No full-text available
To read the full-text of this research,
you can request a copy directly from the authors.
Regular languages in NC1
Abstract
We give several characterizations, in terms of formal logic, semigroup theory, and operations on languages, of the regular languages in the circuit complexity class AC0, thus answering a question of Chandra, Fortune, and Lipton. As a by-product, we are able to determine effectively whether a given regular language is in AC0 and to solve in part an open problem originally posed by McNaughton. Using recent lower-bound results of Razborov and Smolensky, we obtain similar characterizations of the family of regular languages recognized by constant-depth circuit families that include unbounded fan-in mod p addition gates for a fixed prime p along with unbounded fan-in boolean gates. We also obtain logical characterizations for the class of all languages recognized by nonuniform circuit families in which mod m gates (where m is not necessarily prime) are permitted. Comparison of this characterization with our previous results provides evidence for a conjecture concerning the regular languages in this class. A proof of this conjecture would show that computing the bit sum modulo p, where p is a prime not dividing m, is not AC0-reducible to addition mod m, and thus that MAJORITY is not AC0-reducible to addition mod m. Peer Reviewed http://deepblue.lib.umich.edu/bitstream/2027.42/30017/1/0000385.pdf
... Another important application is the class of languages definable by first-order logic with modular predicates FO(<, ). This class is known to have decidable membership [4]. Moreover, it is the star-free closure of the class consisting of the languages counting the length of words modulo some number. ...
... Proposition 2. 4. Let C be a prevariety and consider finitely many languages 1 , . . . ...
... Remark 3. 4. No language theoretic definition of GR is known. ...
We introduce an operator on classes of regular languages, the star-free closure. Our motivation is to generalize standard results of automata theory within a unified framework. Given an arbitrary input class C, the star-free closure operator outputs the least class closed under Boolean operations and language concatenation, and containing all languages of C as well as all finite languages. We establish several equivalent characterizations of star-free closure: in terms of regular expressions, first-order logic, pure future and future-past temporal logic, and recognition by finite monoids. A key ingredient is that star-free closure coincides with another closure operator, defined in terms of regular operations where Kleene stars are allowed in restricted~contexts. A consequence of this first result is that we can decide membership of a regular language in the star-free closure of a class whose separation problem is decidable. Moreover, we prove that separation itself is decidable for the star-free closure of any finite class, and of any class of group languages having itself decidable separation (plus mild additional properties). We actually show decidability of a stronger property, called covering.
... This observation naturally leads to the task of recognising the complexity of the word problem for a given regular language. The circuit and descriptive complexity of regular languages was investigated by Barrington (1989), Barrington et al. (1992), Straubing (1994) who established an AC 0 /ACC 0 /NC 1 trichotomy, gave algebraic characterisations of languages in these classes (implying that the trichotomy is decidable) and also in terms of extensions of FO. Namely, the regular languages L in AC 0 are definable by FO(<, ≡)-sentences with unary predicates x ≡ 0 (mod n); those in ACC 0 are definable by FO(<, MOD)sentences with quantifiers ∃ n x ψ(x) checking whether the number of positions satisfying ψ is divisible by n; and all regular languages L are definable in FO(RPR) with relational primitive recursion (Compton & Laflamme, 1990). ...
... Let L be one of the languages FO(<), FO(<, ≡) or FO(<, MOD). First, using the algebraic characterisation results of Barrington (1989), Barrington et al. (1992), Straubing (1994), we obtain criteria for the L-definability of the language L(A) of any given DFA A in terms of a limited part of the transition monoid of A (Theorem 6). Then, using our criteria and generalising the construction of Cho and Huynh (1991), we show that deciding L-definability of L(A) for any minimal DFA A is PSpace-hard (Theorem 8). ...
... , n} ordered by <, in which S w |= a(i) iff a = a i , for 0 ≤ i ≤ n. Table 3 summarises the known results that connect definability of a regular language L with properties of the syntactic monoid M (L) and syntactic morphism η L (Barrington et al., 1992) and with its circuit complexity under a reasonable binary encoding of L's alphabet (e.g., Bernátsky, 1997, Lemma 2.1) and the assumption that ACC 0 ̸ = NC 1 . We also remind the reader that a regular language is FO(<)-definable iff it is star-free (Straubing, 1994), and that AC 0 ⫋ ACC 0 ⊆ NC 1 (Straubing, 1994;Jukna, 2012). ...
Our concern is the problem of determining the data complexity of answering an ontology-mediated query (OMQ) formulated in linear temporal logic LTL over (Z,<) and deciding whether it is rewritable to an FO(<)-query, possibly with some extra predicates. First, we observe that, in line with the circuit complexity and FO-definability of regular languages, OMQ answering in AC0, ACC0 and NC1 coincides with FO(<,≡)-rewritability using unary predicates x ≡ 0 (mod n), FO(<,MOD)-rewritability, and FO(RPR)-rewritability using relational primitive recursion, respectively. We prove that, similarly to known PSᴘᴀᴄᴇ-completeness of recognising FO(<)-definability of regular languages, deciding FO(<,≡)- and FO(<,MOD)-definability is also PSᴘᴀᴄᴇ-complete (unless ACC0 = NC1). We then use this result to show that deciding FO(<)-, FO(<,≡)- and FO(<,MOD)-rewritability of LTL OMQs is ExᴘSᴘᴀᴄᴇ-complete, and that these problems become PSᴘᴀᴄᴇ-complete for OMQs with a linear Horn ontology and an atomic query, and also a positive query in the cases of FO(<)- and FO(<,≡)-rewritability. Further, we consider FO(<)-rewritability of OMQs with a binary-clause ontology and identify OMQ classes, for which deciding it is PSᴘᴀᴄᴇ-, Π2p- and coNP-complete.
... The connection to boolean circuit complexity is this: F O[N ] is identical to the circuit complexity class AC 0 consisting of languages recognized by constantdepth polynomial-size boolean circuit families in which the AN D-and OR-gates are permitted to have unbounded fan-in. 2 This equivalence was shown by Immerman [7], and, independently, Gurevich and Lewis [5]. Low-depth circuit complexity is one of the very few parts of computational complexity theory for which we possess unconditional superpolynomial lower bounds: the principal result in this vein is the theorem of Furst, Saxe and Sipser [4] that PARITY-the set of bit strings in which the number of 1's is even-is not in AC 0 . ...
... Barrington, et. al. [2], showed that the regular languages in AC 0 are precisely those in F ...
... However, Barrington et. al. [2] also discuss the analogous identity when one allows both ordinary and modular quantifiers: ...
We give a simple new proof that regular languages defined by first-order sentences with no quantifier alteration can be defined by such sentences in which only regular atomic formulas appear. Earlier proofs of this fact relied on arguments from circuit complexity or algebra. Our proof is much more elementary, and uses only the most basic facts about finite automata.
... As this latter class is decidable, this leads to a decision procedure to test if a regular language belongs to WLAC 0 . This comes short of providing an answer to the above open problems, but this shows in particular that Open Problem 1 admits a positive answer iff FO 2 [arb] has the Straubing property, the name stemming from the central conjecture of Straubing in his textbook [19]: ...
... Since μ(A) can be seen as an element of the powerset monoid of M, there is an integer s > 0 such that μ(A) s = (μ(A) s ) 2 ; the smallest such s is the stability index of μ. By construction, μ(A) s is closed under multiplication and we write S(μ) for that set with possibly an identity element added so that it is a monoid. ...
