class 20,21,22,23,24 summary

class_20_03102023/summary.txt

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+Resolution
+Examples 
+Soundness of resolution rule.
+
+Res*(F) 
+
+Proposition 1: 
+If \emptyset \in Res*(F) then F is unsatisfiable. 
+Proof : (by contradiction and using soundness of resolution rule) 
+
+Proposition 2: 
+If  F is unsatisfiable  then  \emptyset \in Res*(F)  
+Proof : (by induction on #prop_variables appearing in F
+Base case : when F has only one variable. 
+
+

class_21_05102023/demorgan.cpp

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+#include <iostream>
+#include <sstream>
+#include<vector>
+#include"z3++.h"
+
+using namespace z3;
+
+
+/**
+   De Morgan's Duality Law: {e not(x and y) <-> (not x) or ( not y) }
+*/
+ 
+ 
+int main(){
+
+context c; 
+expr  x  = c.bool_const("x"); 
+expr  y  = c.bool_const("y"); 
+expr conjecture = !(x && y) == (!x || !y); 
+
+//create a solver 
+solver s(c); 
+
+//assert the negation of conjecture 
+s.add(!conjecture); 
+
+std::cout << s << "\n";
+std::cout << s.to_smt2() << "\n"; 
+
+//check if the result if unsat
+switch (s.check()) {
+case unsat: std::cout << "demorgan is valid\n"; break;
+case sat: std::cout << "demorgan is not valid\n"; break;
+case unknown: std::cout <<"unknown\n"; break; 
+}
+} //end of function 
+
+

class_21_05102023/expr_vector.cpp

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+/*++
+Copyright (c) 2015 Microsoft Corporation
+--*/
+
+#include <iostream>
+#include <sstream>
+#include<vector>
+#include"z3++.h"
+
+using namespace z3;
+ 
+ 
+ 
+void expr_vector_example() {
+    std::cout << "expr_vector example\n";
+    context c;
+    const unsigned N = 10;
+
+    expr_vector x(c);
+
+    for (unsigned i = 0; i < N; i++) { 
+        std::stringstream x_name; 
+        x_name << "x_" << i;
+        x.push_back(c.int_const(x_name.str().c_str()));
+    }
+    
+    solver s(c);
+    for (unsigned i = 0; i < N; i++) {
+        s.add(x[i] >= 1);
+    }
+
+    std::cout << s << "\n" << "solving...\n" << s.check() << "\n";
+    model m = s.get_model();
+    std::cout << "solution\n" << m;
+}
+
+int main() {
+
+    try {
+        expr_vector_example(); std::cout << "\n";
+        std::cout << "done\n";
+    }
+    catch (exception & ex) {
+        std::cout << "unexpected error: " << ex << "\n";
+    }
+    return 0;
+}

class_21_05102023/getmodel.cpp

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+#include <iostream>
+#include <sstream>
+#include<vector>
+#include"z3++.h"
+
+using namespace z3;
+
+
+/**
+find values of expressions that satisfy the following ineaualities 
+x >= 1 
+y < x +3 
+*/
+ 
+int main(){
+context c; 
+expr  x  = c.int_const("x"); 
+expr  y  = c.int_const("y"); 
+//create a solver 
+solver s(c); 
+//adding the constraints
+s.add(x >=1); 
+s.add(y < x+3); 
+
+std::cout << s.check() << "\n\n";
+
+model m = s.get_model(); 
+std::cout << m << "\n";
+
+//traversing the model 
+std::cout << "traversing the model" << "\n";
+for (unsigned i =0; i < m.size(); i++){
+	func_decl v = m[i];
+	assert(v.arity() ==0);
+	std::cout << v.name()<<"="<<m.get_const_interp(v) << "\n";
+}
+ 
+// evaluating the expressions in the model.
+std::cout << "evaluating the expressions in the model" << "\n";
+std::cout << "x+y+1= "<< m.eval(x+y+1)<<"\n";
+
+return 0;  
+} //end of function 
+
+

class_22_09102923/summary.txt

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+Introduction to FOL
+
+-- socrates example.
+   Why do we need FOL to deduct that Socrates is mortal. 
+-- Predicates are relating the objects and properties. 
+-- Signature of FOL (constants, functions, relations). 
+-- we need variables, logical symbols and non-logical symbols, 
+   also to define wffs. 
+
+
+-- Example of Number Theory and some formulas 
+-- Example of set theory and some examples
+-- Example of min FOL (with only =). 
+

class_23_10102923/summary.txt

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+-- Syntax of FOL 
+-- step (1) terms : constant, variabls, or succussive applications to functions. 
+   -- examples of terms in number theory, set theory, min FOL 
+
+-- step (2) atomic formulas : Relations over terms, = exists unless
+specified.  
+   -- examples of atomic formulas in number theory, set theory, min FOL 
+-- step (3) wffs : atomic formulas, or formulas constructed from
+smaller formulas using negation, implication, \forall, \exists. 
+

class_24_12102023/summary.txt

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+Revision of syntax of FOL. 
+Semantics of FOL
+Structure.
+Example of  structure of natural numbers
+EXamples of formulas from number theory 
+
+Assignment functions S: Variables --> \universe
+
+Extenstion of S to give meaning to terms, atomic formulas and wffs. 
+