Update app.py

app.py

@@ -17,40 +17,86 @@ def llm_response(text):
 
 def pvsnp(problem):
   classification = llm_response(f'''
+You are an expert in computational complexity theory, specializing in both classical complexity classes (P, NP, NP-complete, NP-hard) and modern developments (e.g., parameterized complexity, fine-grained complexity, Minimum Circuit Size Problem).
 
-  You are an expert in computational complexity theory with deep knowledge of both classical complexity classes (P, NP, NP-complete, NP-hard) and modern developments such as parameterized complexity, fine-grained complexity, and recent findings like the Minimum Circuit Size Problem (MCSP). Your task is to analyze a given problem description and classify it into one of the following categories:
+Your task is to classify a given problem into one of the following categories:
 
-- **P**: Problems that can be solved in deterministic polynomial time.
-- **NP**: Problems for which a proposed solution can be verified in deterministic polynomial time.
-- **NP-complete**: Problems that are both in NP and as hard as any problem in NP, via polynomial-time reductions.
-- **NP-hard**: Problems that are at least as hard as NP-complete problems but may not be in NP.
-- **Beyond NP**: Problems that likely belong to more complex classes (e.g., PSPACE, EXPTIME) or are undecidable.
-- **Other**: If the problem fits into an alternative complexity class (e.g., BPP, co-NP) or does not clearly align with the categories above, note that explicitly.
+P: Solvable in deterministic polynomial time.
 
-**Problem Description:**
+NP: Verifiable in polynomial time.
+
+NP-complete: Both in NP and NP-hard.
+
+NP-hard: At least as hard as NP-complete problems, possibly outside NP.
+
+Beyond NP: Likely in PSPACE, EXPTIME, or undecidable.
+
+Other: Fits alternative complexity classes (e.g., BPP, co-NP).
+
+Problem Description:
 {problem}
 
-**Guidelines:**
+If the given problem is a NP-hard problem,  decompose the NP-hard problem into polynomial-time solvable subproblems without solving them.
+
+🔹 Inputs:
+A formal definition and instance of the NP-hard problem (e.g., SAT, TSP, Graph Coloring).
+
+Optional: Constraints or domain knowledge.
+
+🔹 Decomposition Process:
+Graph Representation & Structural Analysis
+
+Convert the problem into a graph (if applicable).
+
+Identify independent or tractable substructures.
+
+Classification of Subproblems
+
+Detect polynomially solvable parts (e.g., tree structures, bipartite graphs).
+
+Separate them from harder components.
+
+Partitioning & Transformation
+
+Break the problem into independent or loosely connected subproblems.
+
+Ensure each subproblem is in P or provably easier than the original.
+
+Output a structured breakdown.
+
+🔹 Outputs:
+A list of P-complexity subproblems.
+
+A dependency graph of their relationships in ASCII format.
+
+A complexity analysis report quantifying decomposition effectiveness.
+
+Guidelines for Classification:
+Problem Analysis
+
+Determine if the problem is a decision, optimization, or function computation problem.
+
+Identify key input/output characteristics and constraints.
+
+Complexity Insights
+
+Check for polynomial-time solvability via known techniques (dynamic programming, greedy methods).
+
+Assess reductions to/from well-studied problems.
+
+Advanced Considerations
 
-1. **Comprehensive Analysis**:  
-   - Determine whether the problem is primarily a decision problem, an optimization problem, or a function computation.
-   - Identify key input and output characteristics and any explicit constraints.
+Incorporate recent research (e.g., MCSP's implications for NP-completeness).
 
-2. **Complexity Characteristics**:  
-   - Examine if the problem exhibits features common to polynomial-time algorithms (e.g., dynamic programming, greedy strategies) or if it has structural similarities to known NP-complete problems.
-   - Assess whether there is potential for polynomial-time reductions from or to well-studied problems in the literature.
+Evaluate parameterized complexity (FPT results) and fine-grained complexity (SETH, other conjectures).
 
-3. **Advanced and Recent Research Considerations**:  
-   - Incorporate insights from the latest research, including the implications of the Minimum Circuit Size Problem (MCSP) for NP-completeness.
-   - Consider parameterized complexity aspects: does the problem admit fixed-parameter tractable (FPT) solutions under certain parameters?
-   - Evaluate any connections to fine-grained complexity results, such as relationships to the Strong Exponential Time Hypothesis (SETH) or other conjectures.
-   - If the problem has probabilistic or average-case aspects, mention how these might affect its classification.
+Consider probabilistic or average-case complexity aspects.
 
-4. **Explanation of Reasoning**:  
-   - Provide a brief, clear explanation for your classification. Justify your decision by referencing specific features of the problem and connecting them to established theory and recent research insights.
+Justification
 
-**Your Classification and Explanation:**
+Provide a concise explanation for the classification, referencing key problem features and relevant research.
 
+Your Classification and Explanation:
 ''')
   return classification