... • QA. This is Q applied to A and was historically one of the first lm-varieties of stamps studied [2]. It holds that QA = FO[reg] = AC 0 ∩ Reg. ...
In this paper, the regular languages of wire linear AC0are characterized as the languages expressible in the two-variable fragment of first-order logic with regular predicates, FO2[reg]. Additionally, they are characterized as the languages recognized by the algebraic class QLDA. The class is shown to be decidable and examples of languages in and outside of it are presented.
... FO(<)-definable regular languages were proven to be the same as star-free languages [19], and their algebraic characterisation as languages with aperiodic syntactic monoids was obtained in [22]. Algebraic characterisations of FOdefinability in other signatures, and circuit and descriptive complexity of regular languages were investigated in [3,4,25], which established an AC 0 /ACC 0 /NC 1 trichotomy. In particular, the regular languages decidable in AC 0 are definable by FO(<, ≡)-sentences with unary predicates x ≡ 0 (mod n); those in ACC 0 are definable by FO(<, MOD)-sentences with quantifiers ∃ n x ψ(x) checking whether the number of positions satisfying ψ is divisible by n; and all regular languages are definable in FO(RPR) with relational primitive recursion [11]; see Table 1. ...
... The problem of deciding whether the language of a given DFA A is FO(<)definable is known to be PSpace-complete [7,10,24] (which is also a special case of general results on finite monoids [5,13]). As shown in [4], the algebraic criteria of Table 1 yield algorithms deciding whether a given regular language is in AC 0 and FO(<, ≡)-definable, or in ACC 0 and FO(<, MOD)-definable, or NC 1complete and is not FO(<, MOD)-definable (unless ACC 0 = NC 1 ). However, these 'brute force' algorithms are not optimal, requiring the generation of the Table 1. ...
... Let L be one of the languages FO(<, ≡) or FO(<, MOD). First, using the algebraic characterisation results of [3,4,25], we obtain criteria for the L-definability of the language L(A) of any given DFA A in terms of a limited part of the transition monoid of A (Theorem 1). Then, by using our criteria and generalising the construction of [10], we show that deciding L-definability of L(A) for any minimal DFA A is PSpace-hard (Theorem 2). ...
We prove that, similarly to known PSpace-completeness of recognising FO(<)-definability of the language L(A) of a DFA A, deciding both FO(<,C)- and FO(<,MOD)-definability are PSpace-complete. (Here, FO(<,C) extends the first-order logic FO(<) with the standard congruence modulo n relation, and FO(<,MOD) with the quantifiers checking whether the number of positions satisfying a given formula is divisible by a given n>1. These FO-languages are known to define regular languages that are decidable in AC0 and ACC0, respectively.) We obtain these results by first showing that known algebraic characterisations of FO-definability of L(A) can be captured by `localisable' properties of the transition monoid of A. Using our criterion, we then generalise the known proof of PSpace-hardness of FO(<)-definability, and establish the upper bounds not only for arbitrary DFAs but also for two-way NFAs.
... Our results demonstrate a class of formulas of unbounded depth with exponential size lower bound against the permanent and can be seen as an exponential improvement over the multilinear formula size lower bounds given by Raz [2] for a sub-class of multilinear and non-multilinear formulas. Our proof techniques are built on the one developed by Raz [2] and later extended by Kumar et al. [1]. Our proofs exploit the structural weakness of CF-ROPs against random partitions This is a joint work with C. Ramya. ...
... wcol r (G) ≤ (r + 1) 3( t−1 2 ) · (2r + 1). These results improve earlier results published in [1]. In modern data management scenarios, data is subject to frequent changes. ...
... A width w and length grid graph G is simply a subgraph of the width w and length full grid. In [1] it was shown that deciding whether a perfect matching exists for constant width grid graphs can be decided in ACC 0 . We will now describe an open question originating from that paper, although not appearing there explicitly. ...
... Currently, no results are known for the separation and covering problems associated to fragments enriched with modular predicates. However, earlier results considered the status of membership when a fragment of FO(<) is enriched with modular predicates: Barrington et al. showed that FO(<, MOD)-membership is decidable [BCST92] (see also [Str94]). Chaubard et al. [CPS06] show the same result for the extensions of Σ 1 (<) and BΣ 1 (<): Σ 1 (<, MOD) and BΣ 1 (<, MOD). ...
... Our results apply to enrichment by modular predicates and follow the same scheme, with an interesting twist. We investigate an operation on classes of languages whose origins can be found implicitly or explicitly in [BCST92,CPS06], written C → C • MOD (again, see Section 4 for definitions). Then we show two properties which are similar to those of C • SU. ...
For every class of word languages, one may associate a decision problem called -separation. Given two regular languages, it asks whether there exists a third language in containing the first language, while being disjoint from the second one. Usually, finding an algorithm deciding -separation yields a deep insight on . We consider classes defined by fragments of first-order logic. Given such a fragment, one may often build a larger class by adding more predicates to its signature. In the paper, we investigate the operation of enriching signatures with modular predicates. Our main theorem is a generic transfer result for this construction. Informally, we show that when a logical fragment is equipped with a signature containing the successor predicate, separation for the stronger logic enriched with modular predicates reduces to separation for the original logic. This result actually applies to a more general decision problem, called the covering problem.
... In this way, although, for example, the unresolved strict containment in NC 1 of the class ACC 0 , defined from bounded-depth polynomial-size unbounded fan-in circuits over {AND, OR, MOD}, remains a barrier since the work of Smolensky [Smo87], significant progress was made in (1) understanding the power of the BIT predicate and the related circuit uniformity issues [BIS90], (2) describing the regular languages within subclasses of NC 1 [BCST92,PMT91], and (3) identifying the allimportant role of the interplay between arbitrary and regular numerical predicates in the status of the ACC 0 versus NC 1 question [Str94,p. 169, Conjecture IX.3.4]. ...
... Finally, ever since the regular languages in AC 0 and in ACC 0 were characterized (the latter modulo a natural conjecture [BCST92]), one might have wondered about a similar characterization for the context-free languages in these classes, and in NC 1 . A unified treatment of LOGCFL subclasses under the banner of first-order logic might constitute a useful step towards being able to answer these questions. ...
Building upon the known generalized-quantifier-based first-order characterization of LOGCFL, we lay the groundwork for a deeper investigation. Specifically, we examine subclasses of LOGCFL arising from varying the arity and nesting of groupoidal quantifiers. Our work extends the elaborate theory relating monoidal quantifiers to NC1 and its subclasses. In the absence of the BIT predicate, we resolve the main issues: we show in particular that no single outermost unary groupoidal quantifier with FO can capture all the context-free languages, and we obtain the surprising result that a variant of Greibach's ``hardest context-free language'' is LOGCFL-complete under quantifier-free BIT-free projections. We then prove that FO with unary groupoidal quantifiers is strictly more expressive with the BIT predicate than without. Considering a particular groupoidal quantifier, we prove that first-order logic with majority of pairs is strictly more expressive than first-order with majority of individuals. As a technical tool of independent interest, we define the notion of an aperiodic nondeterministic finite automaton and prove that FO translations are precisely the mappings computed by single-valued aperiodic nondeterministic finite transducers.
... While many star-free languages in the sample show length-generalization, star-freeness does not overall account for the observed behavior ( Figure 4). Within the star-free languages, a standard complexity metric is dot-depth ( Figure 2); this again does not accurately predict length-generalization: it succeeds for a language with dot depth 12 but fails for a language with dot depth 2. We also considered the circuit complexity of regular languages (Barrington et al., 1992). All regular languages included in the sample are in the class TC 0 ; most are also in AC 0 , a smaller class sometimes compared to transformers (Hao et al., 2022;Barcelo et al., 2024). ...
