Large Aperiodic Semigroups

The syntactic complexity of a regular language is the size of its syntactic semigroup. This semigroup is isomorphic to the transition semigroup of the minimal deterministic finite automaton accepting the language, that is, to the semigroup generated by transformations induced by non-empty words on the set of states of the automaton. In this paper we search for the largest syntactic semigroup of a star-free language having $n$ left quotients; equivalently, we look for the largest transition semigroup of an aperiodic finite automaton with $n$ states. We introduce two new aperiodic transition semigroups. The first is generated by transformations that change only one state; we call such transformations and resulting semigroups unitary. In particular, we study complete unitary semigroups which have a special structure, and we show that each maximal unitary semigroup is complete. For $n \ge 4$ there exists a complete unitary semigroup that is larger than any aperiodic semigroup known to date. We then present even larger aperiodic semigroups, generated by transformations that map a non-empty subset of states to a single state; we call such transformations and semigroups semiconstant. In particular, we examine semiconstant tree semigroups which have a structure based on full binary trees. The semiconstant tree semigroups are at present the best candidates for largest aperiodic semigroups. We also prove that $2^n-1$ is an upper bound on the state complexity of reversal of star-free languages, and resolve an open problem about a special case of state complexity of concatenation of star-free languages.Comment: 22 pages, 1 figure, 2 table

Symmetric Groups and Quotient Complexity of Boolean Operations

The quotient complexity of a regular language L is the number of left quotients of L, which is the same as the state complexity of L. Suppose that L and L' are binary regular languages with quotient complexities m and n, and that the transition semigroups of the minimal deterministic automata accepting L and L' are the symmetric groups S_m and S_n of degrees m and n, respectively. Denote by o any binary boolean operation that is not a constant and not a function of one argument only. For m,n >= 2 with (m,n) not in {(2,2),(3,4),(4,3),(4,4)} we prove that the quotient complexity of LoL' is mn if and only either (a) m is not equal to n or (b) m=n and the bases (ordered pairs of generators) of S_m and S_n are not conjugate. For (m,n)\in {(2,2),(3,4),(4,3),(4,4)} we give examples to show that this need not hold. In proving these results we generalize the notion of uniform minimality to direct products of automata. We also establish a non-trivial connection between complexity of boolean operations and group theory

Syntactic Complexity of R- and J-Trivial Regular Languages

The syntactic complexity of a regular language is the cardinality of its syntactic semigroup. The syntactic complexity of a subclass of the class of regular languages is the maximal syntactic complexity of languages in that class, taken as a function of the state complexity n of these languages. We study the syntactic complexity of R- and J-trivial regular languages, and prove that n! and floor of [e(n-1)!] are tight upper bounds for these languages, respectively. We also prove that 2^{n-1} is the tight upper bound on the state complexity of reversal of J-trivial regular languages.Comment: 17 pages, 5 figures, 1 tabl

Checking Whether an Automaton Is Monotonic Is NP-complete

An automaton is monotonic if its states can be arranged in a linear order that is preserved by the action of every letter. We prove that the problem of deciding whether a given automaton is monotonic is NP-complete. The same result is obtained for oriented automata, whose states can be arranged in a cyclic order. Moreover, both problems remain hard under the restriction to binary input alphabets.Comment: 13 pages, 4 figures. CIAA 2015. The final publication is available at http://link.springer.com/chapter/10.1007/978-3-319-22360-5_2

The Proper Appellate Standard of Review for PTAB Factual Findings Made Incidental to Claim Construction

