Finite Model Theory and Proof Complexity Revisited: Distinguishing Graphs in Choiceless Polynomial Time and the Extended Polynomial Calculus

This paper extends prior work on the connections between logics from finite model theory and propositional/algebraic proof systems. We show that if all non-isomorphic graphs in a given graph class can be distinguished in the logic Choiceless Polynomial Time with counting (CPT), then they can also be distinguished in the bounded-degree extended polynomial calculus (EPC), and the refutations have roughly the same size as the resource consumption of the CPT-sentence. This allows to transfer lower bounds for EPC to CPT and thus constitutes a new potential approach towards better understanding the limits of CPT. A super-polynomial EPC lower bound for a Ptime-instance of the graph isomorphism problem would separate CPT from Ptime and thus solve a major open question in finite model theory. Further, using our result, we provide a model theoretic proof for the separation of bounded-degree polynomial calculus and bounded-degree extended polynomial calculus

Choiceless Polynomial Time, Symmetric Circuits and Cai-F\"urer-Immerman Graphs

Choiceless Polynomial Time (CPT) is currently the only candidate logic for capturing PTIME (that is, it is contained in PTIME and has not been separated from it). A prominent example of a decision problem in PTIME that is not known to be CPT-definable is the isomorphism problem on unordered Cai-F\"urer-Immerman graphs (the CFI-query). We study the expressive power of CPT with respect to this problem and develop a partial characterisation of solvable instances in terms of properties of symmetric XOR-circuits over the CFI-graphs: The CFI-query is CPT-definable on a given class of graphs only if: For each graph $G$, there exists an XOR-circuit $C$, whose input gates are labelled with edges of $G$, such that $C$ is sufficiently symmetric with respect to the automorphisms of $G$ and satisfies certain other circuit properties. We also give a sufficient condition for CFI being solvable in CPT and develop a new CPT-algorithm for the CFI-query. It takes as input structures which contain, along with the CFI-graph, an XOR-circuit with suitable properties. The strongest known CPT-algorithm for this problem can solve instances equipped with a preorder with colour classes of logarithmic size. Our result implicitly extends this to preorders with colour classes of polylogarithmic size (plus some unordered additional structure). Finally, our work provides new insights regarding a much more general problem: The existence of a solution to an unordered linear equation system $A \cdot x = b$ over a finite field is CPT-definable if the matrix $A$ has at most logarithmic rank (with respect to the size of the structure that encodes the equation system). This is another example that separates CPT from fixed-point logic with counting

The Model-Theoretic Expressiveness of Propositional Proof Systems

We establish new, and surprisingly tight, connections between propositional proof complexity and finite model theory. Specifically, we show that the power of several propositional proof systems, such as Horn resolution, bounded width resolution, and the polynomial calculus of bounded degree, can be characterised in a precise sense by variants of fixed-point logics that are of fundamental importance in descriptive complexity theory. Our main results are that Horn resolution has the same expressive power as least fixed-point logic, that bounded width resolution captures existential least fixed-point logic, and that the (monomial restriction of the) polynomial calculus of bounded degree solves precisely the problems definable in fixed-point logic with counting

Limitations of Game Comonads via Homomorphism Indistinguishability

Abramsky, Dawar, and Wang (2017) introduced the pebbling comonad for k-variable counting logic and thereby initiated a line of work that imports category theoretic machinery to finite model theory. Such game comonads have been developed for various logics, yielding characterisations of logical equivalences in terms of isomorphisms in the associated co-Kleisli category. We show a first limitation of this approach by studying linear-algebraic logic, which is strictly more expressive than first-order counting logic and whose k-variable logical equivalence relations are known as invertible-map equivalences (IM). We show that there exists no finite-rank comonad on the category of graphs whose co-Kleisli isomorphisms characterise IM-equivalence, answering a question of \'O Conghaile and Dawar (CSL 2021). We obtain this result by ruling out a characterisation of IM-equivalence in terms of homomorphism indistinguishability and employing the Lov\'asz-type theorems for game comonads established by Dawar, Jakl, and Reggio (2021). Two graphs are homomorphism indistinguishable over a graph class if they admit the same number of homomorphisms from every graph in the class. The IM-equivalences cannot be characterised in this way, neither when counting homomorphisms in the natural numbers, nor in any finite prime field.Comment: Minor corrections in Section

