Resolution Matrix Semantics (RMS) introduces a novel truth-value-based framework for modal logic, providing a substantive alternative to Kripke’s relational semantics of possible worlds. Drawing inspiration from Y. Ivlev’s substantive semantics, RMS utilizes a 4-valued structure—necessary truth (tn), contingent truth (tc), contingent false (fc), and necessary false (fn)—augmented by indeterminate values (t, f, t/f) to define modal systems Km, KDm, KTm, S4m, and S5m, analogous to Kripke’s K, KD, T, S4, and S5. By directly assigning determined and indeterminate truth values via an interpretation function, RMS validates formulas without relying on accessibility relations, as demonstrated through soundness and completeness proofs and a tailored tableau method. The framework’s versatility extends to applications in deontic logic, artificial intelligence, and quantum computing, enabling context-dependent reasoning with computational efficiency. RMS thus bridges philosophical logic and practical domains, offering a truth-centric perspective on modal reasoning’s complexities.

Let $f(n)$ be the maximum number of edges in a graph on $n$ vertices in which no two cycles have the same length. P. Erdos raised the problem of determining $f(n)$ (see J.A. Bondy and U.S.R. Murty, Graph Theory with Applications (Macmillan, New York, 1976), p.247, Problem 11). We present the problems, conjectures related to this problems and we summarize the know results. We make the following conjecture: \par \noindent{\bf Conjecture } $$\lim\sb {n \to \infty} {f(n)-n \over \sqrt n} = \sqrt {2 + \frac{4}{9}}.$$

We show that the order of the cardinality of maximal complete $1$-systems of loops on non-orientable surfaces is $\sim |\chi|^{2}$. In particular, we determine the exact cardinality of maximal complete $1$-systems of loops on punctured projective planes. To prove these results, we show that the cardinality of maximal systems of arcs pairwise-intersecting at most once on a non-orientable surface is $2|\chi|(|\chi|+1)$.

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Let $K_{m}-H$ be the graph obtained from $K_{m}$ by removing the edges set $E(H)$ of the graph $H$ ($H$ is a subgraph of $K_{m}$). We use the symbol $Z_4$ to denote $K_4-P_2.$ A sequence $S$ is potentially $K_{m}-H$-graphical if it has a realization containing a $K_{m}-H$ as a subgraph. Let $\sigma(K_{m}-H, n)$ denote the smallest degree sum such that every $n$-term graphical sequence $S$ with $\sigma(S)\geq \sigma(K_{m}-H, n)$ is potentially $K_{m}-H$-graphical. In this paper, we determine the values of $\sigma (K_{r+1}-Z, n)$ for $n\geq 5r+19, r+1 \geq k \geq 5,$ $j \geq 5$ where $Z$ is a graph on $k$ vertices and $j$ edges which contains a graph $Z_4$ but not contains a cycle on $4$ vertices. We also determine the values of $\sigma (K_{r+1}-Z_4, n)$, $\sigma (K_{r+1}-(K_4-e), n)$, $\sigma (K_{r+1}-K_4, n)$ for $n\geq 5r+16, r\geq 4$.

Let $K_{m}-H$ be the graph obtained from $K_{m}$ by removing the edges set $E(H)$ of the graph $H$ ($H$ is a subgraph of $K_{m}$). We use the symbol $Z_4$ to denote $K_4-P_2.$ A sequence $S$ is potentially $K_{m}-H$-graphical if it has a realization containing a $K_{m}-H$ as a subgraph. Let $\sigma(K_{m}-H, n)$ denote the smallest degree sum such that every $n$-term graphical sequence $S$ with $\sigma(S)\geq \sigma(K_{m}-H, n)$ is potentially $K_{m}-H$-graphical. In this paper, we determine the values of $\sigma (K_{r+1}-U, n)$ for $n\geq 5r+18, r+1 \geq k \geq 7,$ $j \geq 6$ where $U$ is a graph on $k$ vertices and $j$ edges which contains a graph $K_3 \bigcup P_3$ but not contains a cycle on $4$ vertices and not contains $Z_4$.

Let $K_k$, $C_k$, $T_k$, and $P_{k}$ denote a complete graph on $k$ vertices, a cycle on $k$ vertices, a tree on $k+1$ vertices, and a path on $k+1$ vertices, respectively. Let $K_{m}-H$ be the graph obtained from $K_{m}$ by removing the edges set $E(H)$ of the graph $H$ ($H$ is a subgraph of $K_{m}$). A sequence $S$ is potentially $K_{m}-H$-graphical if it has a realization containing a $K_{m}-H$ as a subgraph. Let $\sigma(K_{m}-H, n)$ denote the smallest degree sum such that every $n$-term graphical sequence $S$ with $\sigma(S)\geq \sigma(K_{m}-H, n)$ is potentially $K_{m}-H$-graphical. In this paper, we determine the values of $\sigma (K_{r+1}-H, n)$ for $n\geq 4r+10, r\geq 3, r+1 \geq k \geq 4$ where $H$ is a graph on $k$ vertices which contains a tree on $4$ vertices but not contains a cycle on $3$ vertices. We also determine the values of $\sigma (K_{r+1}-P_2, n)$ for $n\geq 4r+8, r\geq 3$.

"A sequence $S$ is potentially $K_4-e$ graphical if it has a realization containing a $K_4-e$ as a subgraph. Let $\sigma(K_4-e,n)$ denote the smallest degree sum such that every $n$ -term graphical sequence $S$ with $\sigma(S)\geq\sigma(K_4-e,n)$ is potentially $K_4-e$ graphical. Gould, Jacobson, Lehel raised the problem of determining the value of $\sigma(K_4-e,n)$ . In this paper, we prove that $\sigma(K_4-e,n)=2[(3n-1)/2]$ for $n\geq7$ and $n=4,5$ , and $\sigma(K_4-e,6)=20$ .''

"In this paper we consider a variation of the classical Turán-type extremal problems. Let $S$ be an $n$ -term graphical sequence, and $\sigma(S)$ be the sum of the terms in $S$ . Let $H$ be a graph. The problem is to determine the smallest even $l$ such that any $n$ -term graphical sequence $S$ having $\sigma(S)\geq l$ has a realization containing $H$ as a subgraph. Denote this value $l$ by $\sigma(H,n)$ . We show $\sigma(C_{2m+1},n)=m(2n-m-1)+2$ , for $m\geq 3$ , $n\geq 3m$ ; $\sigma(C_{2m+2},n)=m(2n-m-1)+4$ , for $m\geq 3$ , $n\geq 5m-2$ .''

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