... By the results ofBarrington et al. (1992), regular languages outside of AC 0 are all, informally speaking, at least as PARITY, and indeed they provably are not in C-RASP[periodic, local]. ...
A major challenge for transformers is generalizing to sequences longer than those observed during training. While previous works have empirically shown that transformers can either succeed or fail at length generalization depending on the task, theoretical understanding of this phenomenon remains limited. In this work, we introduce a rigorous theoretical framework to analyze length generalization in causal transformers with learnable absolute positional encodings. In particular, we characterize those functions that are identifiable in the limit from sufficiently long inputs with absolute positional encodings under an idealized inference scheme using a norm-based regularizer. This enables us to prove the possibility of length generalization for a rich family of problems. We experimentally validate the theory as a predictor of success and failure of length generalization across a range of algorithmic and formal language tasks. Our theory not only explains a broad set of empirical observations but also opens the way to provably predicting length generalization capabilities in transformers.
... However, such a construction does not work at finite precision, because rounding may make it impossible to extract even the second-most-significant bit. 5 We avoid this problem simply by taking A = 1/4, effectively utilizing only every two digits in the binary expansion of h t , ensuring that the second-last input can be read out at a constant margin. We now provide the formal proof: ...
... The set of regular languages known to be in TC 0 is the set of regular languages whose syntactic monoid contains no non-solvable groups [5]. These languages are recognized by cascade products of set-reset automata and automata perfoming modular counting [64]. ...
Recently, recurrent models based on linear state space models (SSMs) have shown promising performance in language modeling (LM), competititve with transformers. However, there is little understanding of the in-principle abilities of such models, which could provide useful guidance to the search for better LM architectures. We present a comprehensive theoretical study of the capacity of such SSMs as it compares to that of transformers and traditional RNNs. We find that SSMs and transformers have overlapping but distinct strengths. In star-free state tracking, SSMs implement straightforward and exact solutions to problems that transformers struggle to represent exactly. They can also model bounded hierarchical structure with optimal memory even without simulating a stack. On the other hand, we identify a design choice in current SSMs that limits their expressive power. We discuss implications for SSM and LM research, and verify results empirically on a recent SSM, Mamba.
... " Logic came to the rescue by giving a descriptional tool for circuits. This started with the work of Barrington et al. [3] and Straubing [29] who showed that AC 0 (= AC 0 ) is equivalent to first-order logic. This means that for any AC 0 circuit family, there is a first-order formula, with quantifiers over positions, that recognizes the same language. ...
... If we assume that ′ ∈ Π 2 [arb], we have to show that the complement of ′ is not in Σ 2 [<]. This time, we pick ( ) 3 as the idempotent, and as the subword. Then ( ) 9 ∉ ′ but ( ) 3 ( ) ( ) 3 ∈ ′ , hence the complement of ′ is not in Σ 2 [<]. ...
The regular languages with a neutral letter expressible in first-order logic with one alternation are characterized. Specifically, it is shown that if an arbitrary formula defines a regular language with a neutral letter, then there is an equivalent formula that only uses the order predicate. This shows that the so-called Central Conjecture of Straubing holds for over languages with a neutral letter, the first progress on the Conjecture in more than 20 years. To show the characterization, lower bounds against polynomial-size depth-3 Boolean circuits with constant top fan-in are developed. The heart of the combinatorial argument resides in studying how positions within a language are determined from one another, a technique of independent interest.
... We denote by A * QA the family of languages over A * definable in this logic. Languages in A * QA arise as the regular languages definable in the circuit complexity class AC 0 (see [11]). Each A * QA is a boolean algebra closed under quotients, however QA is not a variety of languages: To see this, consider the morphism {a, b} * → {a} * that maps a to a and b to the empty string. ...
... Once again we look not just at the syntactic monoid of a language L, but at the additional structure provided by the syntactic morphism η L . It is known that L ∈ A * QA if and only if for every k ≥ 0, η L (A k ) contains no nontrivial groups [11]. The family QA of morphisms from free monoids onto finite monoids with this property forms a kind of pseudovariety with respect to appropriately modified definitions of direct product and division. ...
... We also investigate the non-uniform variant of this problem (in the sense that the regular language L is fixed). In the late 1980s and early 1990s, several connections between classes of regular languages and circuit complexity classes were established [6,8,9]. For example, it was shown that all regular languages belong to NC 1 , all star-free languages belong to AC 0 , all regular languages whose syntactic monoid is solvable belong to ACC 0 , and there are regular languages that are NC 1 -complete. ...
... Together with Proposition 23 and a well-known classification of the complexity of the membership problem for regular languages [6], this yields the following corollary. ...
For a formal language L, the problem of language enumeration asks to compute the length-lexicographically smallest word in L larger than a given input w ∈ Σ∗ (henceforth called the L-successor of w). We investigate this problem for regular languages from a computational complexity and state complexity perspective. We first show that if L is recognized by a DFA with n states, then 2Θ(nlog n) states are (in general) necessary and sufficient for an unambiguous finite-state transducer to compute L-successors. As a byproduct, we obtain that if L is recognized by a DFA with n states, then 2Θ(nlog n) states are sufficient for a DFA to recognize the subset S(L) of L composed of its lexicographically smallest words. We give a matching lower bound that holds even if S(L) is represented as an NFA. It has been known that L-successors can be computed in polynomial time, even if the regular language is given as part of the input (assuming a suitable representation of the language, such as a DFA). In this paper, we refine this result in multiple directions. We show that if the regular language is given as part of the input and encoded as a DFA, the problem is in NL. If the regular language L is fixed, we prove that the enumeration problem of the language is reducible to deciding membership to the Myhill-Nerode equivalence classes of L under DLOGTIME-uniform AC0 reductions. In particular, this implies that fixed star-free languages can be enumerated in AC0, arbitrary fixed regular languages can be enumerated in NC1 and that there exist regular languages for which the problem is NC1-complete.
... On the other hand, any non-trivial monoid M p-recognizes LEN 2 using the sequence (P n ) n∈N in which P n is a single instruction, (1, f : {a, b} → M ), where f (a) = f (b) are set to an accepting element if n is even and to a rejecting element if n is odd. It is known [BCST92] that when V = A, this haphazard modular counting ability of polynomial length programs over monoids from V translates algebraically into "quasi-V power" being necessary and sufficient for morphisms to simulate those programs on regular languages. Our main result shows that the same holds when V = DA. ...
... As A is local [Til87, Example 15.5] and an sp-variety, it follows from Proposition 3.5 that the regular languages in P(A), hence in AC 0 , are precisely those in L(QA), which is the characterization of the regular languages in AC 0 obtained by Barrington, Compton, Straubing and Thérien [BCST92]. ...
The program-over-monoid model of computation originates with Barrington's proof that it captures the complexity class NC. Here we make progress in understanding the subtleties of the model. First, we identify a new tameness condition on a class of monoids that entails a natural characterization of the regular languages recognizable by programs over monoids from the class. Second, we prove that the class known as DA satisfies tameness and hence that the regular languages recognized by programs over monoids in DA are precisely those recognizable in the classical sense by morphisms from QDA. Third, we show by contrast that the well studied class of monoids called J is not tame. Finally, we exhibit a program-length-based hierarchy within the class of languages recognized by programs over monoids from DA.
