatics Schloss Dagstuhl-Leibniz-Zentrum für Informatik   unpublished
Motivated by the search for a logic for polynomial time, we study rank logic (FPR) which extends fixed-point logic with counting (FPC) by operators that determine the rank of matrices over finite fields. While FPR can express most of the known queries that separate FPC from Ptime, nearly nothing was known about the limitations of its expressive power. In our first main result we show that the extensions of FPC by rank operators over different prime fields are incomparable. This solves an open
more » ... is solves an open question posed by Dawar and Holm and also implies that rank logic, in its original definition with a distinct rank operator for every field, fails to capture polynomial time. In particular we show that the variant of rank logic FPR * with an operator that uniformly expresses the matrix rank over finite fields is more expressive than FPR. One important step in our proof is to consider solvability logic FPS which is the analogous extension of FPC by quantifiers which express the solvability problem for linear equation systems over finite fields. Solvability logic can easily be embedded into rank logic, but it is open whether it is a strict fragment. In our second main result we give a partial answer to this question: in the absence of counting, rank operators are strictly more expressive than solvability quantifiers. 1 Introduction "Le roi est mort, vive le roi!" has been the traditional proclamation, in France and other countries, to announce not only the death of the monarch, but also the immediate installment of his successor on the throne. The purpose of this paper is to kill the rank logic FPR, in the form in which it has been proposed in [7], as a candidate for a logic for Ptime. The logic FPR extends fixed-point logic by operators rk p (for every prime p) which compute the rank of definable matrices over the prime field F p with p elements. Although rank logic is well-motivated, as a logic that strictly extends fixed-point logic with counting by the ability to express important properties of linear algebra, most notably the solvability of linear equation systems over finite fields, our results show that the choice of having a separate rank operator for every prime p leads to a significant deficiency of the logic. Indeed, it follows from our main theorem that even the uniform rank problem, of computing the rank of a given matrix over an arbitrary prime, cannot be expressed in FPR and thus separates FPR from Ptime. This also reveals that a more general variant of rank logic, which has already been proposed in [14, 15, 17] and which is based on a rank operator that takes not only the matrix but also the prime p as part of the input, is indeed strictly more powerful than FPR. Our result thus installs this new rank logic, denoted FPR * , as the rightful and distinctly more powerful successor of FPR as a potential candidate for a logic for Ptime. A full version of this paper can be found at [11].
fatcat:wgwebjwkrra6zo76cvith5uugu

A Finite-Model-Theoretic View on Propositional Proof Complexity

Erich Grädel, Martin Grohe, Benedikt Pago, Wied Pakusa
2019
Pakusa Vol. 15:1 Horn-Res. In fact, FO(2-Res) = FO(TC), where FO(TC) is the extension of firstorder logic by a transitive closure operator.  ...  Pakusa Vol. 15:1 takes as input a propositional formula ϕ in conjunctive normal form (CNF), and it refutes the satisfiability of ϕ if there is a derivation of the empty clause from ϕ.  ...  Pakusa Vol. 15:1 are auxiliary variables used to encode this last condition, cf. [10] .  ... 
doi:10.18154/rwth-2019-00950 fatcat:3pbaatf56vctfmvsbhhwo4ncae