The America Invents Act (AIA) represents the most significant change to U.S. patent law since the 1952 Patent Act. Since its passage, the AIA has drawn wide support from the intellectual property community, primarily due to the new post-grant opposition proceedings the Act created. However, certain aspects of the new system created by the AIA are controversial. Specifically, judges and practitioners alike debate which standard of review courts should apply to the factual findings made by the Patent Trial and Appeals Board (PTAB) during these opposition proceedings. While the Federal Circuit has reviewed all factual findings made at the Patent Office under the substantial evidence standard of review per the Administrative Procedures Act (APA), there are others who argue that, due to the adjudicatory nature of PTAB proceedings, and the purpose and intent of the AIA, the substantial evidence standard is too deferential. In the Supreme Court’s decision in Teva v. Sandoz, the Court held that the Federal Circuit should review a district court’s factual findings during claim construction for clear error per the express wording of Rule 52(a)(6) of the Federal Rules of Civil Procedure. Although differences exist between the appellate review of district court decisions and the review of administrative agency decisions, certain members of the intellectual property community have argued that the clear error standard of review for district court factual findings related to claim construction should also be the standard of review for the PTAB’s factual findings relating to claim construction. This Comment seeks to examine the arguments for and against whether courts should review factual findings made incidental to claim construction under the substantial evidence standard or the clearly erroneous standard. This Comment argues that courts should review factual findings made during claim construction for clear error

Intramuscular Immunisation with Chlamydial Proteins Induces Chlamydia trachomatis Specific Ocular Antibodies.

BACKGROUND: Ocular infection with Chlamydia trachomatis can cause trachoma, which is the leading cause of blindness due to infection worldwide. Despite the large-scale implementation of trachoma control programmes in the majority of countries where trachoma is endemic, there remains a need for a vaccine. Since C. trachomatis infects the conjunctival epithelium and stimulates an immune response in the associated lymphoid tissue, vaccine regimens that enhance local antibody responses could be advantageous. In experimental infections of non-human primates (NHPs), antibody specificity to C. trachomatis antigens was found to change over the course of ocular infection. The appearance of major outer membrane protein (MOMP) specific antibodies correlated with a reduction in ocular chlamydial burden, while subsequent generation of antibodies specific for PmpD and Pgp3 correlated with C. trachomatis eradication. METHODS: We used a range of heterologous prime-boost vaccinations with DNA, Adenovirus, modified vaccinia Ankara (MVA) and protein vaccines based on the major outer membrane protein (MOMP) as an antigen, and investigated the effect of vaccine route, antigen and regimen on the induction of anti-chlamydial antibodies detectable in the ocular lavage fluid of mice. RESULTS: Three intramuscular vaccinations with recombinant protein adjuvanted with MF59 induced significantly greater levels of anti-MOMP ocular antibodies than the other regimens tested. Intranasal delivery of vaccines induced less IgG antibody in the eye than intramuscular delivery. The inclusion of the antigens PmpD and Pgp3, singly or in combination, induced ocular antigen-specific IgG antibodies, although the anti-PmpD antibody response was consistently lower and attenuated by combination with other antigens. CONCLUSIONS: If translatable to NHPs and/or humans, this investigation of the murine C. trachomatis specific ocular antibody response following vaccination provides a potential mouse model for the rapid and high throughput evaluation of future trachoma vaccines

Complexity in Prefix-Free Regular Languages

We examine deterministic and nondeterministic state complexities of regular operations on prefix-free languages. We strengthen several results by providing witness languages over smaller alphabets, usually as small as possible. We next provide the tight bounds on state complexity of symmetric difference, and deterministic and nondeterministic state complexity of difference and cyclic shift of prefix-free languages.Comment: In Proceedings DCFS 2010, arXiv:1008.127

Linear Parsing Expression Grammars

PEGs were formalized by Ford in 2004, and have several pragmatic operators (such as ordered choice and unlimited lookahead) for better expressing modern programming language syntax. Since these operators are not explicitly defined in the classic formal language theory, it is significant and still challenging to argue PEGs' expressiveness in the context of formal language theory.Since PEGs are relatively new, there are several unsolved problems.One of the problems is revealing a subclass of PEGs that is equivalent to DFAs. This allows application of some techniques from the theory of regular grammar to PEGs. In this paper, we define Linear PEGs (LPEGs), a subclass of PEGs that is equivalent to DFAs. Surprisingly, LPEGs are formalized by only excluding some patterns of recursive nonterminal in PEGs, and include the full set of ordered choice, unlimited lookahead, and greedy repetition, which are characteristic of PEGs. Although the conversion judgement of parsing expressions into DFAs is undecidable in general, the formalism of LPEGs allows for a syntactical judgement of parsing expressions.Comment: Parsing expression grammars, Boolean finite automata, Packrat parsin