... We will also use Reg to denote the set of numerical 760 predicates for which the associated language of structures is regular. In [2] 761 it is shown that ...
... Combining the two directions of generalization -from ordinary to modular 948 quantifiers, and from Π 1 formulas to formulas with more levels of quantifier N C 1 (see [2] and [29]). ...
The notion of a difference hierarchy, first introduced by Hausdorff, plays an important role in many areas of mathematics, logic and theoretical computer science such as descriptive set theory, complexity theory, and the theory of regular languages and automata. Lattice theoretically, the difference hierarchy over a distributive lattice stratifies the Boolean algebra generated by it according to the minimum length of difference chains required to describe the Boolean elements. While each Boolean element is given by a finite difference chain, there is no canonical such writing in general. We show that, relative to the filter completion, or equivalently, the lattice of closed upsets of the dual Priestley space, each Boolean element over the lattice has a canonical minimum length decomposition into a Hausdorff difference chain. As a corollary, each Boolean element over a co-Heyting algebra has a canonical difference chain (and an order dual result holds for Heyting algebras). With a further generalization of this result involving a directed family of closure operators on a Boolean algebra, we give an elementary proof of the fact that if a regular language is given by a Boolean combination of universal sentences using arbitrary numerical predicates then it is also given by a Boolean combination of universal sentences using only regular numerical predicates.
... As a model of parallelism we choose Boolean circuits. The most important reason is that their relation with regular languages is well understood and documented [3,19,20,32], but Boolean circuits have several other advantages. On one hand, they are very close to hardware implementation: from a Boolean circuit of small depth one can directly obtain a hardware description that could be compiled into, say, an FPGA. ...
... The connection between logic and regular languages is a field of research on its own that takes its root into the celebrated results of McNaughton and Papert [25] and Schützenberger [29], who characterized regular languages of FO [<], that is languages definable in first order logic with the linear order over positions. By extending this result to a slightly more complicated fragment, and by using the parity lower bounds for AC 0 , Barrington et al. [3] proved that regular languages in AC 0 are exactly those definable in FO[<, MOD]; that is, in first order logic with (strict) order and the unary modulo predicates of the form x ≡ r mod q for arbitrary r, q ∈ N. Furthermore, this class of regular languages has decidable membership thanks to its algebraic characterization. ...
XML schema validation can be performed in constant memory in the streaming model if and only if the schema admits only trees of bounded depth - an acceptable assumption from the practical view-point. In this paper we refine this analysis by taking into account that data can be streamed block-by-block, rather then letter-by-letter, which provides opportunities to speed up the computation by parallelizing the processing of each block.
For this purpose we introduce the model of streaming circuits, which process words of arbitrary length in blocks of fixed size, passing constant amount of information between blocks.
This model allows us to transfer fundamental results about the circuit complexity of regular languages to the setting of streaming schema validation, which leads to effective constructions of streaming circuits of depth logarithmic in the block size, or even constant under certain assumptions on the input schema.
For nested-relational DTDs, a practically motivated class of bounded-depth XML schemas, we provide an efficient construction yielding constant-depth streaming circuits with particularly good parameters.
... The logic FO[arb], for instance, allows any predicate (e.g., x = y × z), while FO[<, +1, mod] only enriches FO[<] with predicates x = y+1 and x = 0 mod p, for any constant p. A striking fact, conjectured by McNaughton in the 1960s [11] and showed by Barrington, et al. [5], is that the regular languages of FO [arb] are precisely the languages expressible in FO[<, +1, mod]. Techniques of [20] translate the decidability of membership to the alternation hierarchy Σ i [<] to the alternation hierarchy Σ i [<, +1], while an ordered-monoid version of [9] would show the same for Σ i [<, +1, mod]. ...
We show that for any , it is decidable, given a regular language, whether it is expressible in the fragment of first-order logic FO[<]. This settles a question open since 1971. Our main technical result relies on the notion of polynomial closure of a class of languages , that is, finite unions of languages of the form where each is a letter and each a language of . We show that if a class of regular languages with some closure properties (namely, a positive variety) has a decidable separation problem, then so does its polynomial closure Pol(). The resulting algorithm for Pol() has time complexity that is exponential in the time complexity for and we propose a natural conjecture that would lead to a polynomial time blowup instead. Corollaries include the decidability of half levels of the dot-depth hierarchy and the group-based concatenation hierarchy.
... We now have a more or less complete characterization of which regular languages can be recognized by AHATs. Barrington et al. (1992) showed that every regular language L is either in ACC 0 or NC 1 -complete. ...
Previous work has shown that the languages recognized by average-hard attention transformers (AHATs) and softmax-attention transformers (SMATs) are within the circuit complexity class TC. However, these results assume limited-precision arithmetic: using floating-point numbers with O(log n) bits (where n is the length of the input string), Strobl showed that AHATs can be approximated in L-uniform TC, and Merrill and Sabharwal showed that SMATs can be approximated in DLOGTIME-uniform TC. Here, we improve these results, showing that AHATs with no approximation, SMATs with O(poly(n)) bits of floating-point precision, and SMATs with at most absolute error are all in DLOGTIME-uniform TC.
... We have that Mod r p (i) = 1 if and only if i ≡ r (mod p). In fact, by using a characterization given in [3], one can show that the languages definable in FO(Mod) are precisely the regular languages within AC 0 . Then: Corollary 1. ...
We contribute to the study of formal languages that can be recognized by transformer encoders. We focus on two self-attention mechanisms: (1) UHAT (Unique Hard Attention Transformers) and (2) AHAT (Average Hard Attention Transformers). UHAT encoders are known to recognize only languages inside the circuit complexity class AC 0 , i.e., accepted by a family of poly-sized and depth-bounded boolean circuits with unbounded fan-ins. On the other hand, AHAT encoders can recognize languages outside AC 0), but their expressive power still lies within the bigger circuit complexity class TC 0 , i.e., AC 0-circuits extended by majority gates. We first show a negative result that there is an AC 0-language that cannot be recognized by an UHAT encoder. On the positive side, we show that UHAT encoders can recognize a rich fragment of AC 0-languages, namely, all languages definable in first-order logic with arbitrary unary numerical predicates. This logic, includes, for example, all regular languages from AC 0. We then show that AHAT encoders can recognize all languages of our logic even when we enrich it with counting terms. We apply these results to derive new results on the expressive power of UHAT and AHAT up to permutation of letters (a.k.a. Parikh images).
... However, there are a few connections with the topo-algebraic tools of the theory of regular languages. A famous result of Barrington, Compton, Straubing, and Thérien [6] states that a regular language belongs to AC 0 if and only if its syntactic homomorphism is quasi-aperiodic. Although this result relies on the lower-bound results in circuit complexity of [10] and no purely algebraic proof is known, being able to characterize the class of regular languages that are in AC 0 gives some hope that the non-regular classes might be amenable to treatment by the generalized topo-algebraic methods. ...
In the classical theory of regular languages, the concept of recognition by profinite monoids is an important tool. Beyond regularity, Boolean spaces with internal monoids (BiMs) were recently proposed as a generalization. On the other hand, fragments of logic defining regular languages can be studied inductively via the so-called “Substitution Principle.” In this paper, we make the logical underpinnings of this principle explicit and extend it to arbitrary languages using Stone duality. Subsequently, we show how it can be used to obtain topo-algebraic recognizers for classes of languages defined by a wide class of first-order logic fragments. This naturally leads to a notion of semidirect product of BiMs extending the classical such construction for profinite monoids. Our main result is a generalization of Almeida and Weil’s Decomposition Theorem for semidirect products from the profinite setting to that of BiMs. This is a crucial step in a program to extend the profinite methods of regular language theory to the setting of complexity theory.