Approximations of Isomorphism and Logics with Linear-Algebraic Operators

Anuj Dawar, Erich Grädel, Wied Pakusa
2019
Invertible map equivalences are approximations of graph isomorphism that refine the well-known Weisfeiler-Leman method. They are parameterized by a number k and a set Q of primes. The intuition is that two equivalent graphs G ≡ IM k,Q H cannot be distinguished by means of partitioning the set of k-tuples in both graphs with respect to any linear-algebraic operator acting on vector spaces over fields of characteristic p, for any p ∈ Q. These equivalences have first appeared in the study of rank
more » ... the study of rank logic, but in fact they can be used to delimit the expressive power of any extension of fixed-point logic with linear-algebraic operators. We define LA k (Q), an infinitary logic with k variables and all linear-algebraic operators over finite vector spaces of characteristic p ∈ Q and show that ≡ IM k,Q is the natural notion of elementary equivalence for this logic. The logic LA ω (Q) = k∈ω LA k (Q) is then a natural upper bound on the expressive power of any extension of fixed-point logics by means of Q-linear-algebraic operators. By means of a new and much deeper algebraic analysis of a generalized variant, for any prime p, of the CFI-structures due to Cai, Fürer, and Immerman, we prove that, as long as Q is not the set of all primes, there is no k such that ≡ IM k,Q is the same as isomorphism. It follows that there are polynomial-time properties of graphs which are not definable in LA ω (Q), which implies that no extension of fixed-point logic with linear-algebraic operators can capture PTIME, unless it includes such operators for all prime characteristics. Our analysis requires substantial algebraic machinery, including a homogeneity property of CFI-structures and Maschke's Theorem, an important result from the representation theory of finite groups. ACM Subject Classification Theory of computation → Finite Model Theory over-approximates isomorphism in the sense that if G ∼ = H for a pair of graphs G and H, then G ≡ k H for any k. The relations form a refining family in the sense that if G ≡ k H then G ≡ k H for all k > k. Thus, the equivalence relation gets finer with increasing k and approaches isomorphism in the limit. Moreover, if G and H are n-vertex graphs then G ≡ n H if, and only if, G ∼ = H. For each fixed k, the equivalence relation ≡ k is decidable in polynomial time, indeed in time n O(k) . Thus, if there were a fixed k such that ≡ k were the same as isomorphism, we would have a polynomial-time algorithm for graph isomorphism. However, we know this is not the case. Cai, Fürer and Immerman [6] showed that there are pairs of non-isomorphic graphs G and H with O(k) vertices such that G ≡ k H. We call the construction of such graphs the CFI construction. The Weisfeiler-Leman equivalences arise naturally in the study of graphs in many different guises. We have definitions based on combinatorics (such as Babai's original definition, see [6] ); in logic as the equivalences induced by bounded variable fragments of first-order logic with counting; linear programming (see [2, 21] ); and algebra (as in the original definition of Weisfeiler and Leman, extended to dimension k in [13] ). The equivalences have proved to be of central importance in the area of descriptive complexity theory. In particular, they delimit the power of fixed-point logic with counting (FPC), an important logic in the study of symmetric polynomial-time computation (see [9] ). On many important classes of structures, it turns out that there is a fixed k for which k-WL suffices to distinguish all non-isomorphic graphs. Most significantly, Grohe [20] has shown that for any proper minor-closed class C of graphs, there is a k such that ≡ k coincides with isomorphism on graphs in C.
doi:10.18154/rwth-2019-06475 fatcat:fs5edha6xzfnrpgco3gjwnsjuq

Editors: Valentin Goranko and Mads Dam; Article No

Erich Grädel, Benedikt Pago, Wied Pakusa
18 Leibniz International Proceedings in Informatics Schloss Dagstuhl-Leibniz-Zentrum für Informatik   unpublished
The formula Θ(G, H) consists of the following set of clauses: w∈Wi X[v → w] for each i < n, v ∈ V i (1) v∈Vi X[v → w] for each i < n, w ∈ W i (2) ¬(X[v 1 → w 1 ] ∧ X[v 2 → w 2 ]) if {(v 1 , w 1 ), (v 2  ... 
fatcat:cjennofiwvhn5mpq2slnggk7m4

JSL volume 84 issue 1 Cover and Front matter

2019 Journal of Symbolic Logic (JSL)  
ERICH GRÄ DEL and WIED PAKUSA .............................................................. 54 Amalgamable diagram shapes RUIYUAN CHEN .................................................................  ... 
doi:10.1017/jsl.2019.9 fatcat:2rjeahckqne6rlj25pjcjgdmte
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