PENGARUH PENDIDIKAN KELUARGA TERHADAPPERKEMBAGAN KOGNITIF ANAK USIA 2-3 TAHUN DIKECAMATAN MAKASSAR KOTA MAKASSAR

LUSIANA PAGO PASALLI, 2021, Pengaruh Pendidikan Keluarga Terhadap Perkembangan Kognitif Anak Usia 2-3 Tahun di Kecamatan Makassar Kota Makassar, Pembimbing dalam penelitian ini adalah Dr. Kartini Marzuki, M.Pd., dan Dr.Suardi, S.Pd., M.Pd. Pada program Pendidikan Luar Sekolah Fakultas Ilmu Pendidikan Universitas Negeri Makassar. Pendidikan keluarga merupakan lembaga pendidik yang pertama dan utama bagi seorang anak, sebagai pendidikan yang pertama bagi anak keluarga harus memberikan pendidikan yang terbaik untuk membantu perkembangan kognitif anak. Penelitian ini bertujuan untuk mengetahui Pengaruh Pendidikan Keluarga Terhadap Perkembangan Kognitif Anak Usia 2-3 Tahun di Kecamatan Makassar Kota Makassar. Penelitian ini menggunakan pendekatan kuantitatif. Yang menjadi populasi pada penelitian ini yaitu keluarga di kecamatan Makassar yang memiliki anak usia 2-3 tahun yang jumlahnya 551 keluarga, dari populasi tersebut di ambil sampel sebanyak 85 keluarga dengan menggunakan teknik probability sampling yang dimana sampel di ambil secara acak dengan memberikan peluang yang sama kepada populasi untuk menjadi sampel. Teknik pengumpulan data menggunakan angket,metode analisis data menggunakan analisis statistik deskriptif persentase dan analisis regresi linear sederhana. Berdasarkan hasil penelitian pendidikan keluarga di kecamatan Makassar berlangsung dengan baik hal ini berdasarkan rata-rata jawaban responden pada angket penelitian di peroleh data dengan persentase yaitu: 1) variabel pendidikan keluarga terdapat 15% pada kategori tinggi, pada kategori sedang persentase sebesar 68% dan pada kategori rendah persentase 17% dari data tersebut dapat dikatakan bahwa pendidikan keluarga berada pada kategori sedang, 2) variabel perkembangan kognitif anak usia 2-3 tahun terdapat 14% pada kategori tinggi , pada kategori sedang persentasenya sebesar 66% dan pada kategori rendah persentasenya sebesar 20% dari persentase tersebut dapat dikatakan bahwa perkembangan kognitif anak usia 2-3 tahun berada pada kategori sedang/baik. 3) Pengaruh pendidikan keluarga terhadap perkembangan kognitif anak usia 2-3 tahun berpengaruh secara signifikan dengan persentase sebasar 80%. Kata kunci : pendidikan keluarga, perkembangan kognitif anak usia 2-3 tahun

A Finite-Model-Theoretic View on Propositional Proof Complexity

We establish new, and surprisingly tight, connections between propositional proof complexity and finite model theory. Specifically, we show that the power of several propositional proof systems, such as Horn resolution, bounded-width resolution, and the polynomial calculus of bounded degree, can be characterised in a precise sense by variants of fixed-point logics that are of fundamental importance in descriptive complexity theory. Our main results are that Horn resolution has the same expressive power as least fixed-point logic, that bounded-width resolution captures existential least fixed-point logic, and that the polynomial calculus with bounded degree over the rationals solves precisely the problems definable in fixed-point logic with counting. By exploring these connections further, we establish finite-model-theoretic tools for proving lower bounds for the polynomial calculus over the rationals and over finite fields

Deterministic landslide stability analysis: an example from NW Oregon

A slope stability map is created for the Astoria Basin, northwestern Oregon. The stability analysis is based upon a topographically based model, TOPOG model. The model predicts the occurrence of subsurface water-logging which reduces the shear-strength of a slope-profile. Simply stated, subsurface saturated zones occur wherever the local drainage flux exceeds the slope's ability to transmit water. The model is coupled with the infinite slope stability model for analysing the effect of topography on slope stability. The coupled model divides the landscape into four classes based upon its slope stability. The model is simulated using three different rainfall events. The simulations provide a means to observe the behavior of each slope basin at a given rainfall rate. The result of the simulations creates a map that shows the distribution of the slope-stability classes. The map is tested against the active slide occurrences which are mapped in the field and from geological maps. The locations of the active slides seem to be consistent with the unstable slope-classes in the model

Upaya Meningkatkan Hasil Belajar Siswa Dalam Pembelajaran IPA Dengan Metode Demonstrasi Di Kelas IV SDN 14 Ampana

Penelitian Tindakan Kelas (PTK) ini dilaksanakan di kelas IV SDN 14 Ampana dengan jumlah siswa sebanyak 20 orang. Penelitian ini bertujuan untuk meningkatkan hasil belajar siswa dalam pembelajaran IPA dengan metode demonstrasi di kelas IV SDN 14 Ampana, yang dilaksanakan dalam 2 siklus. Data yang diambil dari penelitian ini adalah data kualitatif dan data kuantitatif. Data kualitatif diperoleh dari hasil observasi aktivitas siswa dan aktivitas guru pada saat proses belajar mengajar berlangsung sedangkan data kuantitatif diperoleh dari hasil belajar siswa pada setiap akhir tindakan. Hasil analisa data kualitatif dari lembar observasi siswa pada siklus I menyatakan cukup dan pada siklus II sangat baik. Selanjutnya hasil analisa data kuantitatif hasil tes tindakan siklus I diperoleh siswa yang tuntas 12 orang dari 20 orang siswa dengan persentase daya serap klasikal sebesar 64,0% dan ketuntasan belajar klasikal sebesar 60,0%. Pada siklus II hasil tes akhir tindakan mengalami peningkatan, siswa yang tuntas 18 orang dari 20 orang siswa dengan persentase daya serap klasikal sebesar 86,5% dan ketuntasan belajar klasikal sebesar 90,0%. Dari hasil penelitian ini menunjukkan bahwa hasil yang diperoleh dari siklus I dan II baik hasil observasi aktivitas siswa dan guru maupun hasil tes akhir tindakan mengalami peningkatan. Hal ini membuktikan bahwa penggunaan metode demonstrasi dapat meningkatkan hasil belajar siswa kelas IV SDN 14 Ampana pada mata pelajaran IPA