... As A is local [Til87, Example 15.5] and an sp-variety, it follows from Proposition 3.5 that the regular languages in P(A), hence in AC 0 , are precisely those in L(QA), which is the characterization of the regular languages in AC 0 obtained by Barrington, Compton, Straubing and Thérien [BCST92]. ...
The program-over-monoid model of computation originates with Barrington's
proof that the model captures the complexity class . Here we
make progress in understanding the subtleties of the model. First, we identify
a new tameness condition on a class of monoids that entails a natural
characterization of the regular languages recognizable by programs over monoids
from the class. Second, we prove that the class known as
satisfies tameness and hence that the regular languages recognized by programs
over monoids in are precisely those recognizable in the classical
sense by morphisms from . Third, we show by contrast that the
well studied class of monoids called is not tame. Finally, we
exhibit a program-length-based hierarchy within the class of languages
recognized by programs over monoids from .
... , n} ordered by <, in which S w |= a(i) iff a = a i , for 0 ≤ i ≤ n. Table 3 summarises the known results that connect definability of a regular language L with properties of the syntactic monoid M (L) and syntactic morphism η L ( Barrington et al., 1992) and with its circuit complexity under a reasonable binary encoding of L's alphabet (e.g., Bernátsky, 1997, Lemma 2.1) and the assumption that ACC 0 = NC 1 . We also remind the reader that a regular language is FO(<)-definable iff it is star-free (Straubing, 1994), and that AC 0 ACC 0 ⊆ NC 1 (Straubing, 1994;Jukna, 2012). ...
Our concern is the problem of determining the data complexity of answering an ontology-mediated query (OMQ) formulated in linear temporal logic LTL over (Z,<) and deciding whether it is rewritable to an FO(<)-query, possibly with some extra predicates. First, we observe that, in line with the circuit complexity and FO-definability of regular languages, OMQ answering in AC^0, ACC^0 and NC^1 coincides with FO(<,\equiv)-rewritability using unary predicates x \equiv 0 (mod n), FO(<,MOD)-rewritability, and FO(RPR)-rewritability using relational primitive recursion, respectively. We prove that, similarly to known PSPACE-completeness of recognising FO(<)-definability of regular languages, deciding FO(<,\equiv)- and FO(<,MOD)-definability is also \PSPACE-complete (unless ACC^0 = NC^1). We then use this result to show that deciding FO(<)-, FO(<,\equiv)- and FO(<,MOD)-rewritability of LTL OMQs is EXPSPACE-complete, and that these problems become PSPACE-complete for OMQs with a linear Horn ontology and an atomic query, and also a positive query in the cases of FO(<)- and FO(<,\equiv)-rewritability. Further, we consider FO(<)-rewritability of OMQs with a binary-clause ontology and identify OMQ classes, for which deciding it is PSPACE-, Pi_2^p- and coNP-complete.
... However, there are a few connections with the topo-algebraic tools of the theory of regular languages. A famous result of Barrington, Compton, Straubing, and Thérien [4] states that a regular language belongs to AC 0 if and only if its syntactic homomorphism is quasi-aperiodic. Although this result relies on [6] and no purely algebraic proof is known, being able to characterize the class of regular languages that are in AC 0 gives some hope that the nonregular classes might be amenable to treatment by the generalized topo-algebraic methods. ...
In the classical theory of regular languages the concept of recognition by profinite monoids is an important tool. Beyond regularity, Boolean spaces with internal monoids (BiMs) were recently proposed as a generalization. On the other hand, fragments of logic defining regular languages can be studied inductively via the so-called "Substitution Principle". In this paper we make the logical underpinnings of this principle explicit and extend it to arbitrary languages using Stone duality. Subsequently we show how it can be used to obtain topo-algebraic recognizers for classes of languages defined by a wide class of first-order logic fragments. This naturally leads to a notion of semidirect product of BiMs extending the classical such construction for profinite monoids. Our main result is a generalization of Almeida and Weil's Decomposition Theorem for semidirect products from the profinite setting to that of BiMs. This is a crucial step in a program to extend the profinite methods of regular language theory to the setting of complexity theory.
... Our next theorem provides the matching NC 1 lower bound applicable to different fragments of propositional DatalogMTL. The bound is shown by reduction of the word acceptance problem for a fixed DFA, which is NC 1 -hard due to the existence of NC 1 -complete regular languages (Barrington et al. 1992). Theorem 11. ...
We study DatalogMTL—an extension of Datalog with metric temporal operators—under integer semantics, where the temporal domain of both interpretations and temporal operators consists of integer time points only. This is in contrast to the standard semantics, which is defined over the rational timeline. DatalogMTL under integer semantics is an interesting KR language: on the one hand, one can often assume the integer timeline in applications; on the other hand, it captures prominent temporal extensions of Datalog such as Datalog1S. We show that the choice of integer semantics leads to more favourable computational properties. We first show that reasoning over integers is at most as hard as reasoning over rationals for DatalogMTL and its natural fragments. Then, we investigate fragments of DatalogMTL where adopting the integer semantics makes reasoning easier. In particular, we show that complexity drops from P-hard to NC1-complete for the propositional fragment (where all object variables are grounded), and from TC0-hard to ACC0 for the linear fragment where the past diamond operator is the only metric operator allowed in rule bodies. Thus, reasoning in such fragments is both tractable and highly parallelisable, which suggests their appropriateness for data-intensive applications.
... Recall that AC 0 is the set of unbounded fan-in, polynomial size, constant-depth Boolean circuits. One can show [6,27,28] that a regular language is recognised by a circuit in AC 0 if and only if its syntactic monoid satisfies the equations (x ω−1 y) ω = (x ω−1 y) ω+1 for all words x and y of the same length. It is also possible to give an inequational characterisation of lattices of languages that are not regular [15]. ...
16th International Conference, RAMiCS 2017, Lyon, France, May 15-18, 2017, Proceedings
... Currently, no results are known for the separation and covering problems associated to fragments enriched with modular predicates. However, earlier results considered the status of membership when a fragment of FO(<) is enriched with modular predicates: Barrington et al. showed that FO(<, MOD)-membership is decidable [BCST92] (see also [Str94]). Chaubard et al. [CPS06] show the same result for the extensions of Σ 1 (<) and BΣ 1 (<): Σ 1 (<, MOD) and BΣ 1 (<, MOD). ...
For every class of word languages, one may associate a decision
problem called -separation. Given two regular languages, it asks
whether there exists a third language in containing the first
language, while being disjoint from the second one. Usually, finding an
algorithm deciding -separation yields a deep insight on
.
We consider classes defined by fragments of first-order logic. Given such a
fragment, one may often build a larger class by adding more predicates to its
signature. In the paper, we investigate the operation of enriching signatures
with modular predicates. Our main theorem is a generic transfer result for this
construction. Informally, we show that when a logical fragment is equipped with
a signature containing the successor predicate, separation for the stronger
logic enriched with modular predicates reduces to separation for the original
logic. This result actually applies to a more general decision problem, called
the covering problem.
... The fragment FO [<] captures the class of star-free regular languages [MP71]. The fragment FO [<, mod] also captures a subclass of the regular languages, which enjoys an (effective) algebraic characterization (see [BCST92,Corollary 10]); moreover FO[<, mod] is maximal with respect to regular languages, in the sense that every non trivial extension of the signature {<, mod} with numerical relations allows to define non-regular languages [Pél92]. For more information on this logic, we refer the reader to the book [Str94]. ...
... Ajtai [1] and Furst, Saxe, and Sipser [2] showed some 30 years ago that Parity, the language of words over {0, 1} having an even number of 1, is not computable by families of shallow circuits, namely AC 0 circuits. Since then, a wealth of precise expressiveness properties of AC 0 has been derived from this sole result [3], [4]. Naturally aiming at a better understanding of the core reasons behind this lower bound, a continuous effort has been made to provide alternative proofs of Parity / ∈ AC 0 . ...
First-order logic (FO) over words is shown to be equiexpressive with FO equipped with a restricted set of numerical predicates, namely the order, a binary predicate MSB, and the finite-degree predicates: FO[Arb] = FO[<, MSB, Fin]. The Crane Beach Property (CBP), introduced more than a decade ago, is true of a logic if all the expressible languages admitting a neutral letter are regular. Although it is known that FO[Arb] does not have the CBP, it is shown here that the (strong form of the) CBP holds for both FO[<, Fin] and FO[<, MSB]. Thus FO[<, Fin] exhibits a form of locality and the CBP, and can still express a wide variety of languages, while being one simple predicate away from the expressive power of FO[Arb]. The counting ability of FO[<, Fin] is studied as an application.
We introduce an operator on classes of regular languages, the star-free closure. Our motivation is to generalize standard results of automata theory within a unified framework. Given an arbitrary input class , the star-free closure operator outputs the least class closed under Boolean operations and language concatenation, and containing all languages of as well as all finite languages. We establish several equivalent characterizations of star-free closure: in terms of regular expressions, first-order logic, pure future and future-past temporal logic, and recognition by finite monoids. A key ingredient is that star-free closure coincides with another closure operator, defined in terms of regular operations where Kleene stars are allowed in restricted contexts.
A consequence of this first result is that we can decide membership of a regular language in the star-free closure of a class whose separation problem is decidable. Moreover, we prove that separation itself is decidable for the star-free closure of any finite class, and of any class of group languages having itself decidable separation (plus mild additional properties). We actually show decidability of a stronger property, called covering.
We study Monadic Second-Order Logic (MSO) over finite words, extended with (non-uniform arbitrary) monadic predicates. We show that it defines a class of languages that has algebraic, automata-theoretic and machine-independent characterizations. We consider the regularity question: given a language in this class, when is it regular? To answer this, we show a substitution property and the existence of a syntactical predicate. We give three applications. The first two are to give very simple proofs that the Straubing Conjecture holds for all fragments of MSO with monadic predicates, and that the Crane Beach Conjecture holds for MSO with monadic predicates. The third is to show that it is decidable whether a language defined by an MSO formula with morphic predicates is regular.
This paper provides a theoretical framework that validates and explains the results in the work with Bei Zhou experimentally finding that AlphaZero-style reinforcement learning algorithms struggle to learn optimal play in NIM, a canonical impartial game proposed as an AI challenge by Harvey Friedman in 2017. Our analysis resolves a controversy around these experimental results, which revealed unexpected difficulties in learning NIM despite its mathematical simplicity compared to games like chess and Go. Our key contributions are as follows: We prove that by incorporating recent game history, these limited AlphaZero models can, in principle, achieve optimal play in NIM. We introduce a novel search strategy where roll-outs preserve game-theoretic values during move selection, guided by a specialised policy network. We provide constructive proofs showing that our approach enables optimal play within the complexity class despite the theoretical limitations of these networks. This research demonstrates how constrained neural networks when properly designed, can achieve sophisticated decision-making even in domains where their basic computational capabilities appear insufficient.
As transformers have gained prominence in natural language processing, some researchers have investigated theoretically what problems they can and cannot solve, by treating problems as formal languages. Exploring such questions can help clarify the power of transformers relative to other models of computation, their fundamental capabilities and limits, and the impact of architectural choices. Work in this subarea has made considerable progress in recent years. Here, we undertake a comprehensive survey of this work, documenting the diverse assumptions that underlie different results and providing a unified framework for harmonizing seemingly contradictory findings.
We investigate the complexity of languages that correspond to algebraic real numbers, and we present improved upper bounds on the complexity of these languages. Our key technical contribution is the presentation of improved uniform TC 0 circuits for division, matrix powering, and related problems, where the improvement is in terms of “majority depth” (initially studied by Maciel and Thérien). As a corollary, we obtain improved bounds on the complexity of certain problems involving arithmetic circuits, which are known to lie in the counting hierarchy, and we answer a question posed by Yap.
We prove that, similarly to known \textsc {PSpace}-completeness of recognising -definability of the language of a DFA , deciding both - and -definability (corresponding to circuit complexity in {\textsc {AC}^0} and {\textsc {ACC}^0}) are \textsc {PSpace}-complete. We obtain these results by first showing that known algebraic characterisations of FO-definability of can be captured by ‘localisable’ properties of the transition monoid of . Using our criterion, we then generalise the known proof of \textsc {PSpace}-hardness of -definability, and establish the upper bounds not only for arbitrary DFAs but also for 2NFAs.
We discuss recent attempts to extend the ontology-based data access (aka virtual knowledge graph) paradigm to the temporal setting. Our main aim is to understand when answering temporal ontology-mediated queries can be reduced to evaluating standard first-order queries over timestamped data and what numeric predicates and operators are required in such reductions. We consider two ways of introducing a temporal dimension in ontologies and queries: using linear temporal logic LTL over discrete time and using metric temporal logic MTL over dense time.
A fundamental result about formal languages states:
Theorem 1 A regular language is first-order definable if and only if its syntactic monoid contains no nontrivial groups.
Rest assured, we will explain in the next section exactly what the various terms in the statement mean!
Complexity theory and the theory of regular languages both belong to the branch of computer science where the use of resources in computing is the main focus. However, they operate at different levels. While complexity theory seeks to classify computational problems by resource use, such as space and time, regular language theory remains at the very base of this hierarchy and is concerned with classes of computational problems for which membership is (potentially) decidable.
Transformers are emerging as the new workhorse of NLP, showing great success across tasks. Unlike LSTMs, transformers process input sequences entirely through self-attention. Previous work has suggested that the computational capabilities of self-attention to process hierarchical structures are limited. In this work, we mathematically investigate the computational power of self-attention to model formal languages. Across both soft and hard attention, we show strong theoretical limitations of the computational abilities of self-attention, finding that it cannot model periodic finite-state languages, nor hierarchical structure, unless the number of layers or heads increases with input length. These limitations seem surprising given the practical success of self-attention and the prominent role assigned to hierarchical structure in linguistics, suggesting that natural language can be approximated well with models that are too weak for the formal languages
typically assumed in theoretical linguistics.
Transformers are emerging as the new workhorse of NLP, showing great success across tasks. Unlike LSTMs, transformers process input sequences entirely through self-attention. Previous work has suggested that the computational capabilities of self-attention to process hierarchical structures are limited. In this work, we mathematically investigate the computational power of self-attention to model formal languages. Across both soft and hard attention, we show strong theoretical limitations of the computational abilities of self-attention, finding that it cannot model periodic finite-state languages, nor hierarchical structure, unless the number of layers or heads increases with input length. Our results precisely describe theoretical limitations of the techniques underlying recent advances in NLP.
We investigate the star-free closure, which associates to a class of languages its closure under Boolean operations and marked concatenation. We prove that the star-free closure of any finite class and of any class of groups languages with decidable separation (plus mild additional properties) has decidable separation. We actually show decidability of a stronger property, called covering. This generalizes many results on the subject in a unified framework. A key ingredient is that star-free closure coincides with another closure operator where Kleene stars are also allowed in restricted contexts.
We study Monadic Second-Order Logic (MSO) over finite words, extended with (non-uniform arbitrary) monadic predicates. We show that it defines a class of languages that has algebraic, automata-theoretic, and machine-independent characterizations. We consider the regularity question: Given a language in this class, when is it regular? To answer this, we show a substitution property and the existence of a syntactical predicate.
We give three applications. The first two are to give very simple proofs that the Straubing Conjecture holds for all fragments of MSO with monadic predicates and that the Crane Beach Conjecture holds for MSO with monadic predicates. The third is to show that it is decidable whether a language defined by an MSO formula with morphic predicates is regular.
We assume the reader is familiar with basic topology on the one hand and finite automata theory on the other hand. No proofs are given in this extended abstract.
Two new characterizations of FO[<, mod]-definable sets, i.e. sets of integers definable in first-order logic with the order relation and modular relations, are provided. Those characterizations are used to prove that satisfiability of first-order logic over words with an order relation and a FO[+]-definable set that is not FO[<, mod]-definable is undecidable.
This paper provides efficient algorithms that decide membership for classes of several Boolean hierarchies for which efficiency (or even decidability) were previously not known. We develop new forbidden-chain characterizations for the single levels of these hierarchies and obtain the following results:
• The classes of the Boolean hierarchy over level Σ1 of the dot-depth hierarchy are decidable in NL (previously only the decidability was known). The same remains true if predicates mod d for fixed d are allowed.
• If modular predicates for arbitrary d are allowed, then the classes of the Boolean hierarchy over level Σ1 are decidable.
• For the restricted case of a two-letter alphabet, the classes of the Boolean hierarchy over level Σ2 of the Straubing–Thérien hierarchy are decidable in NL. This is the first decidability result for this hierarchy.
• The membership problems for all mentioned Boolean-hierarchy classes are logspace many-one hard for NL.
• The membership problems for quasi-aperiodic languages and for d-quasi-aperiodic languages are logspace many-one complete for PSPACE.
In the last two decades visibly pushdown languages (VPLs) have found many applications in diverse areas such as formal verification and processing of XML documents. Recently, there has been a significant interest in studying quantitative versions of finite-state systems as well as visibly pushdown systems. In this work, we take forward this study for visibly pushdown systems by considering a functional version of visibly pushdown automata. Our version is formally a generalization of cost register automata (CRA) defined by [Alur et al., 2013]. We observe that our model continues to have all the good properties of the CRAs in spite of being a generalization.
Apart from studying the functional properties of the model, we also study the complexity theoretic aspects. Recently such a study was conducted by [Allender and Mertz, 2014] with respect to CRAs. Here we show that CRAs when appended with a visible stack (i.e. in the model defined here), continue to have the same complexity theoretic upper bounds as are known for CRAs. Moreover, we observe that one of the upper bounds shown by Allender et al. which was not tight for CRAs becomes tight for our model. Hence, it answers one of the questions raised in their work.
We extend the familiar program of understanding circuit complexity in terms of regular languages to visibly counter languages. Like the regular languages, the visibly counter languages are - complete. We investigate what the visibly counter languages in certain constant depth circuit complexity classes are. We have initiated this in a previous work for . We present characterizations and decidability results for various logics and circuit classes. In particular, our approach yields a way to understand , where the regular approach fails.
We show that any language recognized by an NC ’ circuit (fan-in 2, depth O(log n)) can be recognized by a width-5 polynomial-size branching program. As any bounded-width polynomial-size branching program can be simulated by an NC ’ circuit, we have that the class of languages recognized by such programs is exactly nonuniform NC’. Further, following
The purpose of this paper is to study reducibilities that can be computed by combinational logic networks of polynomial size and constant depth containing AND’s, OR’s and NOT’s, with no bound placed on the fan-in of AND-gates and OR-gates. Two such reducibilities are defined, and reductions and equivalences among several common problems such as parity, sorting, integer multiplication, graph connectivity, bipartite matching and network flow are given. Certain problems are shown to be complete, with respect to these reducibilities, in the complexity classes deterministic logarithmic space, nondeterministic logarithmic space, and deterministic polynomial time. New upper bounds on the size-depth (unbounded fan-in) circuit complexity of symmetric Boolean functions are established.
We examine a powerful model of parallel computation: polynomial size threshold circuits of bounded depth (the gates compute threshold functions with polynomial weights). Lower bounds are given to separate polynomial size threshold circuits of depth 2 from polynomial size threshold circuits of depth 3 and from probabilistic polynomial size circuits of depth 2. With regard to the unreliability of bounded depth circuits, it is shown that the class of functions computed reliably with bounded depth circuits of unreliable ∨, ∧, ¬ gates is narrow. On the other hand, functions computable by bounded depth, polynomial-size threshold circuits can also be computed by such circuits of unreliable threshold gates. Furthermore we examine to what extent imprecise threshold gates (which behave unpredictably near the threshold value) can compute nontrivial functions in bounded depth and a bound is given for the permissible amount of imprecision. We also discuss threshold quantifiers and prove an undefinability result for graph connectivity.
This paper is devoted to the languages accepted by constant-depth, polynomial-size families of circuits in which every circuit element computes the sum of its input bits modulo a fixed period q. It has been conjectured that such a circuit family cannot compute the AND function of n inputs. Here it is shown that such circuit families are equivalent in power to polynomial-length programs over finite solvable groups; in particular, the conjecture implies that Barrington’s result on the computational power of branching programs over nonsolvable groups cannot be extended to solvable groups. It is also shown that polynomial-length programs over dihedral groups cannot compute the AND function. Furthermore, it is shown that the conjecture is equivalent to a characterization, in terms of finite semigroups and formal logic, of the regular languages accepted by such circuit families. There is, moreover, considerable evidence for this characterization. This last result is established using new theorems, of independent interest, concerning the algebraic structure of finite categories.
The formalism of regular expressions was introduced by S. C. Kleene [6] to obtain the following basic theorems.
A super-polynomial lower bound is given for the size of circuits of fixed depth computing the parity function. Introducing the notion of polynomial-size, constant-depth reduction, similar results are shown for the majority, multiplication, and transitive closure functions. Connections are given to the theory of programmable logic arrays and to the relativization of the polynomial-time hierarchy.
We show that any language recognized by an NC1 circuit (fan-in 2, depth O(log n)) can be recognized by a width-5 polynomial-size branching program. As any bounded-width polynomial-size branching program can be simulated by an NC1 circuit, we have that the class of languages recognized by such programs is exactly nonuniform NC1. Further, following Ruzzo (J. Comput. System Sci.22 (1981), 365–383) and Cook (Inform. and Control64 (1985) 2–22), if the branching programs are restricted to be ATIME(logn)-uniform, they recognize the same languages as do ATIME(log n)-uniform NC1 circuits, that is, those languages in ATIME(log n). We also extend the method of proof to investigate the complexity of the word problem for a fixed permutation group and show that polynomial size circuits of width 4 also recognize exactly nonuniform NC1.
An alternative definition is given for a family of subsets of a free monoid that has been considered by Trahtenbrot and by McNaughton.
Regular events have been classified by star-height, star-free events by the “dot-depth hierarchy,” and events of dot-depth one by the so-called β-hierarchy. It is shown here that the latter two hierarchies have appealing characterizations in symbolic logic: Referring to a first-order language L in which star-free events are described, an event is shown to be of dot-depth n iff it is defined by a Boolean combination of L-sentences in prenex normal form with a Σn-prefix. The β-hierarchy is characterized by the quantifier-rank of L-sentences with Σ1-prefix. A similar characterization of star-height would require the inclusion of weak monadic second-order quantifiers; it is shown that the natural classification of events depending on these quantifiers will not yield on infinite hierarchy.
Let A be a finite alphabet and A∗ the free monoid generated by A. A language is any subset of A∗. Assume that all the languages of the form {a}, where a is either the empty word or a letter in A, are given. Close this basic family of languages under Boolean operations; let (0) be the resulting Boolean algebra of languages. Next, close (0) under concatenation and then close the resulting family under Boolean operations. Call this new Boolean algebra (1), etc. The sequence (0), (1),…, k,… of Boolean algebras is called the dot-depth hierarchy. The union of all these Boolean algebras is the family of star-free or aperiodic languages which is the same as the family of noncounting regular languages. Over an alphabet of one letter the hierarchy is finite; in fact, (2) = (1). We show in this paper that the hierarchy is infinite for any alphabet with two or more letters.
It is shown that for every k, polynomial-size, depth-k Boolean circuits are more powerful than polynomial-size, depth-(k−1) Boolean circuits. Connections with a problem about Borel sets and other questions are discussed.
The computation of finite semigroups using unbounded fan-in circuits are considered. There are constant-depth, polynomial size circuits for semigroup product iff the semigroup does not contain a nontrivial group as a subset. In the case that the semigroup in fact does not contain a group, then for any primitive recursive function f circuits of size O(nf−1(n)) and constant depth exist for the semigroup product of n elements. The depth depends upon the choice of the primitive recursive function f. The circuits not only compute the semigroup product, but every prefix of the semigroup product. A consequence is that the same bounds apply for circuits computing the sum of two n-bit numbers.
It is well known that every set in P has small circuits [13]. Adleman [1] has recently proved the stronger result that every set accepted in polynomial time by a randomized Turing machine has small circuits. Both these results are typical of the known relationships between uniform and nonuniform complexity bounds. They obtain a nonuniform upper bound as a consequence of a uniform upper bound.The central theme here is an attempt to explore the converse direction. That is, we wish to understand when nonuniform upper bounds can be used to obtain uniform upper bounds.In this section we will define our basic notion of nonuniform complexity. Then we will show how to relate it to more common notions.
We use algebraic methods to get lower bounds for complexity of different functions based on constant depth unbounded fan-in circuits with the given set of basic operations. In particular, we prove that depth k circuits with gates NOT, OR and MODp where p is a prime require Exp(&Ogr;(n1/2k)) gates to calculate MODr functions for any r ≠ pm. This statement contains as special cases Yao's PARITY result [ Ya 85 ] and Razborov's new MAJORITY result [Ra 86] (MODm gate is an oracle which outputs zero, if the number of ones is divisible by m).
We study an extension of first-order logic obtained by adjoining quantifiers that count with respect to an integer modulus. It is shown that the languages definable in this framework are precisely the regular languages whose syntactic monoids contain only solvable groups. We obtain an analogous result for regular ω-languages and establish some connections with complexity theory for fixed-depth families of circuits.
We consider the computation of finite semigroups using unbounded fan-in circuits. There are constant-depth, polynomial size circuits for semigroup product iff the semigroup does not contain a nontrivial group as a subset. In the case that the semigroup in fact does not contain a group, then for any primitive recursive function f, circuits of size O(nf−1(n)) and constant depth exist for the semigroup product of n elements. The depth depends upon the choice of the primitive recursive function f. The circuits not only compute the semigroup product, but every prefix of the semigroup product. A consequence is that the same bounds apply for circuits computing the sum of two n-bit numbers.
Recently a new connection was discovered between the parallel complexity class NC ¹ and the theory of finite automata in the work of Barrington on bounded width branching programs. There (nonuniform) NC ¹ was characterized as those languages recognized by a certain nonuniform version of a DFA. Here we extend this characterization to show that the internal structures of NC ¹ and the class of automata are closely related.
In particular, using Thérien's classification of finite monoids, we give new characterizations of the classes AC ⁰ , depth- k AC ⁰ , and ACC , the last being the AC ⁰ closure of the mod q functions for all constant q . We settle some of the open questions in [3], give a new proof that the dot-depth hierarchy of algebraic automata theory is infinite [8], and offer a new framework for understanding the internal structure of NC ¹ .
A new model, non-uniform deterministic finite automata (NUDFA's) over general finite monoids, has recently been developed as a strong link between the theory of finite automata and low-level parallel complexity. Achievements of this model include the proof that width 5 branching programs recognize exactly the languages in non-uniform NC1, NUDFA characterizations of several important subclasses of NC1, and a new proof of the old result that the dot-dephth hierarchy is infinite, using M. Sipser's (1983, in “Proceedings, 15th ACM Symposium on the Theory of Computing,” Association for Computing Machinery, New York, pp. 61–69) work on constant depth circuits. Here we extend this theory to NUDFA's over solvable groups (NUDFA's over non-solvable groups have the maximum possible computing power). We characterize the power of NUDFA's over nilpotent groups and prove some optimal lower bounds for NUDFA's over certain groups which are solvable but not nilpotent. Most of these results appeared in preliminary form in (D. A. Barrington and D. Thérien, 1987, in “Automata, Languages, and Programming: 14th International Colloquium,” Springer-Verlag, Berlin, pp. 163–173).
A theory of the semidirect product of categories and the derived category of a category morphism is presented. In order to include division (≺) in this theory, the traditional setting of these constructions is expanded to include relational arrows. In this expanded setting, a relational morphism φ : M → N of categories determines an optimal decomposition [Formula: see text] where [Formula: see text] denotes semidirect product and D(φ) is the derived category of φ.
The theory of the semidirect product of varieties of categories, V * W, is developed. Associated with each variety V of categories is the collection [Formula: see text] of relational morphisms whose derived category belongs to V. The semidirect product of varieties and the composition of classes of the form [Formula: see text] are shown to stand in the relationship [Formula: see text] The associativity of the semidirect product of varieties follows from this result.
Finally, it is demonstrated that all the results in the article concerning varieties of categories have pseudovariety and monoidal versions. This allows us to furnish a straightforward proof that [Formula: see text] for both varieties and pseudovarieties of monoids.
this paper, we consider circuits of bounded depth in the basis f; Phig.
Counter-free Automata
- R Mcnaughton
- S Papert
R. MCNAUGHTON AND S. PAPERT, "Counter-free Automata," MIT Press, Cambridge, MA, 1971.
Threshold circuits of bounded depth
- A Hainal
- W Maass
- P Pudlak
- M Szegedy
- And G Turan
A. HAINAL, W. MAASS, P. PUDLAK, M. SZEGEDY, AND G. TURAN, Threshold circuits of bounded
depth, in "Proceedings, 28th IEEE FOCS, 1987," pp. 99-110.
Categories as algebra
